Lecture 15: Validity and Predicate Logic
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1 Lecture 15: Validity and Predicate Logic 1
2 Goals Today Learn the definition of valid and invalid arguments in terms of the semantics of predicate logic, and look at several examples. Learn how to get equivalents in predicate logic, and look at how equivalents can help us show things about validities. 2
3 Recall: Propositional Validity Here s the definition we learned in Lecture 12 for formulas of propositional logic. We say that ϕ1,, ϕn ψ if any truth-table which contains columns for each of ϕ1,, ϕn, ψ has this feature: whenever a row contains a T in each of the ϕ1,, ϕn columns, this row also has a T in the ψ column. 3
4 Defn: Validity in Predicate Logic Suppose ϕ1,, ϕn, ψ, are formulas of predicate logic. Then we say ϕ1,, ϕn ψ if any model M which interprets the language of ϕ1,, ϕn, ψ has this feature: if VM(ϕ1)=T,..., VM(ϕn)=T then VM(ψ)=T. Again, we call ϕ1,, ϕn the premises and we call ψ the conclusion. And the argument with premises ϕ1,, ϕn and conclusion ψ is said to be valid if ϕ1,, ϕn ψ. We may also continue to use the follows from and has a consequence locutions. Again, the way to see the relationship between propositional logic and predicate logic is as follows: individual models correspond to individual rows of the truth-table. 4
5 Example 1 Let s show that x Fx Fa. So we need to show that any model M in the language with an individual constant a and predicate letter F is such that if VM( x Fx)=T then VM(Fa)=T Well, suppose that M is such a model, and suppose that it s the case that VM( x Fx)=T. We have to show that VM(Fa)=T. 5 But, by the clause for in the semantics, VM( x Fx)=T implies that for all individual constants c, one has that VM(Fc)=T In particular, for the individual constant a, we have that VM(Fa)=T which is what we wanted to show.
6 Example 2 Let s show that ( x Ax) ( x Bx) x (Ax Bx) So suppose M is a model with VM(( x Ax) ( x Bx))=T We must show we also have VM( x (Ax Bx)) = T Since VM(( x Ax) ( x Bx))=T, the clause for says that VM( x Ax)=T or VM( x Bx)=T 6 Suppose first that VM( x Ax)=T. Then the clause for says that VM(Ac)=T for each c. Then the clause for implies that VM(Ac Bc)=T for each c. But then the clause for says that VM( x (Ax Bx)) = T. We argue similarly in the case where VM( x Bx)=T. Hence, we conclude that in either case we have VM( x (Ax Bx)) = T, which is what we wanted to show.
7 Definition of Invalidity Suppose ϕ1,, ϕn, ψ, are formulas of predicate logic. Then we say ϕ1,, ϕn ψ if at least one model M interpreting the language of ϕ1,, ϕn, ψ has this feature: VM(ϕ1)=T,..., VM(ϕn)=T and VM(ψ)=F. Again, we call ϕ1,, ϕn the premises and we call ψ the conclusion. And the argument with premises ϕ1,, ϕn and conclusion ψ is said to be invalid if ϕ1,, ϕn ψ. There s an obvious sense in which showing invalidity is easier than showing validity: to show invalidity, you just need to find one model. 7
8 Example 3 Let s show that x (Ax Bx) ( x Ax) ( x Bx) So we just need to find one model M with the property that VM( x (Ax Bx))=T and VM(( x Ax) ( x Bx))=F Well, consider our old friend M with domain natural numbers and with A interpreted as the evens and B interpreted as the odds. 8 Let s check VM( x (Ax Bx))=T. By the clause for, we have that it suffices to show that VM(Ac Bc)=T for all c from M. But every number is even or odd, and so we know this! Let s check that we indeed have VM(( x Ax) ( x Bx))=F Well, VM( x Ax)=F, since 3 is not even, and VM( x Bx)=F, since 2 is not odd. So by clause for, VM(( x Ax) ( x Bx))=F.
9 Example 3 (Venn Diagrams) Let s show that x (Ax Bx) ( x Ax) ( x Bx) So we just need to find one model M with the property that VM( x (Ax Bx))=T and VM(( x Ax) ( x Bx))=F Well, consider our old friend M with domain natural numbers and with A interpreted as the evens and B interpreted as the odds. There s a traditional way of illustrating these models with pictures called Venn diagrams: A 9 M B
10 More Venn-Diagram Examples Most simple relationships between two predicates A,B can be easily visualized in terms of Venn Diagram. Here s an example of a diagram of model M where the following is true: x (Ax Bx) B So in terms of this picture, this sentence represents the containment of A in B. A 10 M
11 More Venn-Diagram Examples Most simple relationships between two predicates A,B can be easily visualized in terms of Venn Diagram. Here s an example of a diagram of model M where the following is true: x (Ax Bx) B A M So in terms of this picture, this sentence represents that A and B overlap. 11
12 More Venn-Diagram Examples So what would a diagram of x Ax look like? Well, it just corresponds to the claim that everything in the domain is in A. A M It s easy to visualize with the following animation. 12
13 More Venn-Diagram Examples So what would a diagram of x Ax look like? Well, it just corresponds to the claim that something is in A. So it corresponds to A just being any shape within M. M A 13
14 Example 4 Let s show that we have x y Rxy y x Rxy Consider the model M, where Rab iff there s an arrow from a to b in the picture: Let s check VM( x y Rxy)=T. We just need to check that for each number, there s an arrow from that number to another number. But we see it s so! Let s check VM( y x Rxy)=F. Suppose it was true, that is, suppose VM( y x Rxy)=T. By the clause for, we would have VM( x Rxc)=T for some c. But c can t be 1 (since...), and it can t be 2 (since.. ), etc.
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