March 10: Quantificational Notions

Interpretations: Interpretations: EveryEvery interpretation interprets interpretation interprets everyevery individual constant, predicate, and sentence letter individual constant, predicate, and sentence letter of of PLPL. Our partial interpretation may not include . Our partial interpretation may not include some predicates or individual constants, but it some predicates or individual constants, but it does interpret them.does interpret them.Lx, Lxy, Lxyz, LwxyzLx, Lxy, Lxyz, Lwxyza, b, c, d, … aa, b, c, d, … a1, b2 …

Full interpretations are infinitely long as Full interpretations are infinitely long as PLPL includes includes an infinite # of predicates, an infinite # of an infinite # of predicates, an infinite # of individual constants, an infinite # of sentence individual constants, an infinite # of sentence letters and an infinite # of UDs. letters and an infinite # of UDs.

March 10: Quantificational NotionsWe use the concept of an interpretation to specify the We use the concept of an interpretation to specify the

quantificational counterparts of truth-functional quantificational counterparts of truth-functional concepts.concepts.

Individual sentences of Individual sentences of PL PL fall into 1 of 3 categories:fall into 1 of 3 categories:A sentence A sentence PP of of PLPL is is quantificationally truequantificationally true IFF IFF PP is true is true

on every interpretation. on every interpretation. A sentence A sentence P P of of PLPL is is quantificationally falsequantificationally false IFF IFF P P is false is false

on every interpretation.on every interpretation.A sentence A sentence P P is is quantificationally indeterminatequantificationally indeterminate IFF IFF PP is is

neither quantificationally true nor quantificationally neither quantificationally true nor quantificationally false.false.

March 10: Quantificational Notions

A sentence A sentence PP of of PLPL is is quantificationally truequantificationally true IFF IFF PP is is true on every interpretation. true on every interpretation.

How could we demonstrate that some sentence How could we demonstrate that some sentence PP is is quantificationally true given that we cannot check quantificationally true given that we cannot check all of the infinite interpretations of that sentence? all of the infinite interpretations of that sentence? We cannot do it with all sentences, but we can do We cannot do it with all sentences, but we can do it for some using reasoning.it for some using reasoning.

Consider for example:Consider for example:((y) (Cy v ~Cy)y) (Cy v ~Cy)

((y) (Cy v ~Cy)y) (Cy v ~Cy)

We may reason as follows:We may reason as follows:Because the sentence is existentially quantified, it is true on Because the sentence is existentially quantified, it is true on

an interpretation just in case at least one member of the an interpretation just in case at least one member of the UD satisfies the conditions specified by ‘Cy v ~Cy’ – that is, UD satisfies the conditions specified by ‘Cy v ~Cy’ – that is, if at least one member of the UD either if at least one member of the UD either isis C or C or is notis not C. C.

Without knowing what the interpretation of C is, we know Without knowing what the interpretation of C is, we know that every member of a UD satisfies this condition – for that every member of a UD satisfies this condition – for every interpretation interprets C, and every member of every interpretation interprets C, and every member of the UD is or is not in the extension of C.the UD is or is not in the extension of C.

And because, by definition, every interpretation has a And because, by definition, every interpretation has a nonempty set as its UD, then the UD for any nonempty set as its UD, then the UD for any interpretation has at least one member and hence at interpretation has at least one member and hence at least one member that satisfies the open sentence. So the least one member that satisfies the open sentence. So the sentence is quanticationally true.sentence is quanticationally true.

In general, to show that some sentence In general, to show that some sentence PP is is quantificationally true, we may use reasoning to quantificationally true, we may use reasoning to show that no matter what the UD is and no matter show that no matter what the UD is and no matter how the individual constants, predicates, and how the individual constants, predicates, and sentence letters are interpreted, the sentence sentence letters are interpreted, the sentence always turns out to be true. So consider:always turns out to be true. So consider:((x) (x) (y) Bxy y) Bxy ( (y) ( y) ( x) Bxyx) Bxy

Whatever the UD and however B is interpreted, we Whatever the UD and however B is interpreted, we know that the antecedent is either true or false. If know that the antecedent is either true or false. If it is true, so is the consequent. If the antecedent is it is true, so is the consequent. If the antecedent is false, the sentence is true. So the sentence is false, the sentence is true. So the sentence is quantificationally true.quantificationally true.

