March 5: more on quantifiers We have encountered formulas and sentences that include multiple...

March 5: more on quantifiers

We have encountered formulas and sentences that We have encountered formulas and sentences that include multiple quantifiers:include multiple quantifiers:

Take, for example:Take, for example:UD: People in Michael’s officeUD: People in Michael’s officeLxy: x likes yLxy: x likes ym: Michaelm: Michael

‘‘There is someone that Michael likes and everyone There is someone that Michael likes and everyone likes Michael”likes Michael”((x) Lmx & (x) Lmx & (y) Lymy) Lym

March 5: more on quantifiers

But there are also sentences that include multiple But there are also sentences that include multiple quantifiers of quantifiers of overlapping scopeoverlapping scope::

UD: People in Michael’s officeUD: People in Michael’s officeLxy: x likes yLxy: x likes ym: Michaelm: MichaelSo we’ve symbolized:So we’ve symbolized:

‘‘Everyone likes Michael’Everyone likes Michael’((x) Lxmx) Lxm

Now suppose we want to symbolizeNow suppose we want to symbolize‘‘Everyone likes someone’Everyone likes someone’

Multiple quantifiers with overlapping scopeAnd suppose our UD is now ‘all people’And suppose our UD is now ‘all people’UD: all peopleUD: all peopleLxy: x likes yLxy: x likes y

‘‘Everyone likes someone’Everyone likes someone’((x) x) there is someone such that…there is someone such that…((x) (x) (y) y) such that…such that…((x) (x) (y) Lxyy) Lxy

Multiple quantifiers with overlapping scopeNow suppose:Now suppose:UD: all peopleUD: all peopleLxy: x likes yLxy: x likes yAnd we want to symbolize:And we want to symbolize:

‘‘Everyone likes everyone’Everyone likes everyone’((x) x) is such that for everyone…is such that for everyone…and we add (and we add (y) for another universal quantifiery) for another universal quantifierso that we have:so that we have:((x) is such that for all y (x) is such that for all y (y)y)((x) (x) (y) x likes yy) x likes y((x) (x) (y) Lxyy) Lxy

Multiple quantifiers with overlapping scope

Suppose we keep:Suppose we keep:UD: all peopleUD: all peopleLxy: x likes yLxy: x likes yAnd now we want to symbolize:And now we want to symbolize:

‘‘Someone likes everyone’Someone likes everyone’((x) such that for everyone y, x…x) such that for everyone y, x…We add ‘(We add ‘(y)’ for that everyone, and then:y)’ for that everyone, and then:((x) (x) (y) Lxyy) Lxy

Multiple quantifiers with overlapping scopeSuppose we keep:Suppose we keep:UD: all peopleUD: all peopleLxy: x likes yLxy: x likes yAnd we want to symbolizeAnd we want to symbolize

‘‘Someone likes someone’Someone likes someone’((x) such that there is someone…x) such that there is someone…We add (We add (y) for ‘there is some y’… and gety) for ‘there is some y’… and get((x) (x) (y) Lxyy) Lxy

Multiple quantifiers with overlapping scopePairs of quantifiers*Pairs of quantifiers* can occur in 4 combinations: can occur in 4 combinations:

((x) (x) (y) There is an x and there is a y such that [or]y) There is an x and there is a y such that [or] There is a pair x and y such that …There is a pair x and y such that …

((x) (x) (y) For each x and for each y… [or] For eachy) For each x and for each y… [or] For each pair x and y …pair x and y …

((x) (x) (y) For each x there is a y such that …y) For each x there is a y such that …((x) (x) (y) There is an x such that for each y … y) There is an x such that for each y … *Although we won’t deal with them, there can be *Although we won’t deal with them, there can be

more than 2 (or more than a pair) of quantifiers of more than 2 (or more than a pair) of quantifiers of overlapping scope.overlapping scope.

(Informal) Semantics of PL

The extension of a 3-place predicate (e.g., Bwxy: w is The extension of a 3-place predicate (e.g., Bwxy: w is between x and y), is between x and y), is a set of 3-ordered objectsa set of 3-ordered objects, and , and the extension of a 4-place predicate (e.g., Twxyz: w the extension of a 4-place predicate (e.g., Twxyz: w between x, y, and z) is between x, y, and z) is a set of 4-ordered objectsa set of 4-ordered objects, , and so forth; each extension is a set of n-tuple sets and so forth; each extension is a set of n-tuple sets (in the first case, a set of 3 ordered objects and, in (in the first case, a set of 3 ordered objects and, in the second, a set of 4 ordered objects).the second, a set of 4 ordered objects).

