1
$\begingroup$

I am confused about the sense in which every statement is vacuously true in an inconsistent arithmetic. My lecture notes say

The first question above asks whether it is possible to prove both a proposition P and its negation ¬P. If this is the case, then we say that arithmetic is inconsistent; otherwise, we say arithmetic is consistent. If arithmetic is inconsistent, meaning there are false statements that can be proved, then the entire arithmetic system will collapse because from a false statement we can deduce anything, so every statement in our system will be vacuously true.

It seems that it should say "every statement is the consequent in a vacuously true statement" instead. I am confused because the statements that it refers to as being vacuously true aren't necessarily implications, yet I thought that vacuous truths are true implications of the form $p \implies q$ where $q$ is false. Is it correct to say, "every statement is the consequent in a vacuously true implication, namely the implication whose antecedent is $P$", rather than saying that every statement is a vacuous truth? Based on the notes's phrasing, a statement like "My name is Sam" seems to be a possible vacuous truth. Does "vacuously true" not need to refer only to implications?

$\endgroup$
4
  • 2
    $\begingroup$ I agree with your perflexity... False statements are false and true ones are true. What an inconsistent theory does is that it is not able to provide us with a "tool" (the proof) to separate the true from the false ones. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Sep 26, 2023 at 5:47
  • 3
    $\begingroup$ I don't think you need to read the use of "vacuous" here so literally or precisely. I think the intended meaning is clear, that every statement is provable (I would not say "true") for vacuous reasons (the principle of explosion). $\endgroup$
    – Qiaochu Yuan
    Commented Sep 26, 2023 at 5:57
  • $\begingroup$ It's perhaps not the best wording if one insists on the rigorous definition of "vacuous truth" given above, but is still technically correct: an inconsistent system proves that every statement is equivalent to an implication $p \rightarrow q$ so that $\neg p$ is true, and is therefore vacuously true. $\endgroup$
    – Z. A. K.
    Commented Sep 26, 2023 at 6:42
  • $\begingroup$ Paraphrasing Bram28's answer: the explicit version of "every statement is vacuously true" is "under the assumption of inconsistent axioms, every statement is vacuously true." -) $\quad$ P.S. This answer might also be of interest: Is every statement of an inconsistent theory true? $\endgroup$
    – ryang
    Commented Sep 26, 2023 at 12:31

1 Answer 1

Reset to default
2
$\begingroup$

I think this is just s function of how we talk about theorems in mathematics in general. That is, whenever we prove something in mathematics we regard that theorem to be ‘true’ … though you are right in that what we really do is ‘merely’ show that that theorem is a consequence of the axioms we are assuming.

In other words, we go from ‘is a consequence of the axioms that we use to define arithmetic’ to ‘is true in the world of arithmetic’ to ‘arithmetically true’ … to just plain ‘true’.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .