Propositional calculus

A logical calculus is a formal, axiomatic system for recursively generating well-formed formulas (wffs). Essentially, it's a definition of a vocabulary, rules for the formation of well-formed formulas (wffs), and rules of inference permitting the generation of all valid argument forms[?] expressible in the calculus.

The propositional calculus is the foundation of symbolic logic; more complex logical calculi are usually defined by adding new operators and rules of transformation to it. It is generally defined as follows:

The vocabulary is composed of:

1. The letters of the alphabet (usually capitalized).
2. Symbols denoted logical operators: ¬, ∧, ∨, →, ↔
3. Parenthesis for grouping a wff as a sub-wff of a compound wffs: (, )

The rules for the formation of wffs:

1. Letters of the alphabet (usually capitalized) are wffs.
2. If φ is a wff, then ¬ φ is a wff.
3. If φ and ψ are wffs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are wffs.

Repeated applications of these three rules permit the generation of complex wffs. For example:

1. By rule 1, A is a wff.
2. By rule 2, ¬ A is a wff.
3. By rule 1, B is a wff.
4. By rule 3, ( ¬ A ∨ B ) is a wff.

The propositional calculus has ten inference rules[?]. The first eight are non-hypothetical, meaning that they do not involve hypothetical reasoning: specifically, the introduction of hypothetical premises is not used; the last two rules are hypothetical. These rules are introduction and elimination rules for each logical operator, used for deriving argument forms.

Double Negative Elimination

From the wff ¬ ¬ φ, we may infer φ

Conjunction Introduction

From any wff φ and any wff ψ, we may infer ( φ ∧ ψ ).

Conjunction Elimination

From any wff ( φ ∧ ψ ), we may infer φ or ψ

Disjunction Introduction

From any wff φ, we may infer the disjunction of φ with any other wff.

Disjunction Elimination

From wffs of the form ( φ ∨ ψ ), ( φ → χ ), and ( ψ → χ ), we may infer χ.

Biconditional Introduction

From wffs of the form ( φ → ψ ) and ( ψ → φ ), we may infer ( φ ↔ ψ ).

Biconditional Elimination

From the wff ( φ ↔ ψ ), we may infer ( φ → ψ ) or ( ψ → φ ).

Modus Ponens

From wffs of the form φ and ( φ → ψ ), we may infer ψ.

Conditional Proof

If ψ can be derived from the hypothesis φ, we may infer ( φ → ψ ) and discharge the hypothesis.

Reductio ad Absurdum

If we can derive both ψ and ¬ ψ from the introduction of the hypothesis φ, we may infer ¬ φ and discharge the hypothesis.

Introducing a hypothesis means adding a wff to a derivation not originally present as a premise; discharging the hypothesis means eliminating the wff justifiably--any wffs correctly derived from the hypothesis justify the introduction of the hypothesis after the fact.

With wffs and rules of inference, it's possible to derive wffs; the derivation[?] is a valid argument form[?], while the derived wff is known as a lemma.

See also:

• Connective
• Boolean algebra

External links

• Metamath (http://us.metamath.org/index): a project to construct mathematics using an axiomatic system based on propositional calculus, predicate calculus, and set theory