Review Test 5Review Test 5You need to know:You need to know:

How to symbolize sentences that include How to symbolize sentences that include quantifiers of overlapping scopequantifiers of overlapping scopeDefinitions:Definitions:Quantificational truth, falsity and Quantificational truth, falsity and indeterminacyindeterminacyQuantificational equivalenceQuantificational equivalenceQuantificational validityQuantificational validityQuantificational consistencyQuantificational consistencyQuantificational entailmentQuantificational entailment

Review Test 5Review Test 5How to symbolize sentences that include How to symbolize sentences that include

quantifiers of overlapping scopequantifiers of overlapping scopeWe have limited the number of quantifiers with We have limited the number of quantifiers with

overlapping scope that you need to know overlapping scope that you need to know how to symbolize 2 for any given sentence:how to symbolize 2 for any given sentence:

((x) (x) (y): For each x and for each y (or for y): For each x and for each y (or for every pair x and y)every pair x and y)

((x) (x) (y): For each x there is some y such thaty): For each x there is some y such that((w) (w) (z): There is some w such that for every z): There is some w such that for every

z…z…((z) (z) (x): There is some z such that for some x x): There is some z such that for some x

(or there is some pair z and x such that…)(or there is some pair z and x such that…)

Review Test 5Review Test 5How to symbolize sentences that include How to symbolize sentences that include

quantifiers of overlapping scopequantifiers of overlapping scope((x) (x) (y): For each x and for each y (or for y): For each x and for each y (or for

every pair x and y)every pair x and y)

1. UD: the set of positive integers1. UD: the set of positive integers Dxy: x is equal to, or smaller than, or larger Dxy: x is equal to, or smaller than, or larger

than y.than y.Symbolize:Symbolize:For every positive integer x, every positive For every positive integer x, every positive

integer y is such that x is equal to, or smaller integer y is such that x is equal to, or smaller than, or larger than ythan, or larger than y

((x) (x) (y) Dxyy) Dxy

Review Test 5Review Test 5((x) (x) (y): For each x there is some y such thaty): For each x there is some y such thatFor each positive integer, there is some For each positive integer, there is some

positive integer that is larger than it.positive integer that is larger than it.2. UD: the set of positive integers2. UD: the set of positive integersLxy: x is larger than yLxy: x is larger than y((x) (x) (y) Lyxy) LyxChange the UD and predicates to:Change the UD and predicates to:3. UD: everything3. UD: everything Px: x is a positive integerPx: x is a positive integer

Lxy: x is larger than yLxy: x is larger than y((x) [Px x) [Px ( (y) (Py & Lyx)] ory) (Py & Lyx)] or((x) (x) (y) [(Px & Py) y) [(Px & Py) Lyx] Lyx]

((w) (w) (z): There is some w such that for every z): There is some w such that for every z…z…

Some positive integer w is such that for every Some positive integer w is such that for every positive integer z, w is equal to or smaller positive integer z, w is equal to or smaller than z. than z.

UD: the set of positive integersUD: the set of positive integersTxy: x is equal to or smaller than yTxy: x is equal to or smaller than y((w) (w) (z) Twz z) Twz Change the UD and predicates:Change the UD and predicates:UD: everythingUD: everythingPx: x is a positive integerPx: x is a positive integerTxy: x is equal to or smaller than yTxy: x is equal to or smaller than y((w) (w) (z) [(Pw & Pz) z) [(Pw & Pz) Twz] orTwz] or((w) [Px & (w) [Px & (z) (Pz z) (Pz Twz)]Twz)]

Review Test 5Review Test 5How to symbolize sentences that include How to symbolize sentences that include

quantifiers of overlapping scopequantifiers of overlapping scope((z) (z) (x): There is some z such that for some x x): There is some z such that for some x

(or there is some pair z and x such that…)(or there is some pair z and x such that…)The sum of some positive integers x and y is 4.The sum of some positive integers x and y is 4.

