Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf ·...
Transcript of Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf ·...
![Page 1: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/1.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
Propositional Logic: Syntax and Semantics
PHIL UA-70: LogicFebruary 24–26, 2015
(∼ P Q)
∼ P
P
Q
∼ (P Q)
(P Q)
P Q
![Page 2: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/2.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 3: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/3.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 4: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/4.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE PLAN
▶ Construct an artificial language, ‘PL’, within which we canbe incredibly precise about which arguments aredeductively valid and which are deductively invalid.
▶ Provide a method for translating statements of Englishinto PL and statements of PL into English
▶ One advantage: we can theorize about relations ofdeductive validity without having to worry about theambiguity of English (e.g., equivocation and amphiboly).
![Page 5: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/5.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE PLAN
▶ Construct an artificial language, ‘PL’, within which we canbe incredibly precise about which arguments aredeductively valid and which are deductively invalid.
▶ Provide a method for translating statements of Englishinto PL and statements of PL into English
▶ One advantage: we can theorize about relations ofdeductive validity without having to worry about theambiguity of English (e.g., equivocation and amphiboly).
![Page 6: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/6.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE PLAN
▶ Construct an artificial language, ‘PL’, within which we canbe incredibly precise about which arguments aredeductively valid and which are deductively invalid.
▶ Provide a method for translating statements of Englishinto PL and statements of PL into English
▶ One advantage: we can theorize about relations ofdeductive validity without having to worry about theambiguity of English (e.g., equivocation and amphiboly).
![Page 7: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/7.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:
▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech
▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t
‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 8: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/8.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t
‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 9: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/9.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech
▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t
‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 10: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/10.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t
‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 11: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/11.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t
‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 12: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/12.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t‘Bubbie makes pickles’ ✓
‘Up bouncy ball door John variously catapult’ ×▶ a way to interpret the meaning of every grammatical
expression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 13: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/13.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 14: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/14.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 15: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/15.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
▶ To specify a language, we need to provide:▶ a vocabulary for the language
▶ e.g., a list of the words of English and their parts of speech▶ a grammar for the language
▶ e.g., rules for saying which strings of words are grammaticaland which aren’t‘Bubbie makes pickles’ ✓‘Up bouncy ball door John variously catapult’ ×
▶ a way to interpret the meaning of every grammaticalexpression of the language
▶ e.g., a dictionary entry for every word of English and rulesfor constructing the meaning of sentences out of themeanings of words
![Page 16: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/16.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
LANGUAGES IN GENERAL
SYNTAX −−−{
1. Vocabulary2. Grammar
SEMANTICS −−3. Meaning
![Page 17: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/17.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 18: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/18.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE VOCABULARY OF PL
▶ The vocabulary of PL includes the following symbols:
1. An infinite number of statement letters:
A,B,C, ...,Y,Z,A1,B1,C1, ...,Y1,Z1,A2,B2,C2, ...
2. logical operators:∼, ,∨,⊃,≡
3. parenthases( , )
▶ Nothing else is included in the vocabulary of PL.
![Page 19: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/19.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE VOCABULARY OF PL
▶ The vocabulary of PL includes the following symbols:1. An infinite number of statement letters:
A,B,C, ...,Y,Z,A1,B1,C1, ...,Y1,Z1,A2,B2,C2, ...
2. logical operators:∼, ,∨,⊃,≡
3. parenthases( , )
▶ Nothing else is included in the vocabulary of PL.
![Page 20: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/20.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE VOCABULARY OF PL
▶ The vocabulary of PL includes the following symbols:1. An infinite number of statement letters:
A,B,C, ...,Y,Z,A1,B1,C1, ...,Y1,Z1,A2,B2,C2, ...
2. logical operators:∼, ,∨,⊃,≡
3. parenthases( , )
▶ Nothing else is included in the vocabulary of PL.
![Page 21: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/21.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE VOCABULARY OF PL
▶ The vocabulary of PL includes the following symbols:1. An infinite number of statement letters:
A,B,C, ...,Y,Z,A1,B1,C1, ...,Y1,Z1,A2,B2,C2, ...
2. logical operators:∼, ,∨,⊃,≡
3. parenthases( , )
▶ Nothing else is included in the vocabulary of PL.
![Page 22: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/22.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE VOCABULARY OF PL
▶ The vocabulary of PL includes the following symbols:1. An infinite number of statement letters:
A,B,C, ...,Y,Z,A1,B1,C1, ...,Y1,Z1,A2,B2,C2, ...
2. logical operators:∼, ,∨,⊃,≡
3. parenthases( , )
▶ Nothing else is included in the vocabulary of PL.
