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A logical calculus is a formal, axiomatic system for recursively generating wellformed formulas (wffs). Essentially, it's a definition of a vocabulary, rules for the formation of wellformed 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 nonhypothetical, 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 justifiablyany 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.
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