Multiple quantifiers with overlapping scope: the difference a UD makes for symbolization

1. 1. UD: personsUD: personsLxy: x likes yLxy: x likes y

‘‘Everyone likes everyone’Everyone likes everyone’((x) (x) (y) Lxyy) Lxy

‘‘Someone likes someone’Someone likes someone’((x) (x) (y) Lxyy) Lxy

2.2.UD: EverythingUD: EverythingLxy: x likes yLxy: x likes yPx: x is a personPx: x is a person

‘‘Everyone likes everyone’Everyone likes everyone’((x) (x) (y) [(Px & Py) y) [(Px & Py) Lxy] Lxy]

‘‘Someone likes someone’Someone likes someone’((x) (x) (y) [(Px & Py) & Lxy]y) [(Px & Py) & Lxy]

Multiple quantifiers with overlapping scopeOne has to “learn to read” the sentences of One has to “learn to read” the sentences of PLPL into into

“quasi-English” to check out a symbolization. In “quasi-English” to check out a symbolization. In doing so, it is crucial to identify the role of each doing so, it is crucial to identify the role of each logical operatorlogical operator. For example, in. For example, in

((x) (x) (y) [(Px & Py) y) [(Px & Py) Lxy] Lxy] ((x) is the main logical operator and (x) is the main logical operator and (y) is they) is the

main logical operator of the subformula:main logical operator of the subformula:(Px & Py) (Px & Py) Lxy Lxy

So we read (So we read (x) first and (x) first and (y) second.y) second.‘‘Every x is such that every y is such that’ [or ‘Every Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’]pair x and y is such that…’]

Multiple quantifiers with overlapping scopeWe read (We read (x) first and (x) first and (y) second.y) second.

‘‘Every x is such that every y is such that’ [or ‘Every Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’]pair x and y is such that…’]

As the main logical operator of the As the main logical operator of the next subformulanext subformula, , (Px & Py) (Px & Py) Lxy, is the Lxy, is the we move to that next: we move to that next: ‘ ‘Every x is such that every y is such that’ [or Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’] ‘Every pair x and y is such that…’] IFIF

Multiple quantifiers with overlapping scope((x) (x) (y) [(Px & Py) y) [(Px & Py) Lxy] Lxy]‘‘Every x is such that every y is such that’ [or ‘Every Every x is such that every y is such that’ [or ‘Every

pair x and y is such that…’] pair x and y is such that…’] IFIF‘ ‘ x is a person and y is a person’x is a person and y is a person’THENTHENLxy (x likes y)Lxy (x likes y)

All together: All together: ‘‘Every x is such that every y is such that, if x is a Every x is such that every y is such that, if x is a

person and y is a person, then x likes y’ [OR]person and y is a person, then x likes y’ [OR]‘‘For every pair x and y, if x is a person and y is a For every pair x and y, if x is a person and y is a

person, then x likes y’person, then x likes y’

(Informal) Semantics of PLIn In SL, SL, the semantic notion we used to determine truth the semantic notion we used to determine truth

status was the truth value assignment. status was the truth value assignment. This worked because the sentences of This worked because the sentences of SLSL are such that are such that

their truth status is a function of truth functional their truth status is a function of truth functional assignments to their atomic components (atomic assignments to their atomic components (atomic sentences) and, in compound sentences, the TVA’s of sentences) and, in compound sentences, the TVA’s of atomic sentences and truth-functional connectives.atomic sentences and truth-functional connectives.

This doesn’t work in This doesn’t work in PLPL, except for those sentences that , except for those sentences that are just the atomic sentences (A, B, C…) of are just the atomic sentences (A, B, C…) of SLSL..

The basic semantic concept of The basic semantic concept of PLPL is is an interpretationan interpretation..

(Informal) Semantics of PLIn In PL, PL, an interpretation interprets:an interpretation interprets:

each individual constant (if any)each individual constant (if any)each predicateeach predicateeach sentence letter of each sentence letter of PLPL

All of these are relative to some universe of All of these are relative to some universe of discoursediscourse

We can take symbolization keys as we have We can take symbolization keys as we have encountered them so far as embodying encountered them so far as embodying interpretations with the following caveat:interpretations with the following caveat:

In an interpretation, a UD is always some non-empty In an interpretation, a UD is always some non-empty set, and so we classify them as sets when set, and so we classify them as sets when specifying an interpretationspecifying an interpretation


There are 2 kinds of atomic sentences of There are 2 kinds of atomic sentences of PLPL::Sentence letters (A through Z with or without Sentence letters (A through Z with or without subscripts: the semantics involve truth functional subscripts: the semantics involve truth functional assignments)assignments)A n-place predicate of A n-place predicate of PL PL followed by n-constants:followed by n-constants:

