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This is from "Introduction to Mathmatical Logic-forth edition" by Elliot Mendelson.

In page 61-62 , the book has propositon (VIII) and (IX). In the solution manual (I mean the "solutions to selected exercises" section of the book) , they used proposition (VIII) to prove proposition (IX).But I am kind of skeptical of the proof.I my point of view , (VIII) can only tell what will will happen if the free variables of a wf $\mathscr B$ occur in a list $x_{i_1},...,x_{i_k}$ .It can't tell anything about what will happen if the free variables of a wf $\mathscr B$ don't occur in a list $x_{i_1},...,x_{i_k}$.

In the case of the closed wf $\mathscr B$ in (IX) where there is no free variables , the free variables don't occur in a list $x_{i_1},...,x_{i_k}$ (or in any arbitrary list).So , we can't use (VIII) to prove that this closed wf $\mathscr B$ that it is either true or false under a given interpretation (this is proposition (IX) ).But the solution manual uses (VIII) for proving (IX).

$1$.Is my reasoning wrong ?
$2$.If my reasoning isn't wrong , can someone show me an alternate proof of (IX) ?

Kripke Platek
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1 Answers1

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The Lemma "sounds" like (in the meta-theory):

for every formula $\mathscr B$, for every sequence $s,s'$, for every variable $x_i$: if $x_i$ occures in the "list" and $s$ and $s'$ "agree", then ($s \vDash \mathscr B \text { iff } s' \vDash \mathscr B)$.

If $\mathscr B$ is closed, there are no variables occurring in it. Thus, we can use the "empty list" for the Lemma.

Whatever variables $x_i$ we consider, we have that $x_i$ does not occur in the (empty) list, and thus the conditional of the Lemma is vacuously satisfied.

If you are uncomfortable with "vacuous truth", you can reverse the approach: if two sequence $s$ and $s'$ disagree on some free variable, they can assign different truth values to the same formula.

Consider my preferred example: $(x_1=0)$ and the two sequences: $s(x_1)=0$ and $s'(x_1)=1$.

The two sequences do not satisfy the antecedent of the Lemma, and thus the result is different.

Now, what happens with a closed formula? There is no free variable such that two sequences can disagree on. Thus, the result will be the same for every sequence.

Mauro ALLEGRANZA
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  • I seems like the lemma (VIII) for the proof of (IX) needs to be vacuously satisfied in order to use it for the proof. But I heard that mathmatical proof don't allow vacuous truths in them from one of the comments of this post how-to-formally-state-and-prove-vacuous-truth. Is this correct? – Kripke Platek Mar 09 '21 at 10:33
  • This still seems to me like vacuous truth (inform me if I am wrong about this). According to your edit , the lemma is " if two sequence $s$ and $s′$ disagree on some free variable, they can assign different truth values to the same formula". If the hypothesis [Two sequence $s$ and $s′$ disagree on some free variable] is false , then we can't tell if the conclusion [they can assign different truth values to the same formula] is true or false. Can you show me other writeups of a proof of (IX) ? – Kripke Platek Mar 09 '21 at 13:21