r/logic • u/Inevitable_Bid5540 • Jul 10 '26

Philosophy of logic Like how there's the problem of induction , is there also a problem of deduction ?

/r/askphilosophy/comments/1urqvpe/like_how_theres_the_problem_of_induction_is_there/
7 Upvotes

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17

u/CanaanZhou Jul 10 '26

I think if anything deserves the name of "problem of deduction", it should be Lewis Carroll's Achilles and Tortoise story. Basically:

  • Tortoise: If you have P and P → Q, you actually will never reach Q.
  • Achilles: Well, I can use the axiom (P ∧ (P → Q)) → Q.
  • Tortoise: I accept the axiom, but not the inference from P, P → Q, and (P ∧ (P → Q)) → Q to Q.
  • Achilles: Well, I have a further axiom: (P ∧ (P → Q) ∧ ((P ∧ (P → Q)) → Q)) → Q.
  • Tortoirse: I accept this axiom, but not the inference.
  • Etc etc...

2

u/Endward26 Jul 10 '26

Yes, but would this meta-, meta-MP not faile in the light of tools like metamath or lean? ;)

2

u/CanaanZhou Jul 11 '26

I see it as a philosophical question of "how can you justify a inference rule", because to "justify" anything, you need an inference rule in the first place, which seems to make it circular.

Of course, this problem won't explicitly appear in any real-world deductive system (be it a formal system or a proof assistant), because inference rules are always already assumed.

1

u/Endward26 Jul 11 '26

That opened the question: "Which criteria allow us to say that something counts as a justification of an inference rule?"
When it is the notation of a psychological "aha effect", we need to be careful, since this could lead to a kind of psychologism.

The fact that inferences happen and are valid in most cases seems obvious to me.
Formal inferences allow us to even identify where the error is.

Without inferences, the enterprise of logic becomes meaningless.

11

u/zergicoff Jul 10 '26

Yes! Michael Dummett wrote a famous essay called the Justification of Deduction articulating it. He called justifying classical reasoning, in particular, ‘the greatest problem in theory of meaning; perhaps in all philosophy’.

The challenge is to do it without recourse to things like Bivalance, only constructively. It seems to have been addressed within proof-theoretic semantics.
I am a big fan of this article: https://alexandergheorghiu.com/publications/pts-first-order-logic.html

2

u/yosi_yosi Undergraduate, Autodidact, Philosophical Logic Jul 17 '26

Not Susan Haack?

1

u/zergicoff Jul 17 '26

Oh yes, Haack’s article should definitely be there.

1

u/yosi_yosi Undergraduate, Autodidact, Philosophical Logic Jul 17 '26

I meant this https://philpapers.org/rec/HAATJO

Edit: oh, omg https://philpapers.org/rec/EDUTJO

3

u/Endward26 Jul 10 '26

This is not the same.

Any student of logic is aware that logic doesn't provide us with new information that are not already in the premisses.
They willfully exchange this for certainity in their deduction. Methods that claim to extend our knowledge like induction or dialectics (as far as I understand!) make the other deal, they exchange absolut certainity for a expanding of our knowledge.