Review Test 5

1. Review Test 5 You need to know: How to symbolize sentences that include quantifiers of overlapping scope Definitions: Quantificational truth, falsity and indeterminacy Quantificational equivalence Quantificational validity Quantificational consistency Quantificational entailment

2. Review Test 5 How to symbolize sentences that include quantifiers of overlapping scope We have limited the number of quantifiers with overlapping scope that you need to know how to symbolize 2 for any given sentence: (x) (y): For each x and for each y (or for every pair x and y) (x) (y): For each x there is some y such that (w) (z): There is some w such that for every z… (z) (x): There is some z such that for some x (or there is some pair z and x such that…)

3. Review Test 5 How to symbolize sentences that include quantifiers of overlapping scope (x) (y): For each x and for each y (or for every pair x and y) 1. UD: the set of positive integers Dxy: x is equal to, or smaller than, or larger than y. Symbolize: For every positive integer x, every positive integer y is such that x is equal to, or smaller than, or larger than y (x) (y) Dxy

4. Review Test 5 (x) (y): For each x there is some y such that For each positive integer, there is some positive integer that is larger than it. 2. UD: the set of positive integers Lxy: x is larger than y (x) (y) Lyx Change the UD and predicates to: 3. UD: everything Px: x is a positive integer Lxy: x is larger than y (x) [Px  (y) (Py & Lyx)] or (x) (y) [(Px & Py)  Lyx]

5. (w) (z): There is some w such that for every z… Some positive integer w is such that for every positive integer z, w is equal to or smaller than z. UD: the set of positive integers Txy: x is equal to or smaller than y (w) (z) Twz Change the UD and predicates: UD: everything Px: x is a positive integer Txy: x is equal to or smaller than y (w) (z) [(Pw & Pz) Twz] or (w) [Px & (z) (Pz Twz)]

6. Review Test 5 How to symbolize sentences that include quantifiers of overlapping scope (z) (x): There is some z such that for some x (or there is some pair z and x such that…) The sum of some positive integers x and y is 4. UD: the set of positive integers Exy: the sum of x and y is 4 (z) (x) Ezy

7. Review Test 5 Pop quiz! Symbolize the following sentences in PL using the following interpretation: UD: the set of all things Px: x is a professor Sxy: x is a student of y Bxy: x bores y Wx: x is wasting his or her time Any student who is bored by all of his or her professors is wasting her or time. If a professor bores all of his or her students, then the professor is wasting his or her time.

8. Review Test 5 You need to know: Definitions: Quantificational truth, falsity and indeterminacy … and so forth The basic semantic notion in predicate logic is an interpretation, and all of quantificational definitions are in terms of one or more interpretations.

9. What an interpretation is: It includes a UD which is a nonempty set (it has at least one member) An interpretation of every predicate of PL An interpretation of every individual constant of PL As there are an infinite number of interpretations of any of the infinite number of predicates of PL and an infinite number of interpretations of the infinite number of individual constants of PL And an infinite number of UD’s So, PL includes an infinite number of interpretations

10. To construct an interpretation so as to demonstrate that some quantificational notion holds or does not (and you cannot use this method to prove all claims but only some!), you need to specify: A UD: a nonempty set (the domain over which predicates and variables range, and members of which individual constants refer to or denote) An interpretation of each (relevant) predicate that helps you to demonstrate that a quantificational notion does or does not hold (except in terms of equivalence when you need 2) An interpretation of any (relevant) individual constants.

11. Review Test 5 You need to be able to: Identify an interpretation that shows that a sentence is not quantificationally true Identify an interpretation that shows that a sentence is not quantificationally false Identify an interpretation that shows that a set of sentences is quantificationally consistent Identify an interpretation that shows that 2 sentences are not quantificationally equivalent Identify 2 interpretations that show that a sentence is quantificationally indeterminate

12. Cases in which identifying one interpretation or two won’t do the work you need: If told to show that a sentence is quantificationally true, provide the reasoning that demonstrates this (no one or more interpretations can show this) If told to show that a sentence is quantificationally false, provide the reasoning that demonstrates this (same as above) If told an argument is quantificationally valid, provide the reasoning that demonstrates this If told a set quantificationally entails some sentence, provide the reasoning that demonstrates this

13. In general: To disprove that some characteristic applies to all and any interpretations (when you are told it does not), identify an interpretation that shows this For example, that P is not quantificationally true or That some set is not quantificationally consistent or That some argument is not quantificationally valid…

14. Pop quiz 2! a. Can you show that a sentence is quantificationally true by identifying an interpretation on which it is true? b. Can you show that a set is quantificationally consistent by citing an interpretation? c. Do you need one or more interpretations, or must you use reasoning, to show that (x) (y) Syx is quantificationally indeterminate?

