Chapter 34 Reasoning about interpretations
34.1 Validities and contradictions
We can show that a sentence is not a validity just by providing one carefully specified interpretation: an interpretation in which the sentence is false. To show that something is a validity, on the other hand, it would not be enough to construct ten, one hundred, or even a thousand interpretations in which the sentence is true. A sentence is only a validity if it is true in every interpretation, and there are infinitely many interpretations. We need to reason about all of them, and we cannot do this by dealing with them one by one!
Sometimes, we can reason about all interpretations fairly easily. For example, we can offer a relatively simple argument that ‘$R(a,a)\vee \mathrm{\neg}R(a,a)$’ is a validity:
Any relevant interpretation will give ‘$R(a,a)$’ a truth value. If ‘$R(a,a)$’ is true in an interpretation, then ‘$R(a,a)\vee \mathrm{\neg}R(a,a)$’ is true in that interpretation. If ‘$R(a,a)$’ is false in an interpretation, then $\mathrm{\neg}R(a,a)$ is true, and so ‘$R(a,a)\vee \mathrm{\neg}R(a,a)$’ is true in that interpretation. These are the only alternatives. So ‘$R(a,a)\vee \mathrm{\neg}R(a,a)$’ is true in every interpretation. Therefore, it is a validity.
This argument is valid, of course, and its conclusion is true. However, it is not an argument in FOL. Rather, it is an argument in English about FOL: it is an argument in the metalanguage.
Note another feature of the argument. Since the sentence in question contained no quantifiers, we did not need to think about how to interpret ‘$a$’ and ‘$R$’; the point was just that, however we interpreted them, ‘$R(a,a)$’ would have some truth value or other. (We could ultimately have given the same argument concerning TFL sentences.)
Let’s have another example. The sentence ‘$\forall x(R(x,x)\vee \mathrm{\neg}R(x,x))$’ should obviously be a validity. However, saying precisely why is quite tricky. We cannot say that ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’ is true in every interpretation, since ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’ is not even a sentence of FOL (remember that ‘$x$’ is a variable, not a name). Instead, we should say something like this:
Consider some arbitrary interpretation. ‘$\forall x(R(x,x)\vee \mathrm{\neg}R(x,x))$’ is true in our interpretation iff ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’ is satisfied by every object of its domain. Consider some arbitrary member of the domain, which, for convenience, we will call Fred. Either Fred satisfies ‘$R(x,x)$’ or it does not. If Fred satisfies ‘$R(x,x)$’, then Fred also satisfies ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’. If Fred does not satisfy ‘$R(x,x)$’, it does satisfy ‘$\mathrm{\neg}R(x,x)$’ and so also ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’.^{1}^{1} 1 We use here the fact that the truth conditions for connectives also apply to satisfaction: $a$ satisfies $\mathcal{A}(\U0001d4cd)\vee \mathcal{B}(\U0001d4cd)$ iff $a$ satisfies $\mathcal{A}(\U0001d4cd)$ or $\mathcal{B}(\U0001d4cd)$, etc. So either way, Fred satisfies ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’. Since there was nothing special about Fred—we might have chosen any object—we see that every object in the domain satisfies ‘$R(x,x)\vee \mathrm{\neg}R(x,x)$’. So ‘$\forall x(R(x,x)\vee \mathrm{\neg}R(x,x))$’ is true in our interpretation. But we chose our interpretation arbitrarily, so ‘$\forall x(R(x,x)\vee \mathrm{\neg}R(x,x))$’ is true in every interpretation. It is therefore a validity.
This is quite longwinded, but, as things stand, there is no alternative. In order to show that a sentence is a validity, we must reason about all interpretations.
34.2 Other cases
Similar points hold of other cases too. Thus, we must reason about all interpretations if we want to show:

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that a sentence is a contradiction (this requires that it is false in every interpretation);

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that two sentences are logically equivalent (this requires that they have the same truth value in every interpretation);

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that some sentences are jointly unsatisfiable (this requires that there is no interpretation in which all of those sentences are true together, i.e., that, in every interpretation, at least one of those sentences is false);

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that an argument is valid (this requires that the conclusion is true in every interpretation where the premises are true);

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that some sentences entail another sentence.
The problem is that, with the tools available to you so far, reasoning about all interpretations is a serious challenge! For a final example, here is a perfectly obvious entailment:
After all, if everything is both $H$ and $J$, then everything is $H$. But we can only establish the entailment by considering what must be true in every interpretation in which the premise $\forall x(H(x)\wedge J(x))$ is true. To show this, we would have to reason as follows:
Consider an arbitrary interpretation in which ‘$\forall x(H(x)\wedge J(x))$’ is true. It follows that ‘$H(x)\wedge J(x)$’ is satisfied by every object in this interpretation. ‘$H(x)$’ will, then, also be satisfied by every object.^{2}^{2} 2 Here again we make use of the fact that any object that satisfies $\mathcal{A}(\U0001d4cd)\wedge \mathcal{B}(\U0001d4cd)$ must satisfy both $\mathcal{A}(\U0001d4cd)$ and $\mathcal{B}(\U0001d4cd)$. So it must be that ‘$\forall xH(x)$’ is true in the interpretation. We’ve assumed nothing about the interpretation except that it was one in which ‘$\forall x(H(x)\wedge J(x))$’ is true. So any interpretation in which ‘$\forall x(H(x)\wedge J(x))$’ is true is one in which ‘$\forall xH(x)$’ is true.
Even for a simple entailment like this one, the reasoning is somewhat complicated. For more complicated entailments, the reasoning can be extremely torturous.
The following table summarizes whether a single interpretation or counterinterpretation suffices, or whether we must reason about all interpretations.
Yes  No  
validity?  all interpretations  one counterinterpretation 
contradiction?  all interpretations  one counterinterpretation 
equivalent?  all interpretations  one counterinterpretation 
satisfiable?  one interpretation  all interpretations 
valid?  all interpretations  one counterinterpretation 
entailment?  all interpretations  one counterinterpretation 
You might want to compare this table with the table at the end of chapter 15. The key difference resides in the fact that TFL concerns truth tables, whereas FOL concerns interpretations. This difference is deeply important, since each truthtable only ever has finitely many lines, so that a complete truth table is a relatively tractable object. By contrast, there are infinitely many interpretations for any given sentence(s), so that reasoning about all interpretations can be a deeply tricky business.