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Logic Intro

Updated: Dec 10, 2022

In your time studying mathematics, you will undoubtedly come across some Mathematician and Philosopher bickering over which subject gets custody of the field of Logic. (It gets heated, so I wouldn't stick around for long.) Personal allegiances aside, its definition can be a bit shaky as a result of this dispute, so I would encourage you to think of it like this:

Maths concerns itself with truths, and Logic is how we connect those truths together to find new truths. Logic is all about proofs: the justifications and demonstrations of mathematical deduction.

Think about it - how do we know anything is true? If we assume that we have some given truths (called axioms in maths), and that statements can only be true or false, we prove something is true by connecting it to those given truths. Specifically, we use propositional logic to connect logical propositions.

So let's start with that idea of truth.

To build truths, we need to be able to reference them, and make conditions from them. When we try to communicate a statement in Maths, we use symbols to save time when writing.

For example,

ø= "the sky is grey"

Ω = "it is raining"

But what if we want to communicate more than one in a new statement? We can use the combinators "AND","OR", and "NOT". You will often see these represented using symbols like ^ v, and a hockey stick lookin' thing, but I would stick to words for clarity.

AND is a conjunction (ø^Ω) that expresses that statements ø and Ω are both true. If the order was flipped, the statement would still be true: a conjunction is commutative. ø and Ω are conjuncts of the conjunction ø^Ω; if they are both true, then the statement ø^Ω will be true, and vice versa.

ø^Ω = the sky is grey AND it is raining

OR is a disjunction (ø V Ω) that expresses that either statements ø and Ω are both true, or one is true.Unlike the English or, Maths uses an inclusive OR. That is, for the statement "ø V Ω" to be true, both don't have to be true.

NOT is a negation (¬ø) of a statement. The negation of a statement cannot be true when the statement is true, and vice versa.

For example

¬Ω "it is NOT raining" cannot be true if it is raining.

It may seem pedantic, but this rigour helps when we want to tackle more complicated problems.

Try making a few yourself from sentences you know!

See you next time.

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