Logical Contradiction

Updated: 21 hours ago

By KP Mohanan and Tara Mohanan

Look at these pairs of sentences:

a) Zeno is tall. Zeno is not tall.

b) It rained in Pune on 26-4-1976. It didn’t rain in Pune on 26-4-1976.

c) Oil floats in water. Oil doesn’t float in water.

In each case, if one of them is true, the other is false. Take the pair in (c). “Oil floats in water,” and “Oil doesn’t float in water,” can’t both be true. Such pairs of statements where one negates or contradicts the other are said to be logically contradictory.

So, a combination of statements forms a logical contradiction if its component statements are logically contradictory. Thus, “Oil floats in water and oil does not float in water,” forms a logical contradiction.

The statements, “Zeno is tall,” and “Zeno is fat,” are not logically contradictory because they can both be true at the same time: Zeno may be both tall and fat. Likewise, “Zeno is not tall,” and “Zeno is not short,” are not logically contradictory because they are both true if Zeno is neither tall nor short.

Take the statements “Apollo killed Zeno,” and “Zeno is alive.” Are they logically contradictory? The answer depends on what we believe about the world. To see why, consider what “X killed Y” requires: For any X and any Y, if X killed Y is true, then it is true that (a) X did something to Y, and (b) as a result of the action, X died.

Hence, from “Apollo killed Zeno,” we infer that Zeno died. In the world we live in, if a person dies, that person cannot be alive any more. If Zeno died in 135 BCE, he cannot be alive in 2016. Hence, from “Zeno died,” we infer that Zeno is not alive. So together, the two statements, “Apollo killed Zeno,” and “Zeno is alive,” yield the logically contradictory inference that Zeno is not alive and Zeno is alive.

Suppose we live in a world where people who die (and hence cease to be alive) can come back to life. In this world, for instance, Zeno can die and come back to life subsequently. In such a world, the statements “Apollo killed Zeno,” and “Zeno is alive,” are not logically contradictory.

How about the statements, “Zeno is an Indian,” and “Zeno is a Pakistani.” Are they logically contradictory? That depends on what we mean by the words ‘Indian’ and ‘Pakistani’, and on the states of affairs in the world. Suppose by ‘Indian’ we mean, “One who holds an Indian passport.” And by ‘Pakistani’ we mean, “One who holds a Pakistani passport.” In today’s world, no one can simultaneously hold an Indian passport and a Pakistani passport. So, the statement that “X is an Indian and X is a Pakistani” forms a logical contradiction.

Here are the relevant premises (P), steps of reasoning and conclusions (C) that yield this result:

Assertion: Zeno is an Indian and Zeno is a Pakistani.

P 1: An Indian is one who holds an Indian passport.

P 2: A Pakistani is one who holds a Pakistani passport.

C1: Given P1 and P2, it follows that Zeno holds an Indian passport and a Pakistani passport.

P 3: No one can simultaneously hold both an Indian passport and a Pakistani passport.

C 2: Given C1 and P3, it follows that Zeno does not hold both an Indian and a Pakistani passport.

Result: C1 and C2 are logically contradictory.

But now let us imagine that by 2030, the governments of India and Pakistan develop a relationship that allows people to simultaneously hold an Indian passport and a Pakistani passport. Given that state of affairs in the world, “Zeno is an Indian and a Pakistani” wouldn’t form a logical contradiction.

What if we use a different meaning for the word ‘Indian’? Suppose by ‘Indian’ we mean, “One who has a sense of personal identity with the peoples of the Indian subcontinent.” Given this situation, does “Zeno is a Pakistani and Zeno is an Indian,” form a logical contradiction? It can be the case that Zeno has a Pakistani passport and hence is a Pakistani, and has a sense of personal identity with peoples of the Indian subcontinent, and hence is an Indian. Since being a Pakistani does not exclude being an Indian, “Zeno is a Pakistani and Zeno is an Indian” is not a logical contradiction.

This set of examples reveals an important point - Given two statements A and B (that are not logically contradictory by themselves) If:

(i) A and B are part of a larger complex of statements; (ii) the presence of both A and B in that complex derives conclusions C-1 and C-2; and (iii) C-1 and C-2 are logically contradictory;

then: combining A and B results in a logical contradiction.

Here is an exercise. Consider the following set of statements. Do they result in a logical contradiction?

1) Micro-organisms are organisms that are invisible to the naked eye. 2) Fungi are microorganisms. 3) Mushrooms are fungi.

Figure this out on your own.

Photo by Holly Chisholm on Unsplash