Statements, Logical Connectives, and Logical Relations

Statements are the fundamental units of arguments and proofs in logic. These tutorials explain how to identify statements and introduce some of the basic ways that statements may be related to one another.

Complete the exercises and check your answers.

Logical connectives

Here are a few basic concepts in logic that you ought to be familar with, whether you are studying symbolic logic or not.

§1. Negation

The negation of a statement α is a statement whose truth-value is necessarily opposite to that of α. So for example, for any English sentence α, you can form its negation by appending "it is not the case that" to α to form the longer statement "it is not the case that α".

In formal logic, the negation of α can be written as "not-α", "~α" or "¬α".

Here are some concrete examples:

Statement (α) Negation (¬α)
It is rainingIt is not the case that it is raining (i.e. It is not raining.)
1+1=2It is not the case that 1+1=2 (i.e. 1+1 is not 2.)
Spiderman loves MaryIt is not the case that Spiderman loves Mary.

There are two points about negation which should be obvious to you:

  • A statement and its negation can never be true together. They are logically inconsistent with each other.
  • A statement and its negation exhaust all logical possibilities - in any situation, one and only one of them must be true.

Exercise #1

§2. Disjunction

A disjunction is a kind of complex sentence typically expressed in English by the word "or", such as:

Either we meet tonight, or we do not meet at all.

The sentence has the structure of "either P or Q", where P and Q are statements

In logic, we often make a distinction between exclusive disjunction and inclusive disjunction.

According to the exclusive interpretation, "P or Q" is true when P is true, or when Q is true, false when P and Q are both true, and also false when P and Q are false. Many people take the exclusive interpretation to be what is intended in for example "You can have tea or you can have coffee", where it is supposed to be implied that you can only have one or the other but not both.

On the inclusive interpretation, "P or Q" is false when P and Q are both false, and it is true in all other situations, including when both P and Q are true.