A sentence A sentence PP is is quantificationally falsequantificationally false IFF IFF PP is false is false on every interpretation.on every interpretation.

((x) Bx & (x) Bx & (z) ~Bzz) ~Bzis quantificationally false. is quantificationally false. If an interpretation makes the left conjunct true, If an interpretation makes the left conjunct true,

then it makes the right conjunct false. If it makes then it makes the right conjunct false. If it makes the left conjunct false, then the sentence is false. the left conjunct false, then the sentence is false.

As every interpretation includes ‘B’ as a predicate, As every interpretation includes ‘B’ as a predicate, and however it interprets it, it cannot be the case and however it interprets it, it cannot be the case that every member of the UD is in the extension of that every member of the UD is in the extension of B but one member is not.B but one member is not.

It is not always as easy to demonstrate that a It is not always as easy to demonstrate that a sentence is quantificationally true or that it is sentence is quantificationally true or that it is quantificationally false. quantificationally false.

But we can often show that a sentence is But we can often show that a sentence is notnot quantificationally true by showing that it is quantificationally true by showing that it is false on at least one interpretation.false on at least one interpretation.

((x) (Wx x) (Wx Mx) Mx) ( (x) Mxx) Mx1. UD: the set of all living things1. UD: the set of all living thingsWx: x is a whaleWx: x is a whaleMx: x is a mammal Mx: x is a mammal The antecedent is true on this interpretation but The antecedent is true on this interpretation but

the consequent is false.the consequent is false.

We can often show that a sentence is We can often show that a sentence is notnot quantificationally false by showing that it is quantificationally false by showing that it is true on at least one interpretation.true on at least one interpretation.

((x) (x) (y) (Syx y) (Syx ~Sxy) ~Sxy)2. UD: the set of positive integers2. UD: the set of positive integersSxy: x is smaller than ySxy: x is smaller than yIF ‘There is a positive integer x such that all IF ‘There is a positive integer x such that all

positive integers are smaller than x…’positive integers are smaller than x…’The antecedent is false on this interpretation (as The antecedent is false on this interpretation (as

no positive integer is smaller than 1), so the no positive integer is smaller than 1), so the sentence is true on this interpretation, and so sentence is true on this interpretation, and so the sentence is not quantificationally false. the sentence is not quantificationally false.

We can often show that a sentence is We can often show that a sentence is notnot quantificationally false by showing that it is true quantificationally false by showing that it is true on at least one interpretation.on at least one interpretation.

((x) Ex & (x) Ex & (x ) ~Exx ) ~Ex3. UD: the set of positive integers3. UD: the set of positive integersEx: x is evenEx: x is evenOr consider:Or consider:~(~Ga & (~(~Ga & (y) Gy)y) Gy)To construct an interpretation on which this To construct an interpretation on which this

sentence is true (and thus not quantificationally sentence is true (and thus not quantificationally false), we must construct an interpretation on false), we must construct an interpretation on which ~Ga & (which ~Ga & (y) Gy is false. And that means y) Gy is false. And that means making one or the other of the conjuncts false.making one or the other of the conjuncts false.

~(~Ga & (~(~Ga & (y) Gy)y) Gy)We must construct an interpretation on which We must construct an interpretation on which

~Ga & (~Ga & (y) Gy is false. And that means making y) Gy is false. And that means making one or the other of the conjuncts false.one or the other of the conjuncts false.