![Page 23: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/23.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ Any sequence of the symbols in the vocabulary of PL is aformula of PL.
((()A23 ⊃⊃ ZP ⊃ (Q) ⊃ ())
(P ⊃ (Q ⊃ (R ⊃ (S ⊃ T))))A B (C ∼D)))
![Page 24: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/24.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ Any sequence of the symbols in the vocabulary of PL is aformula of PL.
((()A23 ⊃⊃ ZP ⊃ (Q) ⊃ ())
(P ⊃ (Q ⊃ (R ⊃ (S ⊃ T))))A B (C ∼D)))
![Page 25: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/25.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ Any sequence of the symbols in the vocabulary of PL is aformula of PL.
((()A23 ⊃⊃ ZP ⊃ (Q) ⊃ ())
(P ⊃ (Q ⊃ (R ⊃ (S ⊃ T))))A B (C ∼D)))
![Page 26: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/26.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.
SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 27: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/27.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.
∼) If ‘p’ is a wff, then ‘∼p’ is a wff.) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.
∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 28: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/28.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 29: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/29.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.
∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 30: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/30.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.
⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 31: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/31.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.
≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 32: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/32.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.
−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 33: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/33.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 34: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/34.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.
▶ They are formulae variables—variables whose potentialvalues are formulae of PL.
![Page 35: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/35.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ We define a well-formed formula (‘wff’) of PL with thefollowing rules.SL) Any statement letter, by itself, is a wff.∼) If ‘p’ is a wff, then ‘∼p’ is a wff.
) If ‘p’ and ‘q’ are wffs, then ‘(p q)’ is a wff.∨) If ‘p’ and ‘q’ are wffs, then ‘(p ∨ q)’ is a wff.⊃) ‘p’ and ‘q’ are wffs, then ‘(p ⊃ q)’ is a wff.≡) ‘p’ and ‘q’ are wffs, then ‘(p ≡ q)’ is a wff.−) Nothing else is a wff.
▶ ‘p’ and ‘q’ do not appear in the vocabulary of PL.▶ They are formulae variables—variables whose potential
values are formulae of PL.
![Page 36: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/36.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 37: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/37.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]
b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 38: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/38.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]
c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 39: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/39.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]
d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 40: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/40.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]
e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 41: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/41.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]
f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 42: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/42.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 43: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/43.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE GRAMMAR OF PL
▶ To show that ‘(∼(P ∨ Q) ⊃ R)’ is a wff:a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P ∨ Q)’ is a wff [from (a) and (b) and (∨)]d) So, ‘∼(P ∨ Q)’ is a wff [from (c) and (∼)]e) ‘R’ is a wff [from (SL)]f) So, ‘(∼(P ∨ Q) ⊃ R)’ is a wff [from (d), (e), and (⊃)]
▶ A convention: we will allow ourselves to omit theoutermost parenthases, writing ‘∼(P ∨ Q) ⊃ R’ instead of‘(∼(P ∨ Q) ⊃ R)’
![Page 44: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/44.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
IN CLASS EXERCISE!
![Page 45: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/45.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?
a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 46: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/46.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?
a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 47: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/47.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?a) ‘P’ is a wff [from (SL)]
b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 48: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/48.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]
c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 49: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/49.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]
d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 50: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/50.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 51: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/51.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ A wff’s main operator is just the operator associated withthe last rule which would have to be applied if we werebuilding the formula up by applying the rules for wffs.
▶ E.g., what is the main operator of ‘∼P Q’?a) ‘P’ is a wff [from (SL)]b) So, ‘∼P’ is a wff [from (a) and (∼)]c) ‘Q’ is a wff [from (SL)]d) So, ‘(∼P Q)’ is a wff [from (b), (c), and ( )]
▶ So, the main operator is ‘ ’
![Page 52: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/52.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 53: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/53.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]
b) ‘Q’ is a wff [from (SL)]c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 54: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/54.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]
c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 55: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/55.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]
d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 56: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/56.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 57: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/57.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MAIN OPERATORS
▶ What if we had applied the rule ( ) first, and then appliedthe rule (∼)?
a) ‘P’ is a wff [from (SL)]b) ‘Q’ is a wff [from (SL)]c) So, ‘(P Q)’ is a wff [from (a), (b), and ( )]d) So, ‘∼(P Q)’ is a wff [from (c) and (∼)]
▶ ‘∼(P Q)’ is not the same wff as ‘(∼P Q)’
![Page 58: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/58.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 59: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/59.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’
▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 60: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/60.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 61: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/61.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 62: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/62.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’
▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 63: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/63.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 64: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/64.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SUBFORMULAE
▶ p is a subformula of q if and only if, in the course of buildingup q by applying the rules for wffs, p appears on a linebefore q.