What are its semantics?What are its semantics?Start with a one-place predicate, Fx, and the atomic Start with a one-place predicate, Fx, and the atomic

sentence:sentence:FaFa


The truth or falsity of ‘Fa’ depends on The truth or falsity of ‘Fa’ depends on an an interpretationinterpretation that interprets: that interprets:the predicate Fxthe predicate Fxthe constant athe constant aand a UD: a set of objects over which the and a UD: a set of objects over which the predicates and variables range, and from which predicates and variables range, and from which constants pick out objectsconstants pick out objects


Atomic sentences of Atomic sentences of PLPL::Consider Fa on interpretation 1:Consider Fa on interpretation 1:1. UD: set of living things1. UD: set of living things

Fx: x is a humanFx: x is a humana: is Socrates (the historical figure)a: is Socrates (the historical figure)

On this interpretation, ‘Fa’ is trueOn this interpretation, ‘Fa’ is true


Change the UD and/or the predicate and/or the Change the UD and/or the predicate and/or the constant, and the truth status of sentence on the constant, and the truth status of sentence on the interpretation also changes. Consider ‘Fa’ on interpretation also changes. Consider ‘Fa’ on interpretation 2:interpretation 2:

2. UD: set of living things2. UD: set of living things Fx: x is a potatoFx: x is a potato

a: Socratesa: SocratesAssuming, again, that ‘a’ denotes the historical Assuming, again, that ‘a’ denotes the historical

person, Socrates, the sentence ‘Fa’ is false on this person, Socrates, the sentence ‘Fa’ is false on this interpretation.interpretation.


Atomic sentences of Atomic sentences of PLPL involving 2-place predicates: involving 2-place predicates:Lxy: x is larger than yLxy: x is larger than y

And say the UD is “the set of positive integers”And say the UD is “the set of positive integers”A two place predicate is a A two place predicate is a relational predicate: relational predicate: itit

picks out picks out sets of pairs of members of the UD whose sets of pairs of members of the UD whose order often matters order often matters – and in the case of this – and in the case of this predicate, order does matter.predicate, order does matter.


3. UD: the set of positive integers3. UD: the set of positive integersLxy: x is larger than yLxy: x is larger than ya: 1a: 1b: 2b: 2Lba is true on interpretation 3. Lba is true on interpretation 3. The The extensionextension of Lxy on interpretation 3 are those of Lxy on interpretation 3 are those

pairs (4, 3; 5, 2; 201, 200; 4000, 3999; and so forth) pairs (4, 3; 5, 2; 201, 200; 4000, 3999; and so forth) of which it is true that the first member of the pair of which it is true that the first member of the pair is larger than the second member of the pair.is larger than the second member of the pair.

So, the pair 2, 3 is So, the pair 2, 3 is notnot a pair that is an extension of a pair that is an extension of Lxy on interpretation 3.Lxy on interpretation 3.

4. The set of all buildings and all people4. The set of all buildings and all peopleLxy: x is larger than yLxy: x is larger than ya: The Empire State Buildinga: The Empire State Buildingb: George Bushb: George BushLba is false on interpretation 4.Lba is false on interpretation 4.An interpretation may assign the same member of the UD to An interpretation may assign the same member of the UD to

more than one constant:more than one constant:5. The set of planets in our solar system5. The set of planets in our solar systemCxy: x is closer to the sun than y.Cxy: x is closer to the sun than y.m: Venusm: Venusn: Venusn: VenusCmn is false on interpretation 5.Cmn is false on interpretation 5.


Some interpretations of 2-place predicates mean Some interpretations of 2-place predicates mean that the extension of a predicate includes pairs in that the extension of a predicate includes pairs in which the 1which the 1stst and 2 and 2ndnd members are members are the samethe same::

6. UD: the set of positive integers6. UD: the set of positive integersExy: x is less than or equal to yExy: x is less than or equal to yThe extension of the predicate Exy (as defined The extension of the predicate Exy (as defined

above) includes not only 2, 3 but also 2, 2; 4, 4; and above) includes not only 2, 3 but also 2, 2; 4, 4; and so forth.so forth.

(Informal) Semantics of PLCompound sentences of Compound sentences of PL PL that do not include quantifiers:that do not include quantifiers:

Cab v CbaCab v CbaAs the main logical operator is ‘v’, this sentence is true or As the main logical operator is ‘v’, this sentence is true or

false on an interpretation depending on whether at least false on an interpretation depending on whether at least either ‘Cab’ or ‘Cba’ is true on that interpretation. We use either ‘Cab’ or ‘Cba’ is true on that interpretation. We use the characteristic truth tables for the connectives to the characteristic truth tables for the connectives to determine whether a truth functional sentence of determine whether a truth functional sentence of PLPL is is true on some interpretation.true on some interpretation.