UD: the set of positive integersUD: the set of positive integersExy: the sum of x and y is 4Exy: the sum of x and y is 4

((z) (z) (x) Ezyx) Ezy

Review Test 5Review Test 5Pop quiz!Pop quiz!Symbolize the following sentences in Symbolize the following sentences in PLPL using the using the

following interpretation:following interpretation:UD: the set of all thingsUD: the set of all thingsPx: x is a professorPx: x is a professorSxy: x is a student of ySxy: x is a student of yBxy: x bores yBxy: x bores yWx: x is wasting his or her timeWx: x is wasting his or her timeAny student who is bored by all of his or her professors is Any student who is bored by all of his or her professors is

wasting her or time.wasting her or time.If a professor bores all of his or her students, then the If a professor bores all of his or her students, then the

professor is wasting his or her time.professor is wasting his or her time.

Review Test 5Review Test 5You need to know: Definitions:You need to know: Definitions:

Quantificational truth, falsity and Quantificational truth, falsity and indeterminacy … and so forthindeterminacy … and so forth

The basic semantic notion in predicate logic is The basic semantic notion in predicate logic is an an interpretationinterpretation, and all of quantificational , and all of quantificational definitions are in terms of one or more definitions are in terms of one or more interpretations.interpretations.

What an interpretation is:What an interpretation is:

It includes a UD which is a nonempty set (it has It includes a UD which is a nonempty set (it has at least one member)at least one member)

An interpretation of every predicate of An interpretation of every predicate of PLPL

An interpretation of every individual constant of An interpretation of every individual constant of PLPL

As there are an infinite number of interpretations of As there are an infinite number of interpretations of any of the infinite number of predicates of any of the infinite number of predicates of PLPL and an infinite number of interpretations of the and an infinite number of interpretations of the infinite number of individual constants of infinite number of individual constants of PLPL

And an infinite number of UD’sAnd an infinite number of UD’s

So, So, PLPL includes an infinite number of includes an infinite number of interpretationsinterpretations

To construct an interpretation so as to To construct an interpretation so as to demonstrate that some quantificational notion demonstrate that some quantificational notion holds or does not (and you holds or does not (and you cannotcannot use this use this method to prove all claims but only some!), you method to prove all claims but only some!), you need to specify:need to specify:

A UD: a nonempty set (the domain over which A UD: a nonempty set (the domain over which predicates and variables range, and members of predicates and variables range, and members of which individual constants refer to or denote)which individual constants refer to or denote)

An interpretation of each (relevant) predicate that An interpretation of each (relevant) predicate that helps you to demonstrate that a quantificational helps you to demonstrate that a quantificational notion does or does not hold (except in terms of notion does or does not hold (except in terms of equivalence when you need 2)equivalence when you need 2)

An interpretation of any (relevant) individual An interpretation of any (relevant) individual constants.constants.

Review Test 5Review Test 5You need to be able to:You need to be able to:

Identify Identify an interpretationan interpretation that shows that a that shows that a sentence is sentence is notnot quantificationally true quantificationally trueIdentify an interpretation that shows that a Identify an interpretation that shows that a sentence is sentence is notnot quantificationally false quantificationally falseIdentify an interpretation that shows that a set Identify an interpretation that shows that a set of sentences of sentences isis quantificationally consistent quantificationally consistentIdentify an interpretation that shows that 2 Identify an interpretation that shows that 2 sentences are not quantificationally sentences are not quantificationally equivalentequivalentIdentify Identify 2 interpretations2 interpretations that show that a that show that a sentence sentence isis quantificationally indeterminate quantificationally indeterminate

Cases in which identifying one interpretation or two Cases in which identifying one interpretation or two won’t do the work you need:won’t do the work you need:

If told to show that a sentence is quantificationally If told to show that a sentence is quantificationally true, provide the reasoning that demonstrates this true, provide the reasoning that demonstrates this (no one or more interpretations can show this)(no one or more interpretations can show this)

If told to show that a sentence is quantificationally If told to show that a sentence is quantificationally false, provide the reasoning that demonstrates false, provide the reasoning that demonstrates this (same as above)this (same as above)

