March 5: more on quantifiers We have encountered formulas and sentences that include multiple quantifiers: Take, for example: UD: People in Michael’s office Lxy: x likes y m: Michael ‘There is someone that Michael likes and everyone likes Michael” (  x) Lmx & (  y) Lym

March 5: more on quantifiers But there are also sentences that include multiple quantifiers of overlapping scope: UD: People in Michael’s office Lxy: x likes y m: Michael So we’ve symbolized: ‘Everyone likes Michael’ (  x) Lxm Now suppose we want to symbolize ‘Everyone likes someone’

Multiple quantifiers with overlapping scope And suppose our UD is now ‘all people’ UD: all people Lxy: x likes y ‘Everyone likes someone’ (  x) there is someone such that… (  x) (  y) such that… (  x) (  y) Lxy

Multiple quantifiers with overlapping scope Now suppose: UD: all people Lxy: x likes y And we want to symbolize: ‘Everyone likes everyone’ (  x) is such that for everyone… and we add (  y) for another universal quantifier so that we have: (  x) is such that for all y (  y) (  x) (  y) x likes y (  x) (  y) Lxy

Multiple quantifiers with overlapping scope Suppose we keep: UD: all people Lxy: x likes y And now we want to symbolize: ‘Someone likes everyone’ (  x) such that for everyone y, x… We add ‘(  y)’ for that everyone, and then: (  x) (  y) Lxy

Multiple quantifiers with overlapping scope Suppose we keep: UD: all people Lxy: x likes y And we want to symbolize ‘Someone likes someone’ (  x) such that there is someone… We add (  y) for ‘there is some y’… and get (  x) (  y) Lxy

Multiple quantifiers with overlapping scope Pairs of quantifiers* can occur in 4 combinations: (  x) (  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) (  y) For each x and for each y… [or] For each pair x and y … pair x and y … (  x) (  y) For each x there is a y such that … (  x) (  y) There is an x such that for each y … *Although we won’t deal with them, there can be more than 2 (or more than a pair) of quantifiers of overlapping scope.

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

Multiple quantifiers with overlapping scope: the difference a UD makes for symbolization 1. UD: persons Lxy: x likes y ‘Everyone likes everyone’ (  x) (  y) Lxy ‘Someone likes someone’ (  x) (  y) Lxy 2. UD: Everything Lxy: x likes y Px: x is a person ‘Everyone likes everyone’ (  x) (  y) [(Px & Py)  Lxy] ‘Someone likes someone’ (  x) (  y) [(Px & Py) & Lxy]

Multiple quantifiers with overlapping scope One has to “learn to read” the sentences of PL into “quasi-English” to check out a symbolization. In doing so, it is crucial to identify the role of each logical operator. For example, in (  x) (  y) [(Px & Py)  Lxy] (  x) is the main logical operator and (  y) is the (  x) is the main logical operator and (  y) is the main logical operator of the subformula: (Px & Py)  Lxy So we read (  x) first and (  y) second. ‘Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’]

Multiple quantifiers with overlapping scope We read (  x) first and (  y) second. ‘Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’] As the main logical operator of the next subformula, (Px & Py)  Lxy, is the  we move to that next: ‘Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’] IF ‘Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’] IF

Multiple quantifiers with overlapping scope (  x) (  y) [(Px & Py)  Lxy] ‘Every x is such that every y is such that’ [or ‘Every pair x and y is such that…’] IF ‘ x is a person and y is a person’ THEN Lxy (x likes y) All together: ‘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] ‘For every pair x and y, if x is a person and y is a person, then x likes y’

(Informal) Semantics of PL In SL, the semantic notion we used to determine truth status was the truth value assignment. This worked because the sentences of SL are such that their truth status is a function of truth functional assignments to their atomic components (atomic sentences) and, in compound sentences, the TVA’s of atomic sentences and truth-functional connectives. This doesn’t work in PL, except for those sentences that are just the atomic sentences (A, B, C…) of SL. The basic semantic concept of PL is an interpretation.

(Informal) Semantics of PL In PL, an interpretation interprets: each individual constant (if any) each predicate each sentence letter of PL All of these are relative to some universe of discourse We can take symbolization keys as we have encountered them so far as embodying interpretations with the following caveat: In an interpretation, a UD is always some non-empty set, and so we classify them as sets when specifying an interpretation

(Informal) Semantics of PL There are 2 kinds of atomic sentences of PL: Sentence letters (A through Z with or without subscripts: the semantics involve truth functional assignments) A n-place predicate of PL followed by n-constants: What are its semantics? Start with a one-place predicate, Fx, and the atomic sentence: Fa

(Informal) Semantics of PL The truth or falsity of ‘Fa’ depends on an interpretation that interprets: the predicate Fx the constant a and a UD: a set of objects over which the predicates and variables range, and from which constants pick out objects

(Informal) Semantics of PL Atomic sentences of PL: Consider Fa on interpretation 1: 1. UD: set of living things Fx: x is a human a: is Socrates (the historical figure) On this interpretation, ‘Fa’ is true

(Informal) Semantics of PL Change the UD and/or the predicate and/or the constant, and the truth status of sentence on the interpretation also changes. Consider ‘Fa’ on interpretation 2: 2. UD: set of living things Fx: x is a potato Fx: x is a potato a: Socrates Assuming, again, that ‘a’ denotes the historical person, Socrates, the sentence ‘Fa’ is false on this interpretation.

