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Home > Propositional calculus

A calculus, or proof theory is that part of a logical system which determines how to construct arguments: to derive conclusions from premises. It is a set of axioms (which may be an empty set), and a set of inference rules for deriving new well-formed formulas (wffs) from a given set of wffs. It must thus include or be defined in terms of a formal grammar, which will state all of the allowable expressions, the wffs, in the language. Any grammar will in general also be given a semantics, which explains those features (truth, implication) that are, presumably, of interest. Ideally the axioms and inference rules of a calculus are chosen such that if the formulas in a set are semantically true then any formulas derivable from them are also true. (Hence a calculus is formulated independently of a semantics, but with the aim of agreeing with it.)

In a propositional calculus the vocabulary consists of atomic sentences and sentential operators or connectives. The wffs are all sentences; they include the atomic sentences and any sentences built up from those and the sentential operators.

In what follows we will outline a standard propositional calculus. Many different such formulations exist which are all more or less equivalent but differ in (1) which sentential operators they allow, (hence, which language or grammar they are designed for); (2) which (if any) axioms, and which inference rules are used; and (3) in what form derivations are presented. There is no limit to the number of such systems that can be devised.

1 Grammar

The vocabulary is composed of:

1. The capital letters of the alphabet. These abbreviate complete sentences which are atomic in the sense that they cannot be decomposed into smaller sentences.
2. Symbols denoting the following connectives (or logical operators): ¬, ∧, ∨, →In logical calculus of mathematics, the logical conditional (also known as the material implication sometimes material conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent and q is a conclu, ↔In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent and q is a conclusion (or consequent . The operator is denoted using a doublehea
3. The left parenthesis and the right parenthesis: (, ).

The set of wffs is recursiveSee: Recursion Recursive function Recursive set Recursively enumerable set Recursively enumerable language Primitive recursive function.ly defined by the following rules:

1. Basis: Letters of the alphabet (usually capitalized such as A, B, etc.) are wffs.
2. Inductive Clause I: If φ is a wff, then ¬ φ is a wff.
3. Inductive Clause II If φ and ψ are wffs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are wffs.
4. Closure Clause: Nothing else is a wff.

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.

2 Calculus

For simplicity, we will use a natural deductionIn mathematical logic, natural deduction is the name given to a class of foundational approaches for two key concepts in logic, propositions and proofs. There is no universal agreement on the proper foundation of these concepts; natural deduction takes th system, which has no axioms; or, equivalently, which has an empty axiom set.

Derivations using our calculus will be laid out in the form of a list of numbered lines, with a single wff and a justification on each line. Any premisses will be at the top, with a "p" for their justification. The conclusion will be on the last line. A derivation will be considered complete if every line follows from previous ones by correct application of a rule. (For a contrasting approach, see proof-trees ).

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Topics: Propositional Calculus Phi Psi Rarr Set Rules Chi Wff True...

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Propositional knowledgePropositional knowledge or declarative knowledge is knowledge that some proposition is either true or false. This distinguishes propositional knowledge from know-how or procedural knowledge, which is the knowledge of how to perform some task. This article Propositional calculusA propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. It is also extensional. Compare, by contrast, predicate logic and modal logic, which also have their own calculi. A calcul Propositional attitudeA propositional attitude is a relational mental state connecting a person to a proposition. They are often assumed to be the simplest components of thought and can express meanings or content that can be true or false. In being a type of attitude they imp