What is the right way to describe an implication whose premise is undefined?

Question

Am just starting in mathematical logic. I learned that if I have an implication $A \rightarrow B$, and $A$ is false, then $A \rightarrow B$ is true.

But what if $A$ does not make sense ? For example, suppose $f:\mathbb{N} \rightarrow \mathbb{N}$ is a partial function and it so happens that for some $c \in \mathbb{N}$, we have $f(c)$ as undefined. If I have the implication $f(c) = 0 \rightarrow 1=0$, I guess $f(c) = 0$ should not be treated as false right ? since there's no way to compare an undefined $f(c)$ with $0$... Is it correct to say that the implication is neither true nor false but simply does not make sense ?

The question is of type "What is the sum of an apple and a skyscraper". Type error, no answer possible. I vote to close. — Martin Brandenburg, Feb 07 '25 at 00:02
@MartinBrandenburg but you kind of understand how one could naturally come to a question like the OP's when starting with mathematical logic, right? It seems to me that your point should be part of the answer rather than a reason to close the question. — Timotej Šujan, Feb 07 '25 at 00:57
Related: Are ill-defined well-formed statements non-meaningful or merely false? — ryang, Feb 07 '25 at 02:05
The notion of "making sense" is too ambiguous for formal logic. For any function $f$, we say that $f(x)$ is undefined or indeterminate if and only if $x$ is not an element of the domain of definition of $f$. It works both ways in everyday mathematical proofs. — Dan Christensen, Feb 07 '25 at 16:16

score 3 · Accepted Answer · answered Feb 06 '25 at 23:43

3

If $f$ is a partial function then the right way to interpret "$f(c) = 0$" is "$f(c)$ is defined, and equals $0$." So if $f(c)$ is not defined the statement is just false, that's it.

In general if $A$ "does not make sense" then neither does $A \to B$. But here $A$ makes sense, correctly interpreted, and is false.

answered Feb 06 '25 at 23:43

Qiaochu Yuan

468,795

sorry, what I meant was suppose in a prior definition I described $f$ such that $f(c)$ is undefined, and then in a succeeding lemma I wrote $f(c) = 0 \rightarrow 1 = 0$, then the lemma is true, but I guess what I wrote as a whole should just be called "not making sense" or "inconsistent" ? – spacewindow Feb 06 '25 at 23:50
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@space: I don't see how that changes the situation. $f(c) = 0$ is false and so is $1 = 0$. So the implication is true but useless. – Qiaochu Yuan Feb 06 '25 at 23:52
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@spacewindow the trick is that a partial function being "undefined" at a particular input is not the same thing as a formula of predicate logic (or whatever logic you are working in) being non-well-formed. In set theory, a function $f$ is merely a set of ordered pairs, and to say that it is undefined at some input $c$ is to say that $\not\exists x [(c, x) \in f]$. But this does not mean that expressions invoking the function are not WFFs of predicate logic. Technically any time you're invoking a function (or any set) by name you have to interpret that as an abbreviation of some kind – M. Sperling Feb 06 '25 at 23:55
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@spacewindow but it may still be well-formed when you expand it out, even if the function is undefined at a particular input that you have given it – M. Sperling Feb 06 '25 at 23:56
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It would be a different situation if your definition of $f$ was in fact not well-formed in some way; that's not what "undefined" means. In that case you simply haven't defined a function, not even a partial one. – Qiaochu Yuan Feb 07 '25 at 00:00

What is the right way to describe an implication whose premise is undefined?

1 Answers1