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