4. UD: Set of positive integers4. UD: Set of positive integersGx: x is evenGx: x is evena: 2a: 2Here the left conjunct is false and so is the Here the left conjunct is false and so is the

conjunction…conjunction…

Again, we can show a sentence is Again, we can show a sentence is notnot quantificationally true, or that it is quantificationally true, or that it is notnot quantificationally false, by constructing a quantificationally false, by constructing a singlesingle interpretation that demonstrates this. interpretation that demonstrates this.

But we cannot construct a single interpretation But we cannot construct a single interpretation that will demonstrate that a sentence that will demonstrate that a sentence isis quantificationally true or that it quantificationally true or that it isis quantificationally false. For some sentences, quantificationally false. For some sentences, we can arrive at such a conclusion by we can arrive at such a conclusion by reasoning; but for many, we cannot.reasoning; but for many, we cannot.

A sentence A sentence PP of of PLPL is is quantificationally quantificationally indeterminateindeterminate IFF IFF PP is neither is neither quantificationally true nor quantificationally quantificationally true nor quantificationally false.false.

We show that a sentence is quantificationally We show that a sentence is quantificationally indeterminate by constructing two indeterminate by constructing two interpretations: one on which it is true (to interpretations: one on which it is true (to show that it is not quantificationally false) and show that it is not quantificationally false) and one on which it is false (to show that it is not one on which it is false (to show that it is not quantificationally true).quantificationally true).

A sentence A sentence PP of of PLPL is is quantificationally quantificationally indeterminateindeterminate IFF IFF PP is neither is neither quantificationally true not quantificationally quantificationally true not quantificationally false.false.

We showed, using interpretation 4, that the We showed, using interpretation 4, that the following sentence is not quantificationally following sentence is not quantificationally false by finding an interpretation on which it is false by finding an interpretation on which it is true: ~(~Ga & (true: ~(~Ga & (y) Gy). Now we need an y) Gy). Now we need an interpretation on which it is false to show that interpretation on which it is false to show that it is also not quantificationally true.it is also not quantificationally true.

This means we need an interpretation on which This means we need an interpretation on which ~Ga & (~Ga & (y) Gy is true. y) Gy is true.

This means we need an interpretation on which This means we need an interpretation on which ~Ga & (~Ga & (y) Gy is true.y) Gy is true.

5. UD: set of positive integers5. UD: set of positive integers Gx: x is oddGx: x is odd

a: 2a: 2On this interpretation, ~Ga is true (for 2 is not odd) On this interpretation, ~Ga is true (for 2 is not odd)

and (and (y) Gy) is true (for there is at least one odd y) Gy) is true (for there is at least one odd positive integer). So the conjunction is true, and positive integer). So the conjunction is true, and its negation is false.its negation is false.

Interpretations 4 and 5 demonstrate that the Interpretations 4 and 5 demonstrate that the sentence ~(~Ga & (sentence ~(~Ga & (y) Gy is quantificationally y) Gy is quantificationally indeterminate.indeterminate.

Finding an interpretation on which a sentence is Finding an interpretation on which a sentence is true or on which it is false takes some ingenuity. true or on which it is false takes some ingenuity.

Determine if the sentence is one whose Determine if the sentence is one whose quantificational status can be settled by an quantificational status can be settled by an interpretation or can be settled by reasoning (or interpretation or can be settled by reasoning (or by neither).by neither).

Guidelines: Examine the kind of sentence it is.Guidelines: Examine the kind of sentence it is.If it is a truth-functional compound, use the truth If it is a truth-functional compound, use the truth

conditions for that kind of compound.conditions for that kind of compound.If the sentence is universally quantified, then the If the sentence is universally quantified, then the

sentence is true IFF the condition specified after sentence is true IFF the condition specified after the quantifier is satisfied by all members of the the quantifier is satisfied by all members of the UD you choose.UD you choose.

If the sentence is existentially quantified, then it will be If the sentence is existentially quantified, then it will be true IFF the condition specified after the quantifier is true IFF the condition specified after the quantifier is satisfied by at least one member of the UD.satisfied by at least one member of the UD.

