[7.] Predicate Logic Semantics

 

 

[7.1.] “Interpretations in Predicate Logic.” [Ch.8:1]

 

In our examination of sentential logic, we used the notion of an interpretation in explaining other important notions, like validity, tautology, inconsistency, etc.

 

To provide an interpretation of a sentence (or set of sentences) in sentential logic, all we had to do was assign truth values to any sentence constants in the sentence. We did this by constructing truth tables.

 

We then explained other notions in terms of such assignments of truth values:

·         valid: an argument having no interpretation on which all premises are true and the conclusion false

·         tautology: a statement that is true on every interpretation

·         contradiction: a statement that is false on every interpretation

·         inconsistent: a set of premises (or other statements) for which there is no interpretation on which they are all true.

 

We will continue to use the notion of interpretation in predicate logic.

 

However, providing an interpretation of a sentence or a set of sentences in predicate logic is more complicated than simply assigning truth values to sentence constants. It requires the following steps:

 

1.       Specify a domain of discourse (either unrestricted or restricted to some specific set of objects)

 

2.       If there are any individual constants in the sentence(s), specify which individual in the domain each one designates. (Note that there may not be any individual constants in the sentence(s)).

 

3.       Specify which property is designated by each property constant, “so it is clear which individuals in the domain have that property and which do not.” (189)

 

Notice that the examples the textbook gives contain no individual constants. Instead, they all contain quantifiers and bound individual variables.

 

Examples: from exercise 8-1, p.191, which requires that you provide one interpretation on which the sentence is clearly true and another on which it is clearly false:

 

2)                  ($x)(Ax · ~Bx)

 

clearly true domain: unrestricted Ax: x is a horse Bx: x can fly

clearly false domain: unrestricted Ax: x is a horse Bx: x is a mammal

 

 

4)         ~(x)(Ax É Bx)

 

clearly true domain: unrestricted Ax: x is a fish Bx: x is a giraffe

clearly false domain: unrestricted Ax: x is a dog Bx: x is a mammal

 

 

6)         ($x)[(Ax · Bx) · ~Cx]

 

clearly true domain: natural numbers Ax: x is greater than 2 Bx: x is less than 4 Cx: x is evenly divisible by 2

clearly false domain: natural numbers Ax: x is greater than 9 Bx: x is less than 11 Cx: x is evenly divisible by 5

 

 

ASSIGN: exercise 8-1 (pp.190-91)

·         do all for next time—be sure to come up with your own examples, as we will go through these at the start of class

 

 

[7.2.] “Proving Invalidity.” [Ch.8:2]

 

Remember that to prove an argument to be invalid, we need to show that it is possible for the premises to be all true and the conclusion false at the same time.

 

In sentential logic, we did this using the short truth table method to identify an interpretation (an assignment of truth values to all the argument’s premises and conclusion) on which all the premises are true and the conclusion false:

	/\

 

 

In predicate logic, we prove invalidity by doing the same thing: providing an interpretation of an argument on which all of the premises are true and the conclusion false.

 

This requires us to do exactly what we did in the previous section: give interpretations of sentences in predicate logic. The only difference is that now, you will give an interpretation that makes each premise of the argument true and the conclusion false.

 

Consider the argument given in your book to illustrate this (p.187):

 

(x)(Ax É Bx)

(x)(Cx É Bx)   / \ (x)(Ax É Cx)

 

You can do this with an unrestricted domain:

Ax = x is a pitbull

Bx = x is a dog

Cx = x is a poodle

 

or with a restricted domain -- e.g., restricted to natural numbers (i.e., positive whole numbers, a.k.a. positive integers):

Ax = x is greater than 10

Bx = x is greater than 5

Cx = x is greater than 20

 

Any interpretation will work, as long as it makes all the premises true and the conclusion false.

 

 

Another example: exercise 8-2, #2 (p.193):

 

2)         1. (x)(Ax É Bx)

2. ($x)~Ax                    / \ ($x)~Bx

 

domain: animals

Ax = x is a dog

Bx = x is an animal

 

 

ASSIGN: exercise 8-2 (p.193)

·         do all for next time; we’ll cover the odds next class

·         remember that two different property constants can stand for the same property

·         note the mistake in #8 (mentioned on errata sheet)

 

 

[7.3.] “Consistency in Predicate Logic.” [Ch. 8:4]

 

The final exercise in chapter 8 requires you to show that a given argument has consistent premises, i.e., to show that it is possible for all the premises in the argument to be true.

 

In sentential logic, we showed this using the short truth-table method.

 

In predicate logic, we show it using part of the method we used to show that an argument is invalid. Rather than finding an interpretation that makes the premises all true and the conclusion false, we simply find an interpretation that makes all the premises true.

 

As when we used the short truth table method to show premises of arguments in sentential logic to be consistent, the conclusion of the argument is irrelevant; you should just ignore it.

 

For example,

 

exercise 8-4, #6:

 

1.       ~(x)[Dx É (~Fx Ú Gx)]

2.       ($x)[Fx · (Dx Ú ~Gx)]         /\ ($x)(Dx · Fx)

 

domain: unrestricted

Dx: x is a dog

Fx: x is a mammal

Gx: x is a goat

 

 

ASSIGN: exercise 8-4 (p.196)

·         do all for next time; we’ll cover the odds next class

 

 

Stopping point for Wednesday March 12. For next time (Monday March 24), do exercises 8-1, 8-2 & 8-4, and read ch.9:1-2 (pp.198-204)