If told an argument is quantificationally valid, If told an argument is quantificationally valid, provide the reasoning that demonstrates thisprovide the reasoning that demonstrates this

If told a set quantificationally entails some If told a set quantificationally entails some sentence, provide the reasoning that sentence, provide the reasoning that demonstrates thisdemonstrates this

In general:In general:

To disprove that some characteristic applies To disprove that some characteristic applies to all and any interpretations (when you are to all and any interpretations (when you are told it does not), identify an interpretation told it does not), identify an interpretation that shows thisthat shows this

For example, that For example, that PP is not quantificationally is not quantificationally true ortrue or

That some set is not quantificationally That some set is not quantificationally consistent orconsistent or

That some argument is not quantificationally That some argument is not quantificationally valid…valid…

Pop quiz 2!Pop quiz 2!

a. Can you show that a sentence is a. Can you show that a sentence is quantificationally true by identifying an quantificationally true by identifying an interpretation on which it is true?interpretation on which it is true?

b. Can you show that a set is b. Can you show that a set is quantificationally consistent by citing an quantificationally consistent by citing an interpretation?interpretation?

c. Do you need one or more interpretations, c. Do you need one or more interpretations, or must you use reasoning, to show thator must you use reasoning, to show that

((x) (x) (y) Syx is quantificationally y) Syx is quantificationally indeterminate?indeterminate?

2 different kinds of question and, so, 2 2 different kinds of question and, so, 2 different kinds of proof:different kinds of proof:

a. Show that the sentence ‘(a. Show that the sentence ‘(y) (y) (x) Gyx’ is x) Gyx’ is not quantificationally false.not quantificationally false.

Try an interpretation with a UD of the set of Try an interpretation with a UD of the set of positive integerspositive integers

Interpret Gxy so that the sentence is Interpret Gxy so that the sentence is truetrue on on that interpretation. And you will have that interpretation. And you will have shown that the sentence is shown that the sentence is notnot quantificationally false. Example:quantificationally false. Example:Gxy: x is greater than yGxy: x is greater than yGxy: x multiplied by y is even (or odd…)Gxy: x multiplied by y is even (or odd…)

2 different kinds of question and, so, 2 different 2 different kinds of question and, so, 2 different kinds of proof:kinds of proof:

b. Show that the sentence (b. Show that the sentence (y) (Ay & ~Ay) is y) (Ay & ~Ay) is quantificationally false.quantificationally false.

As we cannot demonstrate that the sentence is As we cannot demonstrate that the sentence is false on every possible interpretation, we use false on every possible interpretation, we use reasoning to show that reasoning to show that whateverwhatever the UD, and the UD, and howeverhowever A is interpreted, the sentence will A is interpreted, the sentence will always be false – hence, that it is always be false – hence, that it is quantificationally false.quantificationally false.

This means showing that for any y, ‘Ay & ~Ay’ is This means showing that for any y, ‘Ay & ~Ay’ is always false. This formula is truth functional, always false. This formula is truth functional, and to be true it requires that both conjuncts are and to be true it requires that both conjuncts are true. But there is no interpretation of A on which true. But there is no interpretation of A on which some y can be both A and ~A. If Ay is true, ~Ay some y can be both A and ~A. If Ay is true, ~Ay is false, and vice versa. So the sentence (is false, and vice versa. So the sentence (y) y) (Ay & ~Ay) is quantificationally false.(Ay & ~Ay) is quantificationally false.

2 different kinds of question and, so, 2 2 different kinds of question and, so, 2 different kinds of proof:different kinds of proof:

a. Show that the following argument is a. Show that the following argument is notnot quantificationally valid:quantificationally valid:

((x) (Ax x) (Ax Bx) Bx)~ (~ (x) Axx) Ax----------------------------------------~ (~ (x) Bxx) BxThis means we need to identify an This means we need to identify an

interpretation (interpretation (just onejust one) on which each of ) on which each of the premises is true and the conclusion is the premises is true and the conclusion is false.false.

