Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
02—Traditional Logic
CS 3234: Logic and Formal Systems
Martin Henz and Aquinas Hobor
August 19, 2010
Generated on Wednesday 18th August, 2010, 23:10
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
1 Categorical Terms and their MeaningOrigins and GoalsForm, not ContentCategorical TermsMeaning through models
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Traditional Logic
Origins
Greek philosopher Aristotle (384–322 BCE) wrote treatise PriorAnalytics; considered the earliest study in formal logic; widelyaccepted as the definite approach to deductive reasoning untilthe 19thcentury.
GoalExpress relationships between sets; allow reasoning about setmembership
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 1
All humans are mortal.All Greeks are humans.Therefore, all Greeks are mortal.
Makes “sense”, right?
Why?
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 2
All cats are predators.Some animals are cats.Therefore, all animals are predators.
Does not make sense!
Why not?
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 3
All slack track systems are caterpillar systems.All Christie suspension systems are slack tracksystems.Therefore, all Christie suspension systems arecaterpillar systems.
Makes sense, even if you do not know anything aboutsuspension systems.
Form, not contentIn logic, we are interested in the form of valid arguments,irrespective of any particular domain of discourse.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Categorical Terms
Terms refer to setsTerm animals refers to the set of animals,term brave refers to the set of brave persons, etc
Terms
The set Terms contains all terms under consideration
Examples
animals ∈ Terms
brave ∈ Terms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Models
MeaningA modelM fixes what elements we are interested in, and whatwe mean by each term
Fix universe
For a particularM, the universe UM contains all elements thatwe are interested in.
Meaning of termsFor a particularM and a particular term t , the meaning of t inM, denoted tM, is a particular subset of UM.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 1A
For our examples, we haveTerm = {cats,humans,Greeks, . . .}.
First meaningMUM: the set of all living beings,catM the set of all cats,humansM the set of all humans,. . .
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 1B
Consider the same Term = {cats,humans,Greeks, . . .}.
Second meaningM′
UM: A set of 100 playing cards, depicting living beings,catM: all cards that show cats,humansM: all cards that show humans,. . .
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 2A
Consider the following set of terms:Term = {even,odd,belowfour}
First meaningM1
UM1 = N,evenM1 = {0,2,4, . . .},oddM1 = {1,3,5, . . .}, andbelowfourM1 = {0,1,2,3}.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 2B
Consider the same Term = {even,odd,belowfour}
Second meaningM2
UM2 = {a,b, c, . . . , z},evenM2 = {a,e, i ,o,u},oddM2 = {b, c,d , . . .}, andbelowfourM2 = ∅.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, ProofsCategorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
3 Manipulating Terms and Propositions
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Categorical Propositions
All cats are predators
expresses a relationship between the terms cats (subject) andpredators (object).
Intended meaningEvery thing that is included in the class represented by cats isalso included in the class represented by predators.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Four Kinds of Categorical Propositions
Quantityuniversal particular
Qualityaffirmative All t1 are t2 Some t1 are t2negative No t1 are t2 Some t1 are not t2
ExampleSome cats are not brave is a particular, negativeproposition.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Meaning of Universal Affirmative Propositions
In a particular modelM, All Greeks are mortal meansthat GreeksM is a subset of mortalM
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
More formally...
(All subject are object)M =
{T if subjectM ⊆ objectM,F otherwise
Here T and F represent the logical truth values true and false,respectively.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Meaning of Universal Negative Propositions
In a particular modelM, No Greeks are cats means thatthe intersection of GreeksM and of catsM is empty.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
More formally...
(No subject are object)M =
{T if subjectM ∩ objectM = ∅,F otherwise
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Meaning of Particular Affirmative Propositions
In a particular modelM, Some humans are Greeks meansthat the intersection of humansM and of GreeksM is notempty.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
More formally...
(Some subject are object)M =
{T if subjectM ∩ objectM 6= ∅,F otherwise
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Meaning of Particular Negative Propositions
In a particular modelM, Some Greeks are notvegetarians means that the difference of GreeksM andvegetariansM is not empty.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
More formally...
(Some subject are not object)M =
{T if subjectM/objectM 6= ∅,F otherwise
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Axioms
Axioms are propositions that are assumed to hold.
Axiom (HM)
The proposition All humans are mortal holds.
Axiom (GH)The proposition All Greeks are humans holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Graphical Notation
All humans are mortal
[HumansMortality]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Lemmas
Lemmas are affirmations that follow from all known facts.