15. 2 different kinds of question and, so, 2 different kinds of proof: a. Show that the sentence ‘(y) (x) Gyx’ is not quantificationally false. Try an interpretation with a UD of the set of positive integers Interpret Gxy so that the sentence is true on that interpretation. And you will have shown that the sentence is not quantificationally false. Example: Gxy: x is greater than y Gxy: x multiplied by y is even (or odd…)

16. 2 different kinds of question and, so, 2 different kinds of proof: b. Show that the sentence (y) (Ay & ~Ay) is quantificationally false. As we cannot demonstrate that the sentence is false on every possible interpretation, we use reasoning to show that whatever the UD, and however A is interpreted, the sentence will always be false – hence, that it is quantificationally false. This means showing that for any y, ‘Ay & ~Ay’ is always false. This formula is truth functional, and to be true it requires that both conjuncts are true. But there is no interpretation of A on which some y can be both A and ~A. If Ay is true, ~Ay is false, and vice versa. So the sentence (y) (Ay & ~Ay) is quantificationally false.

17. 2 different kinds of question and, so, 2 different kinds of proof: a. Show that the following argument is not quantificationally valid: (x) (Ax  Bx) ~ (x) Ax -------------------- ~ (x) Bx This means we need to identify an interpretation (just one) on which each of the premises is true and the conclusion is false.

18. (x) (Ax  Bx) ~ (x) Ax -------------------- ~ (x) Bx Identify a UD and an interpretation of Ax and Bx so that the premises are true but the conclusion is false. In this case, interpret A first (because if the 2nd premise is true, the first one will be as well) using a predicate that has no extension; then interpret B as a predicate that does. UD: the set of all things Ax: x is a unicorn Bx: x is a mammal As there are no unicorns, the premises are true; but as there are mammals the conclusion is false.

19. b. Show that the following argument is quantificationally valid: ~(x) (Px  Ex) ---------------------- (x) (Px & ~Ex) Reason this way: Part 1: If the premise is true, then it is not the case that each thing in the domain is such that if it is P, it is E. So the conclusion follows: there is something in the domain that is P and is not E.

20. b. Show that the following argument is quantificationally valid: ~(x) (Px  Ex) ---------------------- (x) (Px & ~Ex) Part 2: If the premise is false , then the argument is also valid. As the premise must be true or false, and if it’s true so is the conclusion, and if it’s false then it is not possible for the premise to be true and the conclusion false, the argument is valid and because we assumed no particular interpretation, it is quantificationally valid.

21. Show that the following sentences are not quantificationally equivalent. ~(y) By (y) ~By Here we can use interpretations, but we need two. We need to identify an interpretation on which one is true and an interpretation on which the other is false. So consider some predicate that doesn’t apply to everything, but does apply to some things; and choose a UD accordingly.

22. ~(y) By (y) ~By 1. UD: the set of living things By: y is a mammal On this interpretation, sentence one is true and sentence two is false. So the sentences are not quantificationally equivalent. Another interpretation to show this: 2. UD: the set of positive integers By: y is even. On this interpretation, sentence one is true and sentence two is false. So the sentences are not quantificationally equivalent.

23. Show that the following sentences are quantificationally equivalent. (x) (Wx  Mx) ~(x) (Wx & ~Mx) Again, as this is a claim that covers an infinite number of interpretations, we have to demonstrate it by reasoning. Whatever the UD, and whatever the interpretations of W and M, the first sentence says that anything that is a W is an M. The second sentence says there is nothing that is both a W and an M.

24. (x) (Wx  Mx) ~(x) (Wx & ~Mx) Suppose the first sentence is true on some interpretation. Then every member of the UD which is W is also M. So no member is both W and ~M, so the second sentence is true. Suppose that the first sentence is false on some interpretation. Then some member of the UD is W but not M. So the second sentence is also false (because (x) Wx & ~Mx) is true on that interpretation).

25. Finally, consider quantificational consistency. Here, unlike some earlier cases, to show that some set of sentences is quantificationally consistent, we need only identify one interpretation on which all the members of the set are true. But to show that some set is quantificationally inconsistent, we need to use reasoning to show that it is not possible for all the members of the set to be true on any interpretation.

26. Quantificational consistency. a. Show that the following set is quantificationally consistent: {(y) (Ey  ~Oy), (x) (Ex & Dx), (w) (Ow & ~Dw)} So we need an interpretation on which each sentence is true. The existentially quantified sentences might be the best to begin with (an interpretation on which both are true and an appropriate UD.

27. {(y) (Ey  ~Oy), (x) (Ex & Dx), (w) (Ow & ~Dw)} Again, it is often useful to try a UD of positive integers. UD: the set of positive integers Ex: x is even Dx: x is evenly divisible by 2 Ox: x is odd The two existentially quantified sentences are true. And so it turns out is the universally quantified sentence. So we’ve shown that the set is quantificationally consistent.

28. Suppose we’re asked to show that the following set is not quantificationally consistent. {~ (x) (y) Gxy, (w) (z) Gzw} We cannot check every interpretation to demonstrate that there is none on which all the members of the set are true. So we need to use reasoning. For whatever UD, and whatever interpretation of G, the first sentence says that for any pair x and y, it is not the case that x bears the relationship G to y.

29. {~ (x) (y) Gxy, (w) (z) Gzw} If this is true, then the second sentence is false for it says that there is some z that bears the relationship G to w. And if the first sentence is false, then the second sentence is true. So there is no interpretation on which both sentences can be true and the set is quantificationally inconsistent.