Sometimes, the desired interpretation Sometimes, the desired interpretation can not be can not be foundfound. For example: . For example:

If a sentence is quantificationally true, there is no If a sentence is quantificationally true, there is no interpretation on which it is false, and any attempt to interpretation on which it is false, and any attempt to construct an interpretation on which the sentence is construct an interpretation on which the sentence is false will fail.false will fail.

The same point holds for quantificationally false The same point holds for quantificationally false sentences.sentences.

A set of positive integers is always a good choice for A set of positive integers is always a good choice for your UD as you construct interpretations.your UD as you construct interpretations.

Quantificational notionsQuantificational notions

An argument of An argument of PLPL is is quantificationallyquantificationally validvalid IFF IFF there is no interpretation on which every there is no interpretation on which every premise is true and the conclusion is false.premise is true and the conclusion is false.

An argument of An argument of PLPL is is quantificationally invalidquantificationally invalid IFF the argument is not quantificationally IFF the argument is not quantificationally valid.valid.

((x) (Fx v Gx)x) (Fx v Gx)((x) ~Fxx) ~Fx------------------------------------((x) Gxx) Gxis quantificationally valid.is quantificationally valid.Suppose that on some interpretation both premises Suppose that on some interpretation both premises

are true. If the first premise is true, then some are true. If the first premise is true, then some member x of the UD is either F or G. member x of the UD is either F or G.

If the second premise is true, then no member of If the second premise is true, then no member of the UD is F. Therefore, because the member that the UD is F. Therefore, because the member that is either F or G is not F, it must be G. Thus (is either F or G is not F, it must be G. Thus (x) Gx x) Gx will be true on any such interpretation.will be true on any such interpretation.

Demonstrating that an argument is not Demonstrating that an argument is not guantificationally valid: find an interpretation in guantificationally valid: find an interpretation in which all of its premises are true and its conclusion is which all of its premises are true and its conclusion is false. Consider:false. Consider:

((x) [(x) [(y) Fy y) Fy Fx] Fx]((y) ~Fyy) ~Fy----------------------------------------------~ (~ (x) Fxx) FxWe can make the first premise true by interpreting F so We can make the first premise true by interpreting F so

that at least one member of the UD is in its extension that at least one member of the UD is in its extension – for then that object will satisfy the condition – for then that object will satisfy the condition specified by (specified by (y) Fy y) Fy Fx beause it will satisfy its Fx beause it will satisfy its consequent.consequent.

((x) [(x) [(y) Fy y) Fy Fx] Fx]((y) ~Fyy) ~Fy----------------------------------------------~ (~ (x) Fxx) FxThe second premise will be true if at least one The second premise will be true if at least one

member of the UD is not in the extension of F. member of the UD is not in the extension of F. So F will have some, but not all, members of So F will have some, but not all, members of the UD in its extension. And because some the UD in its extension. And because some members will be in the extension, the members will be in the extension, the conclusion will be false.conclusion will be false.

6: UD: set of positive integers6: UD: set of positive integers Fx: x is odd.Fx: x is odd.

Again, there are asymmetries in what we can Again, there are asymmetries in what we can prove.prove.

We can prove an argument is We can prove an argument is validvalid only on only on specific interpretations, not all possible specific interpretations, not all possible interpretations, although some we can prove interpretations, although some we can prove are valid by reasoning.are valid by reasoning.

But we can prove an argument is But we can prove an argument is quantificationally invalidquantificationally invalid by finding a single by finding a single interpretation on which it is not valid (as to be interpretation on which it is not valid (as to be quantificationally validquantificationally valid, an argument must be , an argument must be such that there is such that there is nono interpretation on which interpretation on which all its premises are true and its conclusion is all its premises are true and its conclusion is false).false).

Again, there are asymmetries in what we can Again, there are asymmetries in what we can prove.prove.

We can also prove an argument is not We can also prove an argument is not quantificationally invalid by finding an quantificationally invalid by finding an interpretation on which it is valid.interpretation on which it is valid.