Proof obligationA lemma must be followed by a proof that demonstrates how itfollows from known facts.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Trivial Example of Proof
LemmaThe proposition All humans are mortal holds.
Proof.
All humans are mortal
[HM]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Unusual Models
We can choose any model for our terms, also “unusual” ones.
Example
UM = {0,1}, humansM = {0}, mortalM = {1}
HereAll humans are mortal
does not hold.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs
Asserting Axioms
Purpose of axiomsBy asserting an axiom A, we are focusing our attention to onlythose modelsM for which AM = T .
ConsequenceThe lemmas that we prove while utilizing an axiom only hold inthe models in which the axiom holds.
ValidityA proposition is called valid, if it holds in all models.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and PropositionsComplementConversionContrapositionObversionCombinations
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Complement
We allow ourselves to put non in front of a term.
Meaning of complementIn a modelM, the meaning of non t is the complement of themeaning of t
More formally
In a modelM, (non t)M = UM/tM
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Double Complement
Axiom (NonNon)For any term t, the term non non t is considered equal to t.
· · · t · · ·
· · ·non non t · · ·[NNI]
· · ·non non t · · ·
· · · t · · ·[NNE]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Rule Schema
· · · t · · ·
· · ·non non t · · ·[NNI]
is a rule schema. An instance is:
Some t1 are t2
Some non non t1 are t2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Definitions
We allow ourselves to state definitions that may be convenient.Definitions are similar to axioms; they fix the properties of aparticular item for the purpose of a discussion.
Definition (ImmDef)The term immortal is considered equal to the term nonmortal.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Writing a Proof Graphically
LemmaThe proposition All humans are non immortal holds.
Proof.
All humans are mortal[HM]
All humans are non non mortal[NNI]
All humans are non immortal[ImmDef]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Writing a Text-based Proof
LemmaThe proposition All humans are non immortal holds.
Proof.
1 All humans are mortal HM2 All humans are non non
mortalNNI 1
3 All humans are non immortal ImmDef 2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Conversion switches subject and object
Definition (ConvDef)For all terms t1 and t2, we define
convert(All t1 are t2) = All t2 are t1convert(Some t1 are t2) = Some t2 are t1
convert(No t1 are t2) = No t2 are t1convert(Some t1 are not t2) = Some t2 are not t1
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Which Conversions Hold?
IfAll Greeks are humans
holds in a model, then does
All humans are Greeks
hold?
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Valid Conversions
Axiom (ConvE1)If, for some terms t1 and t2, the proposition
convert(Some t1 are t2)
holds, then the proposition
Some t1 are t2
also holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Valid Conversions
Axiom (ConvE2)
If, for some terms t1 and t2, the proposition
convert(No t1 are t2)
holds, then the proposition
No t1 are t2
also holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
In Graphical Notation
In graphical notation, two rules correspond to the two cases.
convert(Some t1 are t2)
Some t1 are t2
[ConvE1]
convert(No t1 are t2)
No t1 are t2
[ConvE2]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Example
Axiom (AC)The proposition Some animals are cats holds.
LemmaThe proposition Some cats are animals holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Proof
Some animals are cats[AC]
convert(Some cats are animals)
[ConvDef]
Some cats are animals[ConvE1]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Example (text-based proof)
Proof.
1 Some animals are cats AC2 convert(Some cats are
animals)ConvDef 1
3 Some cats are animals ConvE1 2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Contraposition switches and complements
Definition (ContrDef)For all terms t1 and t2, we define
contrapose(All t1 are t2)= All non t2 are non t1
contrapose(Some t1 are t2)= Some non t2 are non t1
contrapose(No t1 are t2)= No non t2 are non t1
contrapose(Some t1 are not t2)= Some non t2 are not non t1
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
For which propositions is contraposition valid?
contrapose(All t1 are t2)
All t1 are t2
[ContrE1]
contrapose(Some t1 are not t2)
Some t1 are not t2
[ContrE2]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Obversion switches quality and complements object
Definition (ObvDef)For all terms t1 and t2, we define
obvert(All t1 are t2) = No t1 are non t2obvert(Some t1 are t2) = Some t1 are not non t2
obvert(No t1 are t2) = All t1 are non t2obvert(Some t1 are not t2) = Some t1 are non t2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Examples
Obversion switches quality and complements object
Example 1obvert(All Greeks are humans)= No Greeks are non humans
Example 2obvert(Some animals are cats)= Some animals are not non cats
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Validity of Obversion
Obversion is valid for all kinds of propositions.
