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The proofs of mathematical statements of the form "If P then Q" usually start with the form:

Assume P...then Q.

What is meanty by "Assume P"? Should we consider P as a true statement? What if P is something like "3 is a negative number"? Should I assume that this statement is true and continue my proof? I was thinking to say that the above statement is true because the premise is false. So it makes sense to "assume" only when P is a possible case like "-3 is a negative number". But this depends on the structure of the statement and in general we must know if there are case where the premise is true. Is there any rigorous definition of what "assume" means in the field of mathematics/logic?

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  • Proofs are independent of what is true or false, they are chains of inferences according to rules that utilize axioms and previously proven statements. "Assume P" means that you are adding P to the utilizable statements for the purposes of proving Q. Whether it is true, or can be true, is really moot, although it is helpful intuitively, and is often done, to think of it as "true" for the sake. This should not be too unsettling, we think of time travel as "true" for the sake of reading The Time Machine, and put its logical problems to one side until the reading is done.
    – Conifold
    Jul 24, 2020 at 4:48
  • See Assertion vs Assumption. The difference is highlighted with symbolism: with ⊢ P → Q I'm asserting formula P → Q, i.e. I've a proof of it. With P ⊢ Q I'm asserting that I've a proof of Q from assumption P. Jul 24, 2020 at 6:24
  • The prototypical example is Euclid's Elements system of axioms for geometry: we assume five postulates regarding primitive geometrical entities (as well as five additional "general" axioms) and we use them to prove theorems. Jul 24, 2020 at 8:09
  • Adding on what Mauro said, if you assume the fifth postulate is NOT true, you don't get contradictions, you simply get different type of geometry. Math can describe many "realities" based on the starting assumptions. Some assumptions happen to match our reality rather well. (why is a diff topic). But that doesn't prevent mathematicians from studying DIFFERENT realities.
    – GettnDer
    Jul 28, 2020 at 4:07
  • If some of the answers below satisfies you, please accept it. Sep 7, 2020 at 14:12

4 Answers 4

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When someone begins a proof with a statement 'Assume P', what they are really doing is, in effect, creating within their imagination an artificial world, in which everything about the real world holds, plus also 'P' holds. IMPORTANT: This is generally (I am tempted to say always) done in situations where the prover -does not know- whether 'P' really holds or not (i.e the prover generally doesn't start by assuming things like '3 is a negative number' are true).

Anyway, having created this imaginary world, the prover then performs derivations within that world, eventually arriving at 'Q'. So the truth of 'Q' necessarily follows from the truth of 'P' (if it is true). The prover then takes a giant step: From the fact that 'Q' follows from 'P' inside this imaginary world, the prover concludes that the compound statement 'P->Q' must hold unconditionally in the real world! And it is this compound statement which the prover is really after!

Once the prover has arrived at 'P->Q', this statement may then be applied in other contexts where it is known that 'P' actually does hold; the prover may then conclude that in these contexts, 'Q' actually does hold also.

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  • 1. Why is it a "giant step"? - 2. Why is it "this compound statement which the prover is really after"? Why or when is it not just Q? Jul 24, 2020 at 18:48
  • This is the only correct answer. Excellent and keep it up!
    – user21820
    Jul 31, 2020 at 9:05
  • @Speakpigeon: It's exaggeration to say "giant step", but it is important to understand that (as PMar said) what you actually prove in the end is the compound statement that is the implication (P⇒Q) rather than just Q. Never is it "just Q". And if you could prove P earlier, you wouldn't have said "assume P" but simply "P".
    – user21820
    Jul 31, 2020 at 9:07
  • @user21820 The question is about assuming P. Once P is assumed, any proof that P implies Q proves Q. If you use the prover to prove P implies Q, then you don't need to assume P to begin with. So, you assume P and you prove Q. The proof of Q is the truth of P implies Q, and the proof of the truth of P implies Q is whatever you believe is a proof of it. In mathematics, P would be the axioms. Q would be some theorem logically derived from the axioms. And P implies Q would be some demonstration accepted by mathematicians as a proper proof that the theorem is true given the axioms. Jul 31, 2020 at 15:30
  • @Speakpigeon: You're clearly unfamiliar with actual mathematical reasoning, nor Fitch-style deductive systems for FOL, where "assume" is used to construct subcontexts exactly as described by PMar. I've nothing else to say here, since I know logic very well and don't need you to tell me what you think you know.
    – user21820
    Jul 31, 2020 at 16:58
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If P then Q means that if, hypothetically, P were true, then that would imply that Q is also true. It doesn't matter whether or not P is true. We're trying to prove that if P is indeed true, then it will result in Q being true as well.

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    You are not addressing the question. Jul 24, 2020 at 14:32
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In Natural Deduction, if a conditional statement of the form P=>Q (that is "P implies Q") may be (syntactically) proven, then this it usually done by demonstrating that: Q can indeed be derived under the assumption of P.

"Assume, for the sake of argument, that P is true.   Under that assumption, Q may be derived using such-and-such valid inferences.   Therefore proving that P=>Q."

|  |_ P      Assumed
|  |  :      Some valid inferences
|  |  Q      Derived somehow.
|  P => Q    Deduced (via rule of '=> Introduction')
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    You are not addressing the question. Jul 24, 2020 at 14:30
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The following considerations will answer your questions.

  • The proposition is the foundational element in logics. A proposition is normally the starting point for any propositional logics procedure. Propositions are true or false. Nevertheless, there's no formal definition of the expression "assume" in logics, since it is equivalent to "suppose", "think", "imagine", etc. In all such cases, a proposition is stated, in order to proceed to the calculus. The expression used to introduce it ("assume that", "let's say"..., etc.) is not of importance.
  • Logical propositions don't need to correspond to reality. "Assume 3 is a negative number" is a valid logical proposition. If you ask "will the operation (3+3+3)/3 be also negative?", the logical process is sound (and the result is true). Although it does not correspond to the reality we normally experience.
  • Assume P effectively means that P is true, and it is the starting point of the process.

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