▶ ‘∼ P’ is a subformula of ‘∼ P Q’▶ ‘∼ P’ is not a subformula of ‘∼ (P Q)’
▶ A formula’s immediate subformulae are those wffs whoselines were appealed to in the final step of building toformula up.
▶ the immediate subformulae of ‘∼P Q’ are ‘∼P’ and ‘Q’▶ the immediate subformula of ‘∼(P Q)’ is ‘(P Q)’
▶ Another way of notating proofs that a formulae of PL iswell-formed: syntax trees
![Page 65: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/65.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SYNTACTIC STRUCTURE
(∼ (P ∨ Q) ⊃ R)
(⊃)
∼ (P ∨ Q)
(∼)
(P ∨ Q)
(∨)
P
(SL)
Q
(SL)
R
(SL)
![Page 66: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/66.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SYNTACTIC STRUCTURE
(∼(P ∨ Q) ⊃ R)
∼(P ∨ Q)
(P ∨ Q)
P Q
R
![Page 67: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/67.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SYNTACTIC STRUCTURE
(∼ P Q)
∼ P
P
Q
∼ (P Q)
(P Q)
P Q
![Page 68: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/68.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
IN CLASS EXERCISE!
![Page 69: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/69.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 70: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/70.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MEANING
▶ Our guiding assumption: what it is to understand themeaning of an expression is just to understand thecircumstances in which it is true
▶ Parenthases just make syntactic structure explicit. They donot make any contribution to meaning beyond that.
▶ So: we must say what the meanings of the statementsletters are and what the meanings of the logical operatorsare
![Page 71: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/71.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MEANING
▶ Our guiding assumption: what it is to understand themeaning of an expression is just to understand thecircumstances in which it is true
▶ Parenthases just make syntactic structure explicit. They donot make any contribution to meaning beyond that.
▶ So: we must say what the meanings of the statementsletters are and what the meanings of the logical operatorsare
![Page 72: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/72.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
MEANING
▶ Our guiding assumption: what it is to understand themeaning of an expression is just to understand thecircumstances in which it is true
▶ Parenthases just make syntactic structure explicit. They donot make any contribution to meaning beyond that.
▶ So: we must say what the meanings of the statementsletters are and what the meanings of the logical operatorsare
![Page 73: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/73.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF THE STATEMENT LETTERS
▶ Each statement letter represents a statement in English.
▶ The statement letter is true if and only if the statement inEnglish is true.
![Page 74: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/74.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF THE STATEMENT LETTERS
▶ Each statement letter represents a statement in English.▶ The statement letter is true if and only if the statement in
English is true.
![Page 75: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/75.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.
▶ A wff whose main operator is the tilde is called a negation.▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL
▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs ofPL
![Page 76: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/76.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.▶ A wff whose main operator is the tilde is called a negation.
▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL
▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs ofPL
![Page 77: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/77.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.▶ A wff whose main operator is the tilde is called a negation.▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL
▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs ofPL
![Page 78: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/78.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.▶ A wff whose main operator is the tilde is called a negation.▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL
▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs ofPL
![Page 79: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/79.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.▶ A wff whose main operator is the tilde is called a negation.▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL
▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs ofPL
![Page 80: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/80.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∼’
▶ ‘∼’ is called the tilde.▶ A wff whose main operator is the tilde is called a negation.▶ Its immediate subformula is called the negand.
p ∼pT FF T
▶ Note: ‘p’ is not a wff of PL▶ we are using ‘p’ and ‘q’ as variables ranging over the wffs of
PL
![Page 81: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/81.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘•’
▶ ‘ ’ is known as the dot.
▶ A wff whose main operator is the dot is known as aconjunction.
▶ Its immediate subformulae are called conjuncts.
p q p qT T TT F FF T FF F F
![Page 82: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/82.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘•’
▶ ‘ ’ is known as the dot.▶ A wff whose main operator is the dot is known as a
conjunction.
▶ Its immediate subformulae are called conjuncts.
p q p qT T TT F FF T FF F F
![Page 83: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/83.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘•’
▶ ‘ ’ is known as the dot.▶ A wff whose main operator is the dot is known as a
conjunction.▶ Its immediate subformulae are called conjuncts.
p q p qT T TT F FF T FF F F
![Page 84: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/84.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘•’
▶ ‘ ’ is known as the dot.▶ A wff whose main operator is the dot is known as a
conjunction.▶ Its immediate subformulae are called conjuncts.
p q p qT T TT F FF T FF F F
![Page 85: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/85.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∨’
▶ The operator ‘∨’ is known as the wedge.