((x) (Ax x) (Ax Bx) Bx)~ (~ (x) Axx) Ax----------------------------------------~ (~ (x) Bxx) BxIdentify a UD and an interpretation of Ax and Bx so that Identify a UD and an interpretation of Ax and Bx so that

the premises are true but the conclusion is false. the premises are true but the conclusion is false. In this case, interpret A first (because if the 2In this case, interpret A first (because if the 2ndnd premise premise

is true, the first one will be as well) using a is true, the first one will be as well) using a predicate that has no extension; then interpret B as predicate that has no extension; then interpret B as a predicate that does. a predicate that does.

UD: the set of all thingsUD: the set of all thingsAx: x is a unicornAx: x is a unicornBx: x is a mammalBx: x is a mammalAs there are no unicorns, the premises are true; but as As there are no unicorns, the premises are true; but as

there are mammals the conclusion is false.there are mammals the conclusion is false.

b. Show that the following argument b. Show that the following argument isis quantificationally valid:quantificationally valid:

~(~(x) (Px x) (Px Ex) Ex)

--------------------------------------------

((x) (Px & ~Ex)x) (Px & ~Ex)

Reason this way: Part 1: If the premise is Reason this way: Part 1: If the premise is true, then it is not the case that each thing true, then it is not the case that each thing in the domain is such that if it is P, it is E.in the domain is such that if it is P, it is E.

So the conclusion follows: there is something So the conclusion follows: there is something in the domain that is P and is not E.in the domain that is P and is not E.

b. Show that the following argument b. Show that the following argument isis quantificationally valid:quantificationally valid:

~(~(x) (Px x) (Px Ex) Ex)

--------------------------------------------

((x) (Px & ~Ex)x) (Px & ~Ex)

Part 2: If the premise is false , then the argument is Part 2: If the premise is false , then the argument is also valid.also valid.

As the premise must be true or false, and if it’s true 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 so is the conclusion, and if it’s false then it is not possible for the premise to be true and the possible for the premise to be true and the conclusion false, the argument is valid and conclusion false, the argument is valid and because we assumed no particular because we assumed no particular interpretation, it is quantificationally valid.interpretation, it is quantificationally valid.

Show that the following sentences are Show that the following sentences are notnot quantificationally equivalent.quantificationally equivalent.

~(~(y) Byy) By((y) ~Byy) ~ByHere we can use interpretations, but we need Here we can use interpretations, but we need

two. We need to identify an interpretation two. We need to identify an interpretation on which one is true and an interpretation on which one is true and an interpretation on which the other is false.on which the other is false.

So consider some predicate that doesn’t So consider some predicate that doesn’t apply to everything, but does apply to apply to everything, but does apply to some things; and choose a UD some things; and choose a UD accordingly.accordingly.

~(~(y) Byy) By((y) ~Byy) ~By1. UD: the set of living things1. UD: the set of living things By: y is a mammalBy: y is a mammalOn this interpretation, sentence one is true and On this interpretation, sentence one is true and

sentence two is false.sentence two is false.So the sentences are not quantificationally So the sentences are not quantificationally

equivalent. Another interpretation to show this:equivalent. Another interpretation to show this:2. UD: the set of positive integers2. UD: the set of positive integers By: y is even.By: y is even.On this interpretation, sentence one is true and On this interpretation, sentence one is true and

sentence two is false.sentence two is false.So the sentences are not quantificationally So the sentences are not quantificationally

equivalent.equivalent.

Show that the following sentences are Show that the following sentences are quantificationally equivalent.quantificationally equivalent.

((x) (Wx x) (Wx Mx) Mx)~(~(x) (Wx & ~Mx)x) (Wx & ~Mx)Again, as this is a claim that covers an infinite Again, as this is a claim that covers an infinite

number of interpretations, we have to number of interpretations, we have to demonstrate it by reasoning. demonstrate it by reasoning.

WhateverWhatever the UD, and the UD, and whateverwhatever the the interpretations of W and M, the first interpretations of W and M, the first sentence says that anything that is a W is sentence says that anything that is a W is an M. The second sentence says there is an M. The second sentence says there is nothing that is both a W and an M. nothing that is both a W and an M.