Quantificational equivalencyQuantificational equivalencySentences Sentences PP and and QQ are are quantificationally quantificationally

equivalentequivalent IFF there is no interpretation on IFF there is no interpretation on which which PP and and QQ have different truth values. have different truth values.

The following are quantificationally equivalent:The following are quantificationally equivalent:((x) Fx x) Fx Ga Ga((x) (Fx x) (Fx Ga) Ga)

((x) Fx x) Fx Ga Ga((x) (Fx x) (Fx Ga) Ga)

We reason as follows:We reason as follows:Suppose that (Suppose that (x) Fx x) Fx Ga is true on some Ga is true on some

interpretation. Then (interpretation. Then (x) Fx is either true or x) Fx is either true or false on this interpretation. false on this interpretation.

If (If (x) Fx is true, then so is Ga. But then since Ga x) Fx is true, then so is Ga. But then since Ga is true, every object x in the UD is such that if is true, every object x in the UD is such that if x is F, then a is G. So (x is F, then a is G. So (x) (Fx x) (Fx Ga) is true. Ga) is true.

((x) Fx x) Fx Ga Ga((x) (Fx x) (Fx Ga) Ga)

If (If (x) Fx is false, then every object x in the UD is x) Fx is false, then every object x in the UD is such that such that ifif x is F (which we assume here it is x is F (which we assume here it is not), not), thenthen a is G and the whole sentence is a is G and the whole sentence is true. Again, ‘(true. Again, ‘(x) (Fx x) (Fx Ga) is also true on Ga) is also true on that interpretation.that interpretation.

((x) Fx x) Fx Ga Ga((x) (Fx x) (Fx Ga) Ga)

Now suppose that (Now suppose that (x) Fx x) Fx Ga is false on some Ga is false on some interpretation. interpretation.

Then (Then (x) Fx is true and Ga is false. But if (x) Fx is true and Ga is false. But if (x) Fx is x) Fx is true, then some object x in the UD is in the true, then some object x in the UD is in the extension of F. This object does not satisfy the extension of F. This object does not satisfy the condition that condition that ifif it is F (which it is), it is F (which it is), thenthen a is G a is G (which it is not on our present assumption). So (which it is not on our present assumption). So ((x) (Fx x) (Fx Ga) is false if ( Ga) is false if (x) Fx x) Fx Ga is false. Ga is false.

Taken together with the result that if one is true, so Taken together with the result that if one is true, so is the other, we have demonstrated that the 2 is the other, we have demonstrated that the 2 sentences are quantificationally equivalent.sentences are quantificationally equivalent.

Quantificational consistencyQuantificational consistency

A set of sentences of A set of sentences of PLPL is is quantificationally quantificationally consistent consistent IFF there is at least one interpretation IFF there is at least one interpretation on which all the members of the set are true.on which all the members of the set are true.

A set of sentences of A set of sentences of PLPL is is quantificationally quantificationally inconsistent inconsistent IFF the set is not quantificationally IFF the set is not quantificationally consistent.consistent.

The set {(The set {(x) ~Bax, ~Bba v (x) ~Bax, ~Bba v (x) ~Bax} is x) ~Bax} is quantificationally consistent.quantificationally consistent.

{({(x) ~Bax, ~Bba v (x) ~Bax, ~Bba v (x) ~Bax} x) ~Bax} 7. UD: set of possible integers7. UD: set of possible integers Bxy: x is less than or equal to yBxy: x is less than or equal to y

a: 1a: 1b: 2b: 2

On this interpretation (On this interpretation (x) ~Bax is true since 1 is less x) ~Bax is true since 1 is less than or equal to every positive integer.than or equal to every positive integer.

So, too, ~Bba is true since 2 is neither less than nor So, too, ~Bba is true since 2 is neither less than nor equal to 1, so ~Bba v (equal to 1, so ~Bba v (x) ~Bax is truex) ~Bax is true