▶ A wff whose main operator is the wedge is known as adisjunction.
▶ Its immediate subformulae are called disjuncts.
p q p ∨ qT T TT F TF T TF F F
![Page 86: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/86.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∨’
▶ The operator ‘∨’ is known as the wedge.▶ A wff whose main operator is the wedge is known as a
disjunction.
▶ Its immediate subformulae are called disjuncts.
p q p ∨ qT T TT F TF T TF F F
![Page 87: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/87.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∨’
▶ The operator ‘∨’ is known as the wedge.▶ A wff whose main operator is the wedge is known as a
disjunction.▶ Its immediate subformulae are called disjuncts.
p q p ∨ qT T TT F TF T TF F F
![Page 88: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/88.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘∨’
▶ The operator ‘∨’ is known as the wedge.▶ A wff whose main operator is the wedge is known as a
disjunction.▶ Its immediate subformulae are called disjuncts.
p q p ∨ qT T TT F TF T TF F F
![Page 89: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/89.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.
▶ A wff whose main operator is the horseshoe is known as amaterial conditional.
▶ The immediate subformulae which precedes the horseshoeis known as the antecedent.
▶ The immediate subformulae which follows the horseshoeis known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 90: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/90.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.
▶ The immediate subformulae which precedes the horseshoeis known as the antecedent.
▶ The immediate subformulae which follows the horseshoeis known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 91: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/91.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.▶ The immediate subformulae which precedes the horseshoe
is known as the antecedent.
▶ The immediate subformulae which follows the horseshoeis known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 92: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/92.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.▶ The immediate subformulae which precedes the horseshoe
is known as the antecedent.▶ The immediate subformulae which follows the horseshoe
is known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 93: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/93.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.▶ The immediate subformulae which precedes the horseshoe
is known as the antecedent.▶ The immediate subformulae which follows the horseshoe
is known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 94: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/94.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.▶ The immediate subformulae which precedes the horseshoe
is known as the antecedent.▶ The immediate subformulae which follows the horseshoe
is known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 95: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/95.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘⊃’▶ The operator ‘⊃’ is known as the horseshoe.▶ A wff whose main operator is the horseshoe is known as a
material conditional.▶ The immediate subformulae which precedes the horseshoe
is known as the antecedent.▶ The immediate subformulae which follows the horseshoe
is known as the consequent.
p q p ⊃ qT T TT F FF T TF F T
▶ Note: this is the only binary operator which is notsymmetric.
▶ ‘p ⊃ q’ does not have the same meaning as ‘q ⊃ p’
![Page 96: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/96.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘≡’
▶ The operator ‘≡’ is known as the triple bar.
▶ A wff whose main operator is the triple bar is known as amaterial biconditional.
▶ The immediate subformula which appears before the triplebar is known as the biconditional’s left hand side
▶ The immediate subformula which appears after the triplebar is known as the biconditional’s right hand side.
p q p ≡ qT T TT F FF T FF F T
![Page 97: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/97.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘≡’
▶ The operator ‘≡’ is known as the triple bar.▶ A wff whose main operator is the triple bar is known as a
material biconditional.
▶ The immediate subformula which appears before the triplebar is known as the biconditional’s left hand side
▶ The immediate subformula which appears after the triplebar is known as the biconditional’s right hand side.
p q p ≡ qT T TT F FF T FF F T
![Page 98: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/98.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘≡’
▶ The operator ‘≡’ is known as the triple bar.▶ A wff whose main operator is the triple bar is known as a
material biconditional.▶ The immediate subformula which appears before the triple
bar is known as the biconditional’s left hand side
▶ The immediate subformula which appears after the triplebar is known as the biconditional’s right hand side.
p q p ≡ qT T TT F FF T FF F T
![Page 99: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/99.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘≡’
▶ The operator ‘≡’ is known as the triple bar.▶ A wff whose main operator is the triple bar is known as a
material biconditional.▶ The immediate subformula which appears before the triple
bar is known as the biconditional’s left hand side▶ The immediate subformula which appears after the triple
bar is known as the biconditional’s right hand side.
p q p ≡ qT T TT F FF T FF F T
![Page 100: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/100.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
THE MEANING OF ‘≡’
▶ The operator ‘≡’ is known as the triple bar.▶ A wff whose main operator is the triple bar is known as a
material biconditional.▶ The immediate subformula which appears before the triple
bar is known as the biconditional’s left hand side▶ The immediate subformula which appears after the triple
bar is known as the biconditional’s right hand side.
