# 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**.