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## Propositional calculus
A
The The vocabulary is composed of: - The letters of the alphabet (usually capitalized).
- Symbols denoted logical operators: ¬, &and, &or, &rarr, &harr
- Parenthesis for grouping a
*wff*as a sub-*wff*of a compound*wff*s: (, )
wffs:
- Letters of the alphabet (usually capitalized) are
*wff*s. - If φ is a
*wff*, then ¬ φ is a*wff*. - If φ and ψ are
*wff*s, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are*wff*s.
wffs. For example:
- By rule 1, A is a
*wff*. - By rule 2, ¬ A is a
*wff*. - By rule 1, B is a
*wff*. - By rule 3, ( ¬ A ∨ B ) is a
*wff*.
; Double Negative Elimination: From the
; Conjunction Introduction: From any
; Conjunction Elimination: From any
; Disjunction Introduction: From any
; Disjunction Elimination: From
; Biconditional Introduction: From
; Biconditional Elimination: From the
; Modus Ponens: From ; 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 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: ## External links- Metamath: a project to construct mathematics using an axiomatic system based on propositional calculus, predicate calculus, and set theory
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