p q p ≡ qT T TT F FF T FF F T
![Page 101: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/101.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false
▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)
[F]
∼P
[F]
P
[T]
Q
[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 102: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/102.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)
[F]
∼P
[F]
P
[T]
Q
[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 103: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/103.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)
[F]
∼P
[F]
P
[T]
Q
[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 104: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/104.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)
[F]
∼P
[F]
P[T]
Q[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 105: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/105.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)
[F]
∼P[F]
P[T]
Q[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 106: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/106.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)[F]
∼P[F]
P[T]
Q[F]
∼(P Q)
[T]
(P Q)
[F]
P
[T]
Q
[F]
![Page 107: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/107.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)[F]
∼P[F]
P[T]
Q[F]
∼(P Q)
[T]
(P Q)
[F]
P[T] Q[F]
![Page 108: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/108.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)[F]
∼P[F]
P[T]
Q[F]
∼(P Q)
[T]
(P Q)[F]
P[T] Q[F]
![Page 109: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/109.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
DETERMINING THE TRUTH-VALUE OF A WFF OF PL
▶ Suppose ‘P’ is true and ‘Q’ is false▶ What are the truth-values of ‘∼ P Q’ and ‘∼(P Q)’?
(∼P Q)[F]
∼P[F]
P[T]
Q[F]
∼(P Q)[T]
(P Q)[F]
P[T] Q[F]
![Page 110: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/110.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P Q
![Page 111: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/111.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P Q
![Page 112: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/112.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P QT TT FF TF F
![Page 113: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/113.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P QT T T TT F T FF T F TF F F F
![Page 114: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/114.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P QT T F T TT F F T FF T T F TF F T F F
![Page 115: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/115.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P QT T F T F TT F F T F FF T T F T TF F T F F F
![Page 116: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/116.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼P Q’:
P Q ∼ P QT T F T F TT F F T F FF T T F T TF F T F F F
![Page 117: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/117.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
![Page 118: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/118.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
![Page 119: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/119.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
T TT FF TF F
![Page 120: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/120.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
T T T TT F T FF T F TF F F F
![Page 121: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/121.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
T T T T TT F T F FF T F F TF F F F F
![Page 122: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/122.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRUTH TABLES
▶ Truth-table for ‘∼(P Q)’:
P Q ∼ (P Q)
T T F T T TT F T T F FF T T F F TF F T F F F
![Page 123: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/123.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 124: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/124.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 125: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/125.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
A TRANSLATION GUIDE FROM PL TO ENGLISH
▶ Submitted for your approval:
∼p −→ It is not the case that pp q −→ Both p and qp ∨ q −→ Either p or qp ⊃ q −→ If p, then qp ≡ q −→ p if and only if q
![Page 126: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/126.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty
▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But
it’s not as bad as you might think.
![Page 127: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/127.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty
▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But
it’s not as bad as you might think.
![Page 128: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/128.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty
▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But
it’s not as bad as you might think.
![Page 129: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/129.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty
▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But
it’s not as bad as you might think.
![Page 130: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/130.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty
▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But
it’s not as bad as you might think.
![Page 131: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/131.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’
▶ But it’s not as bad as you might think.
![Page 132: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/132.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ If ‘p’ is false, then ‘p ⊃ q’ is automatically true, no matterwhat statement q represents
▶ But ‘if John Adams was America’s first president, theneating soap cures cancer’ doesn’t seem true.
▶ If ‘q’ is true, then ‘p ⊃ q is automatically true, no matterwhat statement p represents
▶ But ‘if Oswald didn’t shoot Kennedy, then nobody otherthan Oswald shot Kennedy’ looks false, even if you acceptthat nobody other than Oswald shot Kennedy.
▶ I plead guilty▶ ‘p ⊃ q’ isn’t a perfect translation of ‘if p, then q’▶ But it’s not as bad as you might think.
![Page 133: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/133.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ Dmitri: ‘If it’s a weekday, then I’m on campus.’
▶ Steve: ‘If I’m on campus, then it’s a weekday.’
▶ D := ‘Dmitri is on campus’▶ S := ‘Steve is on campus’▶ W := ‘It is a weekday’
![Page 134: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/134.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ Dmitri: ‘If it’s a weekday, then I’m on campus.’▶ Steve: ‘If I’m on campus, then it’s a weekday.’
▶ D := ‘Dmitri is on campus’▶ S := ‘Steve is on campus’▶ W := ‘It is a weekday’
![Page 135: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/135.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ Dmitri: ‘If it’s a weekday, then I’m on campus.’▶ Steve: ‘If I’m on campus, then it’s a weekday.’