(Informal) Semantics of PL Atomic sentences of PL involving 2-place predicates: Lxy: x is larger than y And say the UD is “the set of positive integers” A two place predicate is a relational predicate: it picks out sets of pairs of members of the UD whose order often matters – and in the case of this predicate, order does matter.

(Informal) Semantics of PL 3. UD: the set of positive integers Lxy: x is larger than y a: 1 b: 2 Lba is true on interpretation 3. The extension of Lxy on interpretation 3 are those pairs (4, 3; 5, 2; 201, 200; 4000, 3999; and so forth) of which it is true that the first member of the pair is larger than the second member of the pair. So, the pair 2, 3 is not a pair that is an extension of Lxy on interpretation 3.

4. The set of all buildings and all people Lxy: x is larger than y a: The Empire State Building b: George Bush Lba is false on interpretation 4. An interpretation may assign the same member of the UD to more than one constant: 5. The set of planets in our solar system Cxy: x is closer to the sun than y. m: Venus n: Venus Cmn is false on interpretation 5.

(Informal) Semantics of PL Some interpretations of 2-place predicates mean that the extension of a predicate includes pairs in which the 1 st and 2 nd members are the same: 6. UD: the set of positive integers Exy: x is less than or equal to y The extension of the predicate Exy (as defined above) includes not only 2, 3 but also 2, 2; 4, 4; and so forth.

(Informal) Semantics of PL Compound sentences of PL that do not include quantifiers: Cab v Cba As the main logical operator is ‘v’, this sentence is true or false on an interpretation depending on whether at least either ‘Cab’ or ‘Cba’ is true on that interpretation. We use the characteristic truth tables for the connectives to determine whether a truth functional sentence of PL is true on some interpretation. 7. UD: the set of persons Cxy: x likes y a: Andrea b: Bruce

(Informal) Semantics of PL Compound sentences of PL whose main logical operator is not a quantifier: (  x) (Wx  Mx) & ~(  x) (Ex & Ox) As the main logical operator is ‘&’, this sentence is true or false on an interpretation if and only if both the right and left conjuncts are true on that interpretation. 8. UD: the set of all things Wx: x is a whale Wx: x is a whale Mx: x is a mammal Ex: x is an even positive integer Ox: x is an odd positive integer Ox: x is an odd positive integer

(Informal) Semantics of PL For quantified sentences of PL: (  x) (Wx  Mx) Here the main logical operator is a universal quantifier. 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. 9. UD: the set of living things Wx: x is a whale Mx: x is a mammal. (  x) (Wx  Mx) is true on this interpretation.

(Informal) Semantics of PL Quantified sentences of PL: (  x) (Wx  Mx) Here the main logical operator is a universal quantifier. 10. UD: the set of all things Wx: x has a brain. Mx: x is a car The sentence is false on this interpretation.

(Informal) Semantics of PL Quantified sentences of PL: (  x) (Ex & Ox) Here the main logical operator is an existential quantifier and the sentence is true on an interpretation if there is at least one thing that it is a E and an O. 11. UD: the set of positive integers Ex: x is even Ex: x is even Ox: x is odd Ox: x is odd (  x) (Ex & Ox) is false on this interpretation. (  x) (Ex & Ox) is false on this interpretation.

(Informal) Semantics of PL Quantified sentences of PL: (  x) (  y) Lxy 12. UD: the set of all persons Lxy: x likes y Lxy: x likes y (  x) (  y) Lxy is true on interpretation 12 if and only if for each person there is someone that person likes. (Everyone likes someone.) It could just be him or herself. 13. UD: the sent of all persons Lxy: x knows y Lxy: x knows y It is likely that (  x) (  y) Lxy is true on interpretation 13.

(Informal) Semantics of PL Quantified sentences of PL: (  x) (  y) Lxy 13. UD: the set of all persons Lxy: x likes y Lxy: x likes y (  x) (  y) Lxy is true on interpretation 13 if and only each person likes every other person. Unlikely! 14. UD: the set of all persons Lxy: x knows y Lxy: x knows y It is also unlikely that (  x) (  y) Lxy is true on interpretation 14.

(Informal) Semantics of PL Summary of informal semantics of PL: The truth conditions for sentences of PL are determined by interpretations. An interpretation consists of the specification of a UD, and the interpretation of each sentence letter, predicate, and individual constant relative to the UD designated. (Individual variables are not interpreted.) Homework: 7.8E Exercise 1 and more to be provided for semantics.