# Does this make sense? Is it a contradiction? If a triangle has 2 sides, then it wouldn't be a triangle

"If a triangle has 2 sides, then it wouldn't be a triangle." I know there is something seriously wrong with this statement, but what exactly is it? Is the statement true? You cannot suppose that a triangle has 2 sides? I mean, it is supposed, it's not illegal to do so...any help would be appreciated, thanking you in advance.

• You cannot "analyze" it in purely propositional logic. In order to consider the correct "logical form" we need predicate logic, that is able to express the definition: "(a plane figure) x is a triangle iff x has three sides". – Mauro ALLEGRANZA Sep 25 '17 at 6:10
• From the def, we derive: (∀x) if x is a triangle, then x has three sides" as well as (∀x) if x has three sides, then x is a triangle". Thus, to assert: "x is a triangle and x has two sides (i.e. x does not have three sides)" is a contradiction. – Mauro ALLEGRANZA Sep 25 '17 at 6:14
• @MauroALLEGRANZA the question is if the whole sentence is a contradiction. – viuser Sep 25 '17 at 6:30
• See Ex falso quodlibet: in "classical logic [and] intuitionistic logic any statement can be proven from a contradiction." Thus, according to the above "analysis", the inference: "if x is a triangle and x has two sides, then x is not a triangle" is correct. – Mauro ALLEGRANZA Sep 25 '17 at 7:24
• So it's like saying if (x is a triangle & x has 2 sides) and also (x is a triangle and not a triangle) there are 2 contradictions! thanks for your help! – Rob Hv Sep 25 '17 at 9:12

## 5 Answers

It's about using the right words. You wouldn't say "If a cat barks, it isn't a cat". Presumably, you'd say "If an animal barks, it isn't a cat".

Therefore, a statement should be "If a shape/an object has 2 sides, it isn't a triangle". It might not be the answer for the question, but i think that kind of statements shouldn't be used whatsoever.

If we interpret “if A then B” as “it can never be the case that A is true and B is false” and reasonable infer from the definition of “triangle” that a triangle being two-sided can never happen, then your sentence is true.

Because if A is always wrong, this alone makes “if A then B” true.

Curiously, with this interpretation, also the following sentence:

If a triangle has 2 sides, then it would be a triangle.

would be true.

The question is of course, if we should interpret “if … then” in this way and not, as it is common, assume some stronger sort of dependence. But what could this dependence which goes beyond the purely logical look like in this case? So from a common-sense standpoint your sentence would simply be nonsensical, but still not a contradiction.

• "If a triangle has 2 sides, then it would be a triangle" is true. Two sides/segments define a triangle. – Randy Buchholz Sep 28 '17 at 4:37

There a multiple defines/conditions for triangle. A triangle can be defined by a minimum: Three points A line segment and a point Two line segments

Three sides is extraneous.

The statement makes sense, but is wrong. Two line segments defines a triangle with two sides.

A "complete" triangle is three line segments or sides. But if we are talking about complete triangles, the problem is the statement doesn't say "only two" sides. All "full" triangles have two sides (and one more).

The issue seems to express linguistics. I would also say semantic as far as philosophy goes. Mathematics does not look at context but like science looks at a result and only cares about the result.

The term triangle is analytically defined as a shape with exactly three sides. I don't know the verbatim definition but trianlges certainly can't have less than tree sides and more than three sides. When words are defined this way it is called logically necessary in case of true propositions. Thus triangles always have three sides is impossible to be false and must be true by definition alone. When we have a definition that always makes a proposition false that is called self contradictory. The proposition "all bald men have hair" is forever false. So in light of this, the claim a triangle having two sides is self contradictory. The same would be a proposition "I am married to myself" or "I am a married bachelor". The definition of the words in language alone is enough to determine the truth value of these types of propositions. We don't need scienctific techniques to discover the truth of these types of propositions.

The problem you're having with that statement is that it's ambiguous.

I suggest the follow two possible interpretations of your statement, one of which is impossible, one is completely valid. I believe to understand it, you need to specify which meaning you mean.

1. There is a triangle, it has two sides, therefore it is not a triangle.

Something cannot be both A and NOT A.

1. There is something which is believed to be a triangle, it has two sides, therefore it is not a triangle.

Something can be both "believed (falsely) to be A" and "NOT A" without any logical paradox.