▶ D := ‘Dmitri is on campus’
▶ S := ‘Steve is on campus’▶ W := ‘It is a weekday’
![Page 136: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/136.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ Dmitri: ‘If it’s a weekday, then I’m on campus.’▶ Steve: ‘If I’m on campus, then it’s a weekday.’
▶ D := ‘Dmitri is on campus’▶ S := ‘Steve is on campus’
▶ W := ‘It is a weekday’
![Page 137: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/137.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
▶ Dmitri: ‘If it’s a weekday, then I’m on campus.’▶ Steve: ‘If I’m on campus, then it’s a weekday.’
▶ D := ‘Dmitri is on campus’▶ S := ‘Steve is on campus’▶ W := ‘It is a weekday’
![Page 138: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/138.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T
didn’t lie T
T F
didn’t lie T
F T
lied F
F F
didn’t lie T
S W if S, then W S ⊃ WT T
didn’t lie T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 139: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/139.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F
didn’t lie T
F T
lied F
F F
didn’t lie T
S W if S, then W S ⊃ WT T
didn’t lie T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 140: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/140.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T
lied F
F F
didn’t lie T
S W if S, then W S ⊃ WT T
didn’t lie T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 141: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/141.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F
didn’t lie T
S W if S, then W S ⊃ WT T
didn’t lie T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 142: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/142.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T
didn’t lie T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 143: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/143.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F
lied F
F T
didn’t lie T
F F
didn’t lie T
![Page 144: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/144.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T
didn’t lie T
F F
didn’t lie T
![Page 145: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/145.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F
didn’t lie T
![Page 146: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/146.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie
T
T F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 147: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/147.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie
T
F T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 148: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/148.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied
F
F F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 149: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/149.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie
T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 150: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/150.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie T
S W if S, then W S ⊃ WT T didn’t lie
T
T F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 151: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/151.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie T
S W if S, then W S ⊃ WT T didn’t lie TT F lied
F
F T didn’t lie
T
F F didn’t lie
T
![Page 152: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/152.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie T
S W if S, then W S ⊃ WT T didn’t lie TT F lied FF T didn’t lie
T
F F didn’t lie
T
![Page 153: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/153.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie T
S W if S, then W S ⊃ WT T didn’t lie TT F lied FF T didn’t lie TF F didn’t lie
T
![Page 154: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/154.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
‘p ⊃ q’ AND ‘IF p, THEN q’
D W if W, then D W ⊃ DT T didn’t lie TT F didn’t lie TF T lied FF F didn’t lie T
S W if S, then W S ⊃ WT T didn’t lie TT F lied FF T didn’t lie TF F didn’t lie T
![Page 155: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/155.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
INCLUSIVE AND EXCLUSIVE ‘OR’
The Inclusive ‘or’In the inclusive sense, ‘p or q’ means ‘Either p or q or both.’
▶ ‘Either the elevator or the escalator is working’
The Exclusive ‘or’In the exclusive sense, ‘p or q’ means ‘Either p or q, but not both.’
▶ ‘Either you clean your room, or you’re grounded’
‘∨’ translates to the inclusive ‘or’
![Page 156: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/156.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
INCLUSIVE AND EXCLUSIVE ‘OR’
The Inclusive ‘or’In the inclusive sense, ‘p or q’ means ‘Either p or q or both.’
▶ ‘Either the elevator or the escalator is working’
The Exclusive ‘or’In the exclusive sense, ‘p or q’ means ‘Either p or q, but not both.’
▶ ‘Either you clean your room, or you’re grounded’
‘∨’ translates to the inclusive ‘or’
![Page 157: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/157.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
INCLUSIVE AND EXCLUSIVE ‘OR’
The Inclusive ‘or’In the inclusive sense, ‘p or q’ means ‘Either p or q or both.’
▶ ‘Either the elevator or the escalator is working’
The Exclusive ‘or’In the exclusive sense, ‘p or q’ means ‘Either p or q, but not both.’
▶ ‘Either you clean your room, or you’re grounded’
‘∨’ translates to the inclusive ‘or’
![Page 158: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/158.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
INCLUSIVE AND EXCLUSIVE ‘OR’
The Inclusive ‘or’In the inclusive sense, ‘p or q’ means ‘Either p or q or both.’
▶ ‘Either the elevator or the escalator is working’
The Exclusive ‘or’In the exclusive sense, ‘p or q’ means ‘Either p or q, but not both.’
▶ ‘Either you clean your room, or you’re grounded’
‘∨’ translates to the inclusive ‘or’
![Page 159: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/159.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
INCLUSIVE AND EXCLUSIVE ‘OR’
The Inclusive ‘or’In the inclusive sense, ‘p or q’ means ‘Either p or q or both.’
