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I know that (1=2 AND NOT 1=2) is a logical contradiction, but what about 1=2 by itself? Is it a logical contradiction, or merely a false statement? And what about something like NOT 1=1?

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    I know this is a banal comment, but you fist need to specify what you mean by "1", "=" and "2". To be more pedantic and annoying, you should say which logic you are working in (is it classical? etc.) – Alex Dec 15 '20 at 16:55
  • This is why (some) computer languages distinguish between assignment and comparison. Thus, in the Pascal language, the comparison 1=2 is a false statement, while the assignment 1:=2 is an impossibility. – RonJohn Dec 16 '20 at 7:32
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In formal logic a Contradiction "consists of a logical incompatibility or incongruity between two or more propositions".

More precisely, in logic we call contradiction a formula that is always false [related: tautology, a formula that is always true].

A simple example is a formula having the "logical form": A ∧ ¬A.

The formula ¬ (1=1) is a contradiction because it is always false: it is the denial of the equality axiom: x=x.

Regarding 1=2, it is not, because in order to refute it we need the Peano axioms, i.e. the first-order axiom for arithmetic.

In it we define: 2=s(1), where s(x) denotes the successor function, and we prove PA⊢∀n(n≠s(n)).

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    Strictly speaking, whether ¬(1=1) is a contradiction depends on whether you are treating identity as part of your logic, or as part of your theory and hence interpretable. – Bumble Dec 15 '20 at 20:34
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    We need to use the axiom 0 ≠ Sx. More precisely, by the successor injectivity axiom, 1 = 2 implies 0 = 1, i.e. 0 = S0. But by the aforementioned axiom, 0 cannot be the successor of any number. – user76284 Dec 15 '20 at 20:54
  • @user76284 Wait, if 0 ≠ Sx, then what have you done with -1? – Brilliand Dec 15 '20 at 22:13
  • @Brilliand -1 is an integer, not a natural number. I assumed we were working in the context of natural numbers. For integers, it would depend on how you're axiomatizing them. If you're encoding them as differences of natural numbers (the usual encoding), you can reduce it to the question of whether 1 = 2 in the naturals, and you'd apply the same argument. – user76284 Dec 15 '20 at 22:21
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    @MauroALLEGRANZA No difference. I was just explicating your answer. It was rather terse, but not incorrect. – Bumble Jan 3 at 10:46
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Contradiction

  1. the act of going against; opposition; denial
  2. a declaration of the opposite or contrary
  3. a statement that is at variance with itself (often in the phrase a contradiction in terms)

If the formula ¬(1 = 1) is a contradiction because it contradicts the formula 1 = 1, which it does, then each and every statement we can possibly think of is a contradiction because, well, obviously, it contradicts its negation.

Usually, we talk of a contradiction in two situations. First, somebody says something and then somebody else, or even the same person, says the opposite, thereby contradicting the first person or contradicting themselves. In this sense, a contradiction is an interaction between people whereby one goes against the other. This is not exactly the sense we are interested in here.

The other sense is when the same statement includes both an assertion and the negation of the same assertion, as indeed is the case in "1 = 2 and not 1 = 2".

Unlike "1 = 2 and not 1 = 2", the statement "1 = 2" is not a contradiction in itself. It is only a contradiction in relation to its opposite, i.e., "Not 1 = 2".

If we wanted to say that the statement "1 = 2" is a contradiction because it contradicts "Not 1 = 2", then we would have to say that "Not 1 = 2" is also a contradiction because it contradicts "1 = 2", and then the whole notion of contradiction would become trivial and uninformative because all statements would have to be called contradictions.

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Is 1=2 a logical contradiction, or merely a false statement?

“1 = 2” is a contradiction. The claim violates two laws of thought: the law of identity (something is what it is) and the law of noncontradiction (something cannot both be and not be).

If 1 = 2, then the very statement illustrates the problem. Assume 2 is equal to 1 + 1; then “1 = 1 + 1” says that 1 is equal to something different than itself, that it is equal to a number larger than itself.

The problem continues. It might be that 1 = 1 + 1, but there is no logical place to stop. The number one can further equal 1 + 1 + 1, and then continues to add 1 until 1 = n, where n is any positive integer. The final conclusion: 1 is equal to 1 + 1 + 1… to infinity.

Really, given the assumption that 1 = 2, there is no such thing as the number 2, because every positive integer is equal to 1. So the original statement, 1 = 2, collapses in on itself.

Plato summed it up:

SOCRATES: …. first, … nothing can become greater or less, either in number or magnitude, while remaining equal to itself—you would agree?

THEAETETUS: Yes.

SOCRATES: Secondly, that without addition or subtraction there is no increase or diminution of anything, but only equality.

THEAETETUS: Quite true.

Source: Plato, Theaetetus (Jowett, trans.) (Project Gutenberg)

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