Logic and propositional calculus is the branch of mathematics that deals with propositions, their truth value, and the rules by which they can be combined and manipulated. It is the foundation of all mathematical reasoning and proof.

Logic in mathematics differs from the everyday use of the word logic. In mathematics we work with precise rules for when a proposition is true or false. A proposition is a statement that is either true or false. The whole propositional calculus is built on this simple principle.

 

 

Propositional calculus as a system

We can combine propositions using logical connectives such as and, or, and not.

 

$$ \large p \land q \quad\; (\text{and}) $$

$$ \large p \lor q \quad\; (\text{or}) $$

$$ \large \lnot p \quad\; (\text{not}) $$

 

When propositions are combined, we can establish rules for the truth value in all possible cases. These rules are collected in truth tables.

Here is an example of a truth table for the conjunction \( \large p \land q \):

 

$$ \begin{array}{|c|c|c|} \hline p & q & p \land q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \\ \hline \end{array} $$

 

We can also extend the language with quantifiers, so that we can talk about all elements in a set or about the existence of at least one element. The universal quantifier expresses for all, while the existential quantifier expresses there exists:

 

$$ \large \forall x \in M : P(x) $$

$$ \large \exists x \in M : P(x) $$

 

 

Logical laws

Propositions can often be rewritten without changing their truth value. This is done using logical laws. Examples are De Morgan's laws, double negation, and distributivity. These rules make it possible to simplify complex propositions and find alternative formulations.

 

 

Logic in mathematics

Logic is used directly in mathematical proofs. A simple example is the statement that an integer is even if and only if it can be written as two times another integer. It can be expressed logically:

 

$$ \large n \text{ is even } \Leftrightarrow \exists k \in \mathbb{Z} : n = 2 \cdot k $$

 

 

Structure and overview

In further study one can examine propositions, logical connectives, truth tables, quantifiers, and logical laws in greater detail.