Proof by Contradiction

Theses

All ideas in the mind are known as Statements. When they are expressed verbally or by script, they are called sentences. Not all ideas are expressed like this, though; some are expressed as body language while others as incoherent noises, and so forth. Some statements, in fact, are never expressed at all. As an example, the idea that “the sky is above us” is a Statement and the text you see on your screen is the expression of that statement as a Sentence.

Now some statements have an extramental reality to which they are associated while others do not. For example, the statement “the sky is above us” is associated to the extramental nature of the sky and its being in the position that it is relative to us. The real sky – not the concept in our mind – is the reality associated with the statement above. Statements that are of this nature are called Propositions or Theses. On the other hand, examples of statements that have no such associated reality are as follows: “is the sky above us?” “come here,” “I wish I was younger” and “what a day!” None of these statements have an associated reality, and notice that they do not claim, posit or assert anything; they do not put forward any claim, unlike Propositions/Theses.

Theses, then, are either true or false. They cannot be neither (Principle of Bivalence) and they cannot be both (Principle of Contravalence). We say that a thesis is true if it correlates to the extramental reality to which it is associated, and false if it does not. For example, the sky is, in actual fact, above us (relative to our orientation on the planet); therefore, we say the statement “the sky is above us” is true.

A point we’d like to make before moving forward is that Theses are statements that make a claim, so they exclude questions, commands, expressions of desire, etc as mentioned hereinabove. But what is also excluded from Theses is nonsensical statements like “this sentence is false.” Notice the recursive, and thus, nonsensical nature of this statement.

Antitheses

An Antithesis (an-ti-tha-sis) is the opposite (or contradiction) of a thesis. More precisely, a statement S is the antithesis of a statement T if both are exactly the same except that the affirmativeness of both is different. For example, the statements “I am alive” and “I am not alive” are contradictory because, all other things being the same, their affirmativeness is different.

It is crucial that all other aspects of the statements be the same. For example, a popular objection against contradictions is the idea that light is both a particle and a wave. So we could say that light is both a particle and not a particle. Granted these statements have opposite affirmativeness, but they are not exactly the same in all other aspects. In reality, light is observed to be a particle under certain experiments, and a wave under other experiments; it is not both simultaneously, as is commonly misconceived.

Some have identified eight aspects in which two statements need to match before their differing affirmativeness can constitute contradiction:

1.       the subject of both statements must be the same

2.       the predicate (i.e. the thing being claimed) must be the same

3.       the place must be the same

4.       the time must be the same

5.       the conditions and circumstances must be the same

6.       both must be in an actual sense or both in a potential sense

7.       both must be talking about either the whole or the part, and if the part, then the same part

8.       relation: all tertiary attributes must be the same

A degree of care must be given in forming the antithesis of a thesis. Let’s consider an example where things can go wrong: “every animal is dangerous.” One may inappropriately form the contradiction of this by saying “every animal is not dangerous.” This is incorrect; forming the contradiction is not as simple as replacing an affirmative copula with a negative one or vice versa. In the thesis above, we are claiming that, given any animal, it will be dangerous. Now the antithesis above is claiming that, given any animal, it will not be dangerous. This is not a proper antithesis. The correct antithesis is “not every animal is dangerous;” in other words, “some animals are not dangerous.”

In the incorrect antithesis (every animal is not dangerous), we made a claim against all animals (are not dangerous) that was the opposite of the claim made against them in the thesis (are dangerous). All we did was contradict the claim, not the statement. Contradicting the statement properly involves incorporating all quantifiers, modalities and compound structures. For more information on how to form a contradiction, visit Indicated Assent.

As a final note of caution, notice that a thesis doesn’t need to be affirmative. We can start with a negative thesis, in which case its antithesis will be affirmative. Moreover, theses can be compound statements, they can be quantified, they can be moderated, and so forth; they do not have to be the simple kinds of statements we’ve been working with so far.

Principle of Non-Contradiction &
Proof by Contradiction

The Principle of Non-Contradiction states that a thesis and its antithesis cannot both be true. Similarly, the Law of Excluded Middle states that either a thesis must be true or its antithesis, and there is no third option (i.e. they cannot both be true, both be false, both true and false, neither true nor false). What these laws are saying , essentially, is that a thesis will be true and its antithesis will be false, or a thesis will be false and its antithesis true, period.

This is an axiom of classical and modern logic. It is a first principle which is very obviously and clearly true and it is used extensively in proofs and in building higher-than-first principle laws. It needs no proof of its own. No rational human denies the veracity of this axiom.

The Principle of Non-Contradiction is the basis of a form of proof known as Proof by Contradiction. This is an air-tight method of deductive reasoning that has been used for thousands of years and its veracity can indeed be taken for granted. It is utilized when we do not, or cannot, prove directly that a proposition T is true. What we do instead is assume that the antithesis of T – let’s call it S – is true and we then look for contradictions or absurdities – things that just cannot be true, no matter what they may be. Once we find an absurdity, we know that our assumption of S being true was incorrect; S is actually false given the absurdity. By the Principle of Non-Contradiction, if S is false, the thesis T must be true. And we have thus proven that T is true.

As a very simple example, let’s try to prove the true claim “I am taller than 5 feet.” We can do this by measuring my height, or etc. But let’s employ proof by contradiction for the sake of pedagogy. Let’s take a countertop which is 5 feet above the ground. We will say: Assume the antithesis–that I am 5 feet or shorter. The countertop is known to be 5 feet above the ground. Since I am no more than 5 feet, I should not be able to see above the counter. Yet I can indeed see above the counter. We have a contradiction. So the assumption that I am no more than 5 feet tall is false. And this means that “I am taller than 5 feet” is true.

Veracity of the Principle of Non-Contradiction

There is no real proof for the veracity of the Principle of Non-Contradiction because it is so axiomatic and basic. It is something that is – or at least should – be grained in every human’s mind as incontestable truth. So instead of proof, we rely on its obviousness. If someone does not vie for its veracity, we cannot engage in any rational discussions with them and we have no common ground for discourse with them. This axiom is something that must be agreed on before any rational discussions can proceed.

If someone does contest the veracity of this axiom, then it is a thing of barbarism and incivility on their part. Recall how exactly we form the antithesis of a thesis; we take the thesis and apply negation to it. We can then optionally move the negation closer into the root of the thesis if there are quantifiers and modalities. So if someone denies that an antithesis and thesis are contradictory, what they are doing, in essence, is denying the meaning of negation (“not”). This is not a denial of any rational argument; it is a denial of semantics and observation. And what can be said of a person who does this?