Axiom (ObvE)If, for some proposition p
obvert(p)
holds, then the proposition p also holds.
obvert(p)
p[ObvE]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Example
Axiom (SHV)The proposition Some humans are vegans holds.
Lemma (NNVeg)The proposition Some humans are not non vegans holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Proof
Some humans are vegans[SHV]
Some humans are non non vegans[NNI]
obvert(Some humans are not non vegans)[ObvDef]
Some humans are not non vegans[ObvE]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Proof (text-based)
Proof.
1 Some humans are vegans SHV2 Some humans are non non
vegansNNI 1
3 obvert(Some humans are notnon vegans)
ObvDef 2
4 Some humans are not nonvegans
ObvE 3
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Another Lemma
Lemma (SomeNon)
For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.
A lemma of the form “If p1 then p2” is valid, if in every model inwhich the proposition p1 holds, the proposition p2 also holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
Proof
Lemma (SomeNon)
For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.
Proof.
1 Some non t1 are non t2 premise2 convert(Some non t2 are non t1) ConvDef 13 Some non t2 are non t1 ConvE1 24 obvert(Some non t2 are not t1) ObvDef 35 Some non t2 are not t1 ObvE 4
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ComplementConversionContrapositionObversionCombinations
“iff” means “if and only if”
Lemma (AllNonNon)For any terms t1 and t2, the proposition All non t1 are nont2 holds iff the proposition All t2 are t1 holds.
All non t1 are non t2
All t2 are t1
All t2 are t1
All non t1 are non t2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and SyllogismsArgumentsSyllogismsBarbaraFun With Barbara
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Argument
An argument has the form
If premises then conclusion
Sometimes also
premises therefore conclusion
Example:
Lemma (SomeNon)For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Syllogisms
A syllogism is an argument with two premises, in which threedifferent terms occur, and in which every term occurs twice, butnever twice in the same proposition.
ExampleAll cats are predators.Some animals are cats.Therefore, all animals are predators.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Barbara
Axiom (B)For all terms minor, middle, and major, if All middle are majorholds, and All minor are middle holds, then All minor aremajor also holds.
All middle are major All minor are middle
All minor are major[B]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Why is Barbara valid?
If the first premise holds, then areas 1 and 4 are empty, and ifthe second premise holds, then areas 2 and 3 are empty. Theconclusion simply states that areas 1 and 2 are empty, whichclearly follows from the premises, regardless what otherproperties the model under consideration has.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Example
LemmaThe proposition All Greeks are mortal holds.
Proof.
1 All Greeks are humans GH2 All humans are mortal HM3 All Greeks are mortal B 1,2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Officers as Poultry?
PremisesNo ducks waltz.No officers ever decline to waltz.All my poultry are ducks.
ConclusionNo officers are my poultry.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Formulation in Term Logic
Lemma (No-Officers-Are-My-Poutry)If
No ducks are things-that-waltz holds,No officers are non things-that-waltz holds,andAll my-poutry are ducks holds,
then No officers are my-poultry also holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof
1 No officers are nonthings-that-waltz
premise
2 obvert(All officers arethings-that-waltz)
ObvDef 1
3 All officers arethings-that-waltz)
ObvE 2
4 No ducks arethings-that-waltz)
premise
5 convert(No things-that-waltzare ducks)
ConvDef 4
6 No things-that-waltz areducks
ConvE2 5
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof (continued)
7 No things-that-waltz are nonnon ducks
NNI 6
8 obvert(All things-that-waltzare non ducks)
ObvDef 7
9 All things-that-waltz arenon ducks
ObvE 8
10 All my-poultry are ducks premise11 All my-poultry are non non
ducksNNI 10
12 All non non my-poultry arenon non ducks
NNI 11
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof (continued)
13 contrapose(All non ducks arenon my-poultry)
ContrDef 12
14 All non ducks are nonmy-poultry
ContrE1 13
15 All things-that-waltz arenon my-poultry
B 9,14
16 All officers are nonmy-poultry
B 3,15
17 obvert(No officers aremy-poultry)
ObvDef 16
18 No officers are my-poultry ObvE 17
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Admin
Assignment 1: out on module homepage; due 26/8,11:00amCoq Homework 1: out on module homepage; due 27/8,9:30pmMonday, Wednesday: Office hoursTuesday: Tutorials (clarification of assignment)Wednesday: Labs (Coq Homework 1; start earlier!)
CS 3234: Logic and Formal Systems 02—Traditional Logic
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