Propositional Logic

Propositional logic, also called zeroth-order logic, is a formal system. Given the properties of a formal system, this means that the rules are entirely regular. The primary application of propositional logic is to model statements in natural language.

Domain

There are only two distinct values: true and false, denoted by T and F respectively.

Syntax

The rules for constructing expressions in propositional logic follow.

Negation

A negation is denoted by the following symbol: $\neg$.

Connective

A connective is one of the following symbols: $\land ,\;\lor$.

Atom

An atom is denoted by a string of letters without spaces. For instance:

$a$

$something$

$thisIsAnAtom$

Clause

A clause is either of the following: an atom; or a clause, followed by a connective, followed by a clause. In the second case, it must be enclosed in parentheses. A clause may be preceded by a negation. For instance:

$a$

$\neg \,a$

$(a\land b)$

$\neg \,(a\land b)$

$(a\land \neg \,(b\lor \neg \,c))$

Semantics

The rules for determining the value of expressions follow.

Atom

An atom can have one of the values T and F, but not both or neither.

Negation

If an atom a is preceded by a negation, the value of the resulting expression is represented by this table:

$a$	$\neg \,a$
T	F
F	T

Clause

The rules for determining the value of clauses follow, where a and b are some given clauses:

$a$	$b$	$a\land b$	$a\lor b$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	F

Thus conclude the rules of propositional logic.