Lecture 21: Interpretations and variable assignments

Recall the definitions from the last lecture, in Section 19.5.

With a single instance / counterexample Only by invoking every interpretation

you can show: you can show:

• An argument is invalid. • An argument is valid.

• A sentence is not a first-order truth (validity) • A sentence is a validity.

• Two sentences are not equivalent. • Two sentences are equivalent.

• Some sentences are jointly satisfiable. • Some sentences are not jointly satisfiable.

So far, the semantic notions were defined for sentences,

i. e., formulas without any free variables.

But, we can extend them also to formulas with free variables,

when we add variable assignments to our considerations.

Notation: Instead of speaking of the interpretation with a variable assignment

I [d/x],

we shall speak of the interpretation I with

the variable assignment that assigns d to x , and write (x) = d .

We will also say: ‘the interpretation I under the variable assignment ’.

20.3 Truth in FOL

Now we can define for FOL formulas, not just sentences:

Definition

A FOL formula that is true (or false) in some interpretation I

under all variable assignments over its domain

is said to be true (or false) in that interpretation I .

Definition

A FOL formula that is true (or false) in any interpretation

is said to be a validity or first-order truth (or falsehood).

20.3.1 Examples

20.3.1.1 A(x) _ ¬A(x))

Consider the formula

(A(x) _ ¬A(x)),

where ‘ A ’ is some basic predicate of the given language.

• Now consider any interpretation I , and let’s say that

that interpretation assigns the property ‘P’ (perhaps, of ‘being an peacock’) to ‘ A ’.

• Now consider any variable assignment . This will assign the object (x) to ‘ x ’.

• Now this object (x) either has property P or it does not have the property P.

a) If (x) has the property P, then the formula is true.

b) If (x) does not have the property P, then it must have

the property not-P (so the property of ‘not being an peacock’).

Either way, in both cases the formula ‘ (A(x) _ ¬A(x)) ’ is true!

The reason for this is just the truth-functional behavior of ‘ _ ’ and ¬ ’.

Conclusion:

• is any variable assignment whatsoever over I , so ‘ (A(x) _ ¬A(x)) ’ is true in I .

• And I is any interpretation, which means that ‘ (A(x)_¬A(x)) ’ is a first-order truth.

20.3.1.2 (A(x) ! A(x))

For similar reasons, ‘ (A(x) ! A(x)) ’ is also a first-order truth.

• To see this, think about how the truth-function ‘ ! ’ behaves.

20.3.1.3 (A(x) ^ ¬A(x))

Again, similar arguments show that ‘ (A(x) ^ ¬A(x)) ’ is a first-order falsehood.

20.3.1.4 8x x = x

Now consider the sentence:

8x x = x.

• Again, consider any interpretation I as well as any variable assignment over I .

• According to our definitions,

‘ 8x x = x ’ is true in I , if ‘ x = x ’ is satisfied by every object in the domain.

In other words, if every variable assignment makes ‘ x = x ’ true.

• But that has to be the case, because this is simply what identity says!

So we have:

• Since any over I makes ‘ x = x ’ true, then ‘ 8x x = x ’ is true in I .

• And since I was chosen arbitrarily, ‘ 8x x = x ’ is true in any interpretation.

Therefore, ‘ 8x x = x ’ is a first-order truth.

20.3.2 Summary

First-order truths (sometimes also called ‘first-order validities’) are

• something like tautologies in the TFL case,

• i. e., roughly speaking, they are ‘always true’, or ‘true in all cases’.

It’s just the sense of ‘case’ which is different here:

• All possible cases are here not given by direct truth-value assignments,

but indirectly by some selection of an interpretation.

And we have far more to consider than just atomic sentences!

20.4 First-order equivalence

Another notion that is somewhat similar to TFL is that of first-order equivalence:

Definition

Two formulas which are true (or false) in exactly the same interpretations and

with just the same variable assignments are said to be first-order equivalent.

20.4.0.1 (A(x) ! B(x)) and (¬A(x) _ B(x))

As before, we can easily check (in much the same way as did with first-order truths)

that the formulas:

(A(x) ! B(x)) and (¬A(x) _ B(x))

are equivalent.

Basically:

• Pick an any interpretation I and any variable assignment . . .

• Then assume that one formula is true and show that the other is also true.

• Then, the same for false, and vice versa.

The reasons for this equivalence is the way the respective truth-functions work,

which should be fairly clear to you at this point.

20.4.0.2 Quantifiers

Now let’s consider the formulas (really: sentences)

¬8x ¬A(x) and 9x A(x),

which are also equivalent. To argue for this informally, we could proceed as follows:

• Suppose ‘ ¬8x ¬A(x) ’ is true in an interpretation in I under a variable assignment .

• That means that ‘ 8x ¬A(x) ’ is false in I under that assignment.

• This can only be, because there is an object in I ’s domain that

does not satisfy ‘ ¬A(x) ’ when we set (x) to be that object.

• But then (x) must satisfy ‘ A(x) ’,

which guarantees that ‘ 9x A(x) ’ is true in I under .

A similar argument will work is the other direction, so we see that ‘ ¬8x¬A(x) ’ and ‘ 9x A(x) ’

are true in precisely the same circumstances and are thus equivalent.

Once you get used to this style of reasoning, it should become easy to see

that the device of variable assignments is really not very complicated.

20.5 Validity of inferences in FOL (Entailment)

Let us turn finally to the notion of validity of inferences in FOL.

Definition

An inference in FOL (all premises and conclusion are FOL formulas) is valid iff

• any variable assignment that makes

• all the premises true (relative to an interpretation over a chosen domain)

• also makes the conclusion of the argument true.

In this case, we write ‘ P1 , ... , Pn |= Q ’.

Some inferences are clearly valid:

• For instance, we have a full analogy with the case of tautologies in TFLs, namely:

if an inference has a first-order truth as its conclusion, it will be valid,

regardless of what the premises are.

This is because one can then never find an interpretation and

a variable assignment over it which makes the conclusion false.

• If we have an inference where the premises are jointly unsatisfiable,

then that inference, too, must be valid.

Because we can never make all the premises true,

no matter how we choose the interpretation and variable assignment.

• Inferences between equivalent formulas are clearly valid.

For example, since ¬8x¬A(x) and 9x A(x) are FOL equivalent, we always have

¬8x ¬A(x) ✏ 9x A(x),

as well as

9x A(x) ✏ ¬8x ¬A(x).

• See also the argument in Section 19.7.1, which was skipped in the last lecture!

In the above examples we’ve been able to argue generally, about all interpretations.

Unfortunately, however, this is an exception. . .

20.6 General difficulty: Too many interpretations!

Recall (from Section 20.2):

1. To show that something is an FOL truth,

we have to look at all interpretations and all assignments.

2. To show that two formulas are FOL equivalent,

we have to look at all interpretations and all assignments

to see if they always assign the same truth-values to both formulas.

3. To show that an argument is valid,

we have to look at all interpretations and all assignments

to see if, having assigned T to the premises, they also assign T to the conclusion.

So in all these cases, we have to survey all interpretations.

• How many are there? Infinitely many. . .

Now, the situation was radically different in the case of TFLs:

• In TFL, if we consider n atomic sentences, there are

only a fixed and finite number of truth-value assignments at issue, namely 2n .

• These various possibilities might be very many,

but are surveyable in principle (if perhaps not in practice):

this is why the truth-table method works.

But, in the case of FOL interpretations the range of possible interpretations is infinite

and thus simply unsurveyable in principle.

Thus, unlike in the case of TFL, to establish FOL truth, equivalence, and validity

there is no method that is guaranteed to give us an answer. . .