Lessons in Logic: Conjunction

When two things join together without a malfunction,
You’re observing the signs of a perfect conjunction

A conjunction in logic is essentially the same thing as a conjunction in language. It’s an operator which creates an “and” statement. It came up briefly in the introduction with the premise “The cat is on the mat and the cat is a brown tabby”. Well that “and” is the operator, that joins the two propositions together. It’s featured in a lot of arguments that we make, and is often symbolized as \land or •. 

The most important things to know about a conjunction is that it can only join two propositions together, and that it can only be true if both of the propositions are true. For the premise “Dinner is delicious and dinner is lasagna” or D \land L to be true, each of those things has to be true separately. In natural language, we’d condense that to “Dinner is delicious lasagna”, but because there’s two facts, two propositions are needed. The two propositions don’t have to be related, either. Dinner might be delicious lasagna, but the statement “Snow is cold and the moon is in space” is also true, even though there’s no connection between one and the other. All that matters about is whether or not each of them is true separately.

The truth of the whole statements is dependent on both of the propositions being true, though. “Snow is cold and the moon is in space” is only true if both of its parts are true. If we look at “Steam is cold and the moon is in space”, steam isn’t cold, so the statement as a whole is false. The truth table for a conjunction reflects this, showing that it can only be true if all of its parts are true.

Conjunctions can interact with other operators, like negation. We could say “Steam is not cold and the moon is in space”, and symbolize it as ~C \land M, applying negation to only a single proposition in the premise. Or we can negate the entire statement by bracketing it like this: ~(C \land M), which would mean “It is not the case that steam is cold and the moon is in space”. This is useful because it lets us indicate when two things cannot be true together, such as “It is not the case that the light is on and the light is off.” A light can’t be both on and off, it has to be one or the other.

Conjunction is also used to express one of the laws of logic, the Law of Noncontradiction, which states that ~(A \land ~A). It cannot be true that A is the case and ~A is the case simultaneously. For example, if we let A stand for “the light is on”, it can’t be true that the light is both on and not on. It’s a contradiction. I’ll talk about contradictions more when we get into rules of inference. For now, thanks for reading, and I hope you’ve enjoyed today’s lesson in logic.


Leave a Reply

Your email address will not be published. Required fields are marked *