RULE OF INFERENCE: CONJUNCTION
RULES OF INFERENCE AND REPLACEMENT
A rule of inference allows you to deduce a certain sentence from one or two others. For example, you can derive a conjunction by conjoining two sentences given as premises. However, there are some rules that are even more powerful than rules of inference, namely, rules of replacement. A rule of replacement allows you to substitute one formula or sentence for another. This replacement is made possible by the fact that the two formulas are equivalent (the formula being replaced and the formula replacing it). The mechanisms behind RULES of INFERENCE and RULES OF REPLACEMENT emerge from the meanings and truth tables of the sentence connectives.
RULE OF INFERENCE: CONJUNCTION
For example, the rules of simplification and conjunction emerge directly from the fact that when two sentences are connected by a conjunction, what’s being asserted is that both conjuncts are true. Thus if a conjunction is true, we can derive either conjunct. In addition, if we have two sentences that are true we can join the two together using a conjunction
SIMPLIFICATION CONJUNCTION
Given p & q, we can deduce Given p and given q, we can
deduce
p
p & q
RULES OF REPLACEMENT: CONJUNCTION
COMMUTATION ASSOCIATION
The formula p & q The formula p & (q & r),
can (1) replace or can (2) can (1) replace or can (2)
be replaced at any time be replaced at any time
during a deduction by during a deduction by
q & p (p & q) & r
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