7. UD: the set of persons7. UD: the set of personsCxy: x likes yCxy: x likes ya: Andreaa: Andreab: Bruceb: Bruce


Compound sentences of Compound sentences of PL PL whose main logical operator is whose main logical operator is not a quantifier:not a quantifier:

((x) (Wx x) (Wx Mx) & ~( Mx) & ~(x) (Ex & Ox)x) (Ex & Ox)As the main logical operator is ‘&’, this sentence is true or As the main logical operator is ‘&’, this sentence is true or

false on an interpretation if and only if both the right and false on an interpretation if and only if both the right and left conjuncts are true on that interpretation.left conjuncts are true on that interpretation.

8. UD: the set of all things8. UD: the set of all things Wx: x is a whaleWx: x is a whale

Mx: x is a mammalMx: x is a mammalEx: x is an even positive integerEx: x is an even positive integer

Ox: x is an odd positive integerOx: x is an odd positive integer


For quantified sentences of For quantified sentences of PLPL::((x) (Wx x) (Wx Mx) Mx)

Here the main logical operator is a universal Here the main logical operator is a universal quantifier.quantifier.

So the sentence is true on some interpretation if and So the sentence is true on some interpretation if and only if it is true that all x’s that are W’s are M’s.only if it is true that all x’s that are W’s are M’s.

9. UD: the set of living things9. UD: the set of living thingsWx: x is a whaleWx: x is a whaleMx: x is a mammal.Mx: x is a mammal.((x) (Wx x) (Wx Mx) is true on this interpretation. Mx) is true on this interpretation.


Quantified sentences of Quantified sentences of PLPL::((x) (Wx x) (Wx Mx) Mx)

Here the main logical operator is a universal Here the main logical operator is a universal quantifier.quantifier.

10. UD: the set of all things10. UD: the set of all thingsWx: x has a brain.Wx: x has a brain.Mx: x is a carMx: x is a carThe sentence is false on this interpretation.The sentence is false on this interpretation.


Quantified sentences of Quantified sentences of PLPL::((x) (Ex & Ox)x) (Ex & Ox)

Here the main logical operator is an existential Here the main logical operator is an existential quantifier and the sentence is true on an quantifier and the sentence is true on an interpretation if there is at least one thing that it is interpretation if there is at least one thing that it is a E and an O.a E and an O.

11. UD: the set of positive integers11. UD: the set of positive integers Ex: x is evenEx: x is even

Ox: x is oddOx: x is odd ((x) (Ex & Ox) is false on this interpretation.x) (Ex & Ox) is false on this interpretation.


Quantified sentences of Quantified sentences of PLPL::((x) (x) (y) Lxyy) Lxy12. UD: the set of all persons12. UD: the set of all persons Lxy: x likes yLxy: x likes y((x) (x) (y) Lxy is true on interpretation 12 if and only if for y) Lxy is true on interpretation 12 if and only if for

each person there is someone that person likes. (Everyone each person there is someone that person likes. (Everyone likes someone.) likes someone.)

It could just be him or herself.It could just be him or herself.13. UD: the sent of all persons13. UD: the sent of all persons

Lxy: x knows yLxy: x knows yIt is likely that (It is likely that (x) (x) (y) Lxy is true on interpretation 13.y) Lxy is true on interpretation 13.


Quantified sentences of Quantified sentences of PLPL::((x) (x) (y) Lxyy) Lxy13. UD: the set of all persons13. UD: the set of all persons Lxy: x likes yLxy: x likes y((x) (x) (y) Lxy is true on interpretation 13 if and only each y) Lxy is true on interpretation 13 if and only each

person likes every other person.person likes every other person.Unlikely! Unlikely! 14. UD: the set of all persons14. UD: the set of all persons

Lxy: x knows yLxy: x knows yIt is also unlikely that (It is also unlikely that (x) (x) (y) Lxy is true on interpretation y) Lxy is true on interpretation

14.14.


Summary of informal semantics of Summary of informal semantics of PLPL::The truth conditions for sentences of The truth conditions for sentences of PLPL are are

determined by determined by interpretationsinterpretations..An interpretation consists of the specification of a An interpretation consists of the specification of a

UD, and the interpretation of each sentence letter, UD, and the interpretation of each sentence letter, predicate, and individual constant relative to the predicate, and individual constant relative to the UD designated. (Individual variables are not UD designated. (Individual variables are not interpreted.)interpreted.)

Homework: 7.8E Exercise 1 and more to be provided Homework: 7.8E Exercise 1 and more to be provided for semantics.for semantics.

March 5: more on quantifiers We have encountered formulas and sentences that include multiple...

Documents

Transcript of March 5: more on quantifiers We have encountered formulas and sentences that include multiple...