You should study Logic

If you are intrigued by logic and have a bachelor’s degree in Mathematics, Philosophy, Computer Science, Linguistics or a related subject, you should consider applying to the Master programme in Logic at University of Gothenburg.

You should study logic iff this sentence is true.

Note that “iff” is a short-hand for “if and only if”. Let’s name the sentence $A$ :

$A$: “You should study logic iff this sentence is true.”

Now, “this sentence” refers to the statement $A$ so we may rewrite $A$ as:

$A$: “You should study logic iff $A$ is true.”

If $A$ is true then “$A$ is true” is true and therefore “you should study logic” is true. But even if $A$ is false “you should study logic” has to be true. Let’s see if we can make sense of this.

Using logic

We can divide $A$ into two less complex statements combined with the operator $\leftrightarrow$:

$A$: “You should study logic” $\leftrightarrow$ “$A$ is true.”

Since $A$ and “$A$ is true” has the same truth conditions we can simplify and rewrite it as

$A$: $B \leftrightarrow A$

where $B$ is “You should study logic.” As $A$ is the statement $B \leftrightarrow A$ the proposition

$A \leftrightarrow (B \leftrightarrow A)$

is true (even without knowing if $A$ or $B$ are true or false.)

Propositional logic is the basic logic used to analyse complex statements in terms of atomic statements. In this case the atomic statements are $A$ and $B$ and these can be either true (T) or false (F):

$$A$$	$$B$$	$$B \leftrightarrow A$$	$$A \leftrightarrow (B \leftrightarrow A)$$
T	T	T	T
T	F	F	F
F	T	F	T
F	F	T	F

The only two cases in which $A \leftrightarrow (B \leftrightarrow A)$ is true is the first and third row. In both of these situations $B$ is true (note that $A$ can be both true and false).

Therefore, in all circumstances $B$ is true and we have proved (without any assumptions) that you should study logic.