▶ ‘Either the elevator or the escalator is working’
The Exclusive ‘or’In the exclusive sense, ‘p or q’ means ‘Either p or q, but not both.’
▶ ‘Either you clean your room, or you’re grounded’
‘∨’ translates to the inclusive ‘or’
![Page 160: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/160.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 161: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/161.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 162: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/162.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 163: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/163.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 164: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/164.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 165: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/165.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 166: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/166.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 167: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/167.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 168: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/168.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 169: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/169.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 170: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/170.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 171: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/171.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 172: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/172.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
CANONICAL LOGICAL FORM
▶ Call the phrases on the right-hand-side of the translationguide the canonical logical expressions of English.
▶ If the logical structure of an English statement is written inthis form, then that statement is in canonical logical form.
▶ E.g.,
If both John loves Andrew and it is not the case thatAndrew loves John, then it is not the case that Johnand Andrew will be friends.
▶ Expressions in this form are easily translated into PL. LetJ = ‘John loves Andrew’, A = ‘Andrew loves John’, andF = ‘John and Andrew will be friends’. Then,
(J ∼A) ⊃ ∼F
![Page 173: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/173.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
ENGLISH SENTENCES NOT IN CANONICAL LOGICAL
FORM
▶ John and Andrew won’t be friends if John lovesAndrew but Andrew doesn’t love him back.
▶ We’ll have to say a bit more about how to translatesentences like this into PL.
![Page 174: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/174.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
ENGLISH SENTENCES NOT IN CANONICAL LOGICAL
FORM
▶ John and Andrew won’t be friends if John lovesAndrew but Andrew doesn’t love him back.
▶ We’ll have to say a bit more about how to translatesentences like this into PL.
![Page 175: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/175.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
ENGLISH SENTENCES NOT IN CANONICAL LOGICAL
FORM
▶ John and Andrew won’t be friends if John lovesAndrew but Andrew doesn’t love him back.
▶ We’ll have to say a bit more about how to translatesentences like this into PL.
![Page 176: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/176.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 177: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/177.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
OUTLINE
THE LANGUAGE PLSyntax for PLSemantics for PL
TRANSLATION FROM PL TO ENGLISH
A Translation GuideCanonical Logical Form
TRANSLATIONS FROM ENGLISH TO PLTranslating into Canonical Logical Form
![Page 178: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/178.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ Consider:
Harry doesn’t like chestnuts
▶ This has the same meaning as
It is not the case that Harry likes chestnuts.
▶ So, if we let H = ‘Harry likes chestnuts’, then we cantranslate it into PL as follows:
∼H
![Page 179: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/179.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ Consider:Harry doesn’t like chestnuts
▶ This has the same meaning as
It is not the case that Harry likes chestnuts.
▶ So, if we let H = ‘Harry likes chestnuts’, then we cantranslate it into PL as follows:
∼H
![Page 180: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/180.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ Consider:Harry doesn’t like chestnuts
▶ This has the same meaning as
It is not the case that Harry likes chestnuts.
▶ So, if we let H = ‘Harry likes chestnuts’, then we cantranslate it into PL as follows:
∼H
![Page 181: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/181.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ Consider:Harry doesn’t like chestnuts
▶ This has the same meaning as
It is not the case that Harry likes chestnuts.
▶ So, if we let H = ‘Harry likes chestnuts’, then we cantranslate it into PL as follows:
∼H
![Page 182: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/182.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ Consider:Harry doesn’t like chestnuts
▶ This has the same meaning as
It is not the case that Harry likes chestnuts.
▶ So, if we let H = ‘Harry likes chestnuts’, then we cantranslate it into PL as follows:
∼H
![Page 183: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/183.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ A general strategy:
▶ for any sentence of English, find another sentence ofEnglish which has the same meaning as the first, and whichis in canonical logical form.
▶ Then, translate the sentence in canonical logical form intoPL following the translation guide:
It is not the case that p −→ ∼pBoth p and q −→ p qEither p or q −→ p ∨ q
If p, then q −→ p ⊃ qp if and only if q −→ p ≡ q
![Page 184: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/184.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ A general strategy:▶ for any sentence of English, find another sentence of
English which has the same meaning as the first, and whichis in canonical logical form.
▶ Then, translate the sentence in canonical logical form intoPL following the translation guide:
It is not the case that p −→ ∼pBoth p and q −→ p qEither p or q −→ p ∨ q
If p, then q −→ p ⊃ qp if and only if q −→ p ≡ q
![Page 185: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/185.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ A general strategy:▶ for any sentence of English, find another sentence of
English which has the same meaning as the first, and whichis in canonical logical form.