((x) (Wx x) (Wx Mx) Mx)

~(~(x) (Wx & ~Mx)x) (Wx & ~Mx)

Suppose the first sentence is true on some Suppose the first sentence is true on some interpretation. Then every member of the interpretation. Then every member of the UD which is W is also M. So no member is UD which is W is also M. So no member is both W and ~M, so the second sentence is both W and ~M, so the second sentence is true. true.

Suppose that the first sentence is false on Suppose that the first sentence is false on some interpretation. Then some member some interpretation. Then some member of the UD is W but not M. So the second of the UD is W but not M. So the second sentence is also false (because (sentence is also false (because (x) Wx & x) Wx & ~Mx) is true on that interpretation).~Mx) is true on that interpretation).

Finally, consider quantificational consistency.Finally, consider quantificational consistency.

Here, unlike some earlier cases, to show that Here, unlike some earlier cases, to show that some set of sentences is quantificationally some set of sentences is quantificationally consistent, we need only identify consistent, we need only identify one one interpretationinterpretation on which all the members of on which all the members of the set are true.the set are true.

But to show that some set is quantificationally But to show that some set is quantificationally inconsistent, we need to use reasoning to inconsistent, we need to use reasoning to show that it is not possible for all the show that it is not possible for all the members of the set to be true on members of the set to be true on anyany interpretation.interpretation.

Quantificational consistency.Quantificational consistency.

a. Show that the following set is a. Show that the following set is quantificationally consistent:quantificationally consistent:

{({(y) (Ey y) (Ey ~Oy), ( ~Oy), (x) (Ex & Dx), (x) (Ex & Dx), (w) (Ow & w) (Ow & ~Dw)}~Dw)}

So we need an interpretation on which each So we need an interpretation on which each sentence is true. The existentially sentence is true. The existentially quantified sentences might be the best to quantified sentences might be the best to begin with (an interpretation on which both begin with (an interpretation on which both are true and an appropriate UD.are true and an appropriate UD.

{({(y) (Ey y) (Ey ~Oy), ( ~Oy), (x) (Ex & Dx), (x) (Ex & Dx), (w) (Ow & ~Dw)}w) (Ow & ~Dw)}

Again, it is often useful to try a UD of positive Again, it is often useful to try a UD of positive integers.integers.

UD: the set of positive integersUD: the set of positive integers

Ex: x is evenEx: x is even

Dx: x is evenly divisible by 2Dx: x is evenly divisible by 2

Ox: x is oddOx: x is odd

The two existentially quantified sentences are true.The two existentially quantified sentences are true.

And so it turns out is the universally quantified And so it turns out is the universally quantified sentence.sentence.

So we’ve shown that the set is quantificationally So we’ve shown that the set is quantificationally consistent.consistent.

Suppose we’re asked to show that the Suppose we’re asked to show that the following set is following set is notnot quantificationally quantificationally consistent. consistent.

{~ ({~ (x) (x) (y) Gxy, (y) Gxy, (w) (w) (z) Gzw} z) Gzw}

We cannot check every interpretation to We cannot check every interpretation to demonstrate that there is none on which demonstrate that there is none on which all the members of the set are true. So we all the members of the set are true. So we need to use reasoning.need to use reasoning.

For whatever UD, and whatever interpretation For whatever UD, and whatever interpretation of G, the first sentence says that for any of G, the first sentence says that for any pair x and y, it is not the case that x bears pair x and y, it is not the case that x bears the relationship G to y.the relationship G to y.

{~ ({~ (x) (x) (y) Gxy, (y) Gxy, (w) (w) (z) Gzw} z) Gzw}

If this is true, then the second sentence is If this is true, then the second sentence is false for it says that there is some z that false for it says that there is some z that bears the relationship G to w.bears the relationship G to w.

And if the first sentence is false, then the And if the first sentence is false, then the second sentence is true.second sentence is true.

So there is no interpretation on which both So there is no interpretation on which both sentences can be true and the set is sentences can be true and the set is quantificationally inconsistent.quantificationally inconsistent.