In chapter 4 of the book it says:

The sentence $a=a$ is necessarily true. So of course, no matter what your initial premises may be, it will be impossible for those premises to be true and for $a=a$ to be false--simply because it is impossible for $a=a$ to be false!

It then goes on to say a few pages later:

it should be clear, however, that not all logically necessary claims are tautologies. The simplest example of a logically necessary claim that is not a tautology is the FOL sentence $a =a$. Since this is an atomic sentence, its truth table would contain one T and one F. The truth table method is too coarse to recognize that the row containing the F does not represent a genuine possibility."

How can this sentence be such that "it is impossible to be false" yet not be a tautology?


edit: nvmd, I see what they're saying.

  • $\begingroup$ If you see what they're saying, you can give that as your own answer to this question. That way, we can give you feedback on what understanding you came to. $\endgroup$ – Semiclassical Oct 4 '16 at 18:33

You have to check the definition of tautology.

Usually, tautology is defined in the context of propositional logic.

For first order logic, a formula is a tautology if it is a formula obtainable from a tautology of propositional logic by replacing (uniformly) each sentence symbol by a formula of the first-order language.

Thus, $\forall x P(x) \to \forall x P(x)$ is a tautology, being an "instance" of $A \to A$, while $x=x$ is not, because it is an "instance" of the single sentence symbol $A$, which is not a tautology.

Of course, $x=x$ is valid, i.e. true in every interpretation.

  • $\begingroup$ This makes sense. Indeed, that's what I realized too. That what they mean by 'a=a' being an atomic sentence is that it is either true or false, just as a literal in propositional logic is either true or false, but not a tautology. That the possibility of falsehood here isn't really genuine isn't relevant. $\endgroup$ – guest Oct 4 '16 at 18:38
  • $\begingroup$ @guest - exactly ! If you start from $∀xP(x)→∀xP(x)$ and replace (uniformly) each atomic formula with a sentence letter, you get $A \to A$, which is a tautology. But from the atomic $\forall x (x=x)$ you get only $A$. $\endgroup$ – Mauro ALLEGRANZA Oct 5 '16 at 6:08

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