▶ Then, translate the sentence in canonical logical form intoPL following the translation guide:
It is not the case that p −→ ∼pBoth p and q −→ p qEither p or q −→ p ∨ q
If p, then q −→ p ⊃ qp if and only if q −→ p ≡ q
![Page 186: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/186.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ A general strategy:▶ for any sentence of English, find another sentence of
English which has the same meaning as the first, and whichis in canonical logical form.
▶ Then, translate the sentence in canonical logical form intoPL following the translation guide:
It is not the case that p −→ ∼pBoth p and q −→ p qEither p or q −→ p ∨ q
If p, then q −→ p ⊃ qp if and only if q −→ p ≡ q
![Page 187: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/187.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:
▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 188: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/188.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.
▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 189: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/189.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.
▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 190: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/190.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.
▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 191: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/191.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.
▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 192: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/192.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 193: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/193.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
TRANSLATING INTO CANONICAL LOGICAL FORM
▶ For instance:▶ I won’t go if John does.▶ If John goes, then it is not the case that I will go.▶ John goes ⊃ it is not the case that I will go.▶ John goes ⊃ ∼I will go.▶ J ⊃∼ I
▶ J = ‘John goes’; and I = ‘I will go’.
![Page 194: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/194.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: NEGATION
▶ I hate not getting what I want and I hate getting what I want.
▶ It is not the case that I hate getting what I want, and I hategetting what I want.
▶ ∼ I hate getting what I want I hate getting what I want.▶
∼H H
![Page 195: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/195.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: NEGATION
▶ I hate not getting what I want and I hate getting what I want.▶ It is not the case that I hate getting what I want, and I hate
getting what I want.
▶ ∼ I hate getting what I want I hate getting what I want.▶
∼H H
![Page 196: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/196.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: NEGATION
▶ I hate not getting what I want and I hate getting what I want.▶ It is not the case that I hate getting what I want, and I hate
getting what I want.▶ ∼ I hate getting what I want I hate getting what I want.
▶
∼H H
![Page 197: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/197.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: NEGATION
▶ I hate not getting what I want and I hate getting what I want.▶ It is not the case that I hate getting what I want, and I hate
getting what I want.▶ ∼ I hate getting what I want I hate getting what I want.▶
∼H H
![Page 198: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/198.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: NEGATION
H ∼ H HT F T F TF T F F F
![Page 199: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/199.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: CONJUNCTION
▶ Hannes loves peaches and he loves apples.Hannes loves peaches but he loves apples.
▶ P = ‘Hannes loves peaches’ and A = ‘Hannes loves apples’▶
P A
![Page 200: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/200.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: CONJUNCTION
▶ Hannes loves peaches and he loves apples.Hannes loves peaches but he loves apples.
▶ P = ‘Hannes loves peaches’ and A = ‘Hannes loves apples’▶
P A
![Page 201: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/201.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: CONJUNCTION
▶ Hannes loves peaches and he loves apples.Hannes loves peaches but he loves apples.
▶ P = ‘Hannes loves peaches’ and A = ‘Hannes loves apples’
▶
P A
![Page 202: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/202.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: CONJUNCTION
▶ Hannes loves peaches and he loves apples.Hannes loves peaches but he loves apples.
▶ P = ‘Hannes loves peaches’ and A = ‘Hannes loves apples’▶
P A
![Page 203: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/203.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: CONJUNCTION
p and qp, but q
p; however, qp, though q
p as well as q
−→ p q
![Page 204: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/204.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: DISJUNCTION
p or qp unless q
}−→ p ∨ q
![Page 205: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/205.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: THE MATERIAL CONDITIONAL
If p, then qp only if q
q if pq is true whenever p is
p is sufficient for qq is necessary for p
−→ p ⊃ q
![Page 206: Propositional Logic: Syntax and Semanticspitt.edu/~jdg83/teaching/pdfs/Logic SP 2015/6.pdf · Propositional Logic: Syntax and Semantics PHIL UA-70: Logic February 24–26, 2015 (˘](https://reader033.fdocuments.in/reader033/viewer/2022053012/5f0fd94b7e708231d4463012/html5/thumbnails/206.jpg)
THE LANGUAGE PL TRANSLATION FROM PL TO ENGLISH TRANSLATIONS FROM ENGLISH TO PL
SOME ISSUES: THE MATERIAL BICONDITIONAL
p if and only if qp is necessary and sufficient for qp is true when and only when q is
−→ p ≡ q