If I know one thing, it’s that I know nothing

Ever had someone say something that doesn’t seem to make sense? Something like:

“This is the beginning of the end”

“If I know one thing, it’s that I know nothing”

“Less is more”

“I always tell the truth, even when I lie”

Well that might be because what you’re hearing is an example of a paradox. A paradox is a statement which is just wrong. It is a statement that despite apparently valid reasoning from true premises, leads to seemingly self-contradictory or logically unacceptable conclusions. Sounds overly complicated right?

But don’t despair, as mathematicians have figured out a way to help us identify these statements using Logic theory. In logic theory, sentences and words in everyday language are represented by mathematical notation. This blog post is aimed at giving a brief introduction to this area of mathematics and how it might relate to paradoxes and contradictions

A conjunction or “and” is represented by $\wedge$ and replaces the word “and” in a sentence. Let’s say $P$ and $Q$ are statements then $P \wedge Q$ is called a conjunction. For example, “I like pasta and that dog is black” are two statements “I like pasta”, “that dog is black” joined by a conjunction or an “and” so you could write this in logic notation:

$P:$ I like pasta

$Q:$ that dog is black

$P \wedge Q$ : I like pasta and that dog is black

A disjunction or “or” is represented by this symbol $\vee$ and replaces the word “or” in a sentence. So with the statements $P$ and $Q$ above, $P \vee Q$ is called a disjunction and the equivalent English statement would be

$P \vee Q$ : I like pasta or that dog is black

A negation or “Not” is represented by this symbol $\neg$ and replaces the word “not” in a sentence. So with the statement $P$ above, $\neg P$ is called a negation of $P$ and the equivalent English statement would be

$\neg P:$ I don’t like pasta

So how does this all relate to paradoxes or contradictory statements? Well let’s consider the scenario where I said a statement $P$ which was true. Such as $P:$ “I like cats“, and in reality I do really like cats – here’s a picture of my cat below:

Well then if I now said the negation of this statement $\neg P:$ “I don’t like cats”, then this statement would be false. So in general we can say that the negation of a true statement is a false statement.

How does this idea of a true and false statement now relate to conjugations? Well If I say $P \wedge Q$ then for this statement to be true, both statements $P$ and $Q$ have to be true. For example, let’s so I told you I don’t like dogs, and then said the statement:

“I like cats and I like dogs“

Well you’d know in this case that this statement is false because the second half of the statement is false so for a conjugation to be true, we need both statements either side of the conjugation to be true, i.e. both $P$ and $Q$ need to be true for $P \wedge Q$ to be true.

So how does this relate to paradoxes? Consider the statement

$P \wedge \neg P$