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?
In formal logic a Contradiction "consists of a logical incompatibility or incongruity between two or more propositions".
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)).
- the act of going against; opposition; denial
- a declaration of the opposite or contrary
- 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.
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?
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)