A branch of Logic which studies the ways of joining and/or modifying entire propositions, statements or sentences in order to form more complex propositions, statements or sentences.
It also deals with the logical relationships and properties that are derived by combining or altering the statements.
Statements consist of a set of promitives (atomic components), grammar rules which define well defined formulae and inference rules in order to make deductions.
Syntax Formation Rules
Propositional Logic does not consider smaller parts of statements (such as words), instead treating simple statements as indivisible wholes.
The Language uses uppercase letters A, B, C etc. to represent complete statements. The logical signs ∧, ∨, →, ↔, and ¬ are used in place of the truth-functional operators, AND, OR, IF THEN, IF AND ONLY IF, and NOT, respectively.
Sentences
Sentences are concatenations of one or more primitives, those without any free variables are known as statements (or propositions). When a sentence follows grammar rules we say it is a well formed formulae (WFF).
Statements
Statements can be true or false (but not both nor neither). However, the truthfulness of a statement does not indicade its validity. It is crucial to investigate the structure of statements and the arguments. This is not the same as saying whether a statement is empirically true, the statement (proposition) is well formed but it may not be true.
Connectives
Logical connectives are used to combine atomic statements (propositions) into compound (more complex) statements. The five common connectives include:
∧=AND∨=OR→=IF THEN↔=IF AND ONLY IF¬=NOT
AND
A conjunction (dyadic operator) which allows for compound statements such as: p AND q is the same as p ∧ q.
OR
A dysjunction (dyadic operator) which allows for compound statements such as: p OR q is the same as p ∨ q.
NOT
A negation (monadic operator) which allows for compound statements such as: NOT p sometimes written as ~p or ¬p
IMPLIES
Implication is another example of a dyadic operator which allows for compound statements, it implies that if p is true then q must also be true.
IFF
If and only if is another dyadic operator written as p ↔ q which implies that if p is true then so is q and if p is false then so is q.
See also Truth Tables
Weaknesses to Propositional Logic
Since propositional Logic deals with facts that are either true or false it becomes difficult when trying to use it in scenarios where that might not always be the case. There is also no way to indicate that terms in different sentences are referring to the same object. It also lacks a way to talk about relations between objects.