4

Examples of use/ mention fallacies one often encounters are toy examples, such as :

(1) The Morning Star is Venus. (2) Venus is a 5 letters word. (3) Therefore the Morning Star is a 5 letters word.

(1) Not every body knows oneself. (2) Therefore, not everybody knows oneself.

It is sometimes alledged that " Many logical fallacies arise due to the confusion of use and mention, even though the topic is pretty simple". (https://cs.lmu.edu/~ray/notes/usemention/)

Could more serious examples of use/mention fallacies be pointed to?

I mean, examples of reasonings that really appear as valid.

  • 1
    Haddocks' Eyes – Mauro ALLEGRANZA Nov 18 '20 at 10:13
  • 1
    In real life contexts it is hard to imagine that a speaker cannot manage the distinction between using a word to speak about the world and using it to speak about the word itself. – Mauro ALLEGRANZA Nov 18 '20 at 10:14
  • There is no fallacy by the name use / mention. What you are describing has a name already so there is no need to invent one. The fallacy you are describing is named EQUIVOCATION. There is also another phrase used which is the fallacy of four terms. This means you can't use a word in more than one context & then claim that you have reasoned correctly. You seem to be trying hard to make the reasoning deductive BUT you are copying style here. If you are analyzing an argument then YOU MUST use correct form & provide all premises hidden or not. There should be no "you will get it" type of stuff. – Logikal Nov 18 '20 at 17:44
  • 1
    @Floridus Floridi As a practical example of a failure to observe the distinction having bad consequences, take a look at this: itre.cis.upenn.edu/~myl/languagelog/archives/005349.html – Bumble Nov 19 '20 at 9:29
3

I will give an example from a recent paper That We See That Some Diagrammatic Proofs Are Perfectly Rigorous by Azzouni. It has been a common opinion since Pasch and Hilbert that proofs relying on diagrams, such as proofs in Euclid, lack in rigor because certain assumptions used in them are left implicit in the proof, and do not come up in the course of it. They are "wrapped into" the diagram, as it were. But diagrams can be misleading (All Triangles Are Isosceles fallacy is a typical illustration since Klein), the argument goes, so this tacit reliance constitutes a logical gap.

But diagrammatic proofs do not amount to visual reports about the looks of a diagram. There are rules in place for inferring from diagrams, just as there are rules in place for inferring from strings of symbols in formal proofs, which are held up as the standard of rigor. And those rules are also implicit, they are metamathematical. And if they are broken by oversight (e.g. if one confuses a conditional with its converse or reuses a variable in a universal instantiation) fallacies result.

Azzouni argues that the idea that diagrammatic proofs are non-rigorous is a case of the use/mention error. That some geometric assumptions are not mentioned explicitly when manipulating diagrams is taken to mean that they are not used there either, and there is a gap. The reason this error is appealing is that mathematicians generally tend to convert implicit metamathematical rules into explicit mathematical axioms, which creates the impression that something was missing before it was done. But the "missing" content actually has no effect on rigor. As an illustration, Azzouni uses the pictorial proof of the sum of the geometric series by subdividing a unit square into rectangles, see Proofs without Words:

"A charge that’s more likely to be sustained is that something (a great deal, maybe) in the diagram is tacit. In particular, what’s tacit are the various conventions about (two-dimensional) space that the success of the diagram is trading on: that divisions of the square correspond to the fractions of 1 that are so labeled in the diagram.

[...] So my suggestion is this: the respect in which the mathematical content in a visual proof procedure is tacit amounts only to the fact that the content in question is used and not mentioned in that proof procedure. I’m also suggesting (and I admit this may seem odd) that the tendency to see diagrammatic methods as not rigorous (apart from worries about generality described in §2) ultimately boils down to a use/mention error.

The way the error proceeds is this: One senses, or is otherwise aware of, the substantial mathematical content being used in a diagrammatic proof. One then assumes, since this content obviously isn’t explicit (isn’t explicated by axioms, for example), that the proof in question must involve missing steps involving that content. But this isn’t true if that content is playing a role enabling the proof procedure (if that content is actually metamathematical).

When one thinks mathematical content is missing from these diagrammatic proofs, one mixes up use and mention: the only thing missing is meta-mathematical content. But meta-mathematical content is also missing from ordinary language proofs—which almost everyone agrees are rigorous. Related to this point is another one. We tacitly use our visualization faculties to appreciate the proof-theoretic elements in diagrammatic proofs. What goes unremarked is that we use exactly the same visualization faculties (tacitly) to appreciate the proof-theoretic elements in language proofs as well."

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.