Phi 270
Fall 2013
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5.4.s. Summary

1

The law for the conditional as a premise applies only to reductio arguments and provides a way of rejecting a conditional by deriving its antecedent φ from the premises and reducing its consequent to absurdity given the premises. Rejecting a Conditional (RC) is the corresponding derivation rule.

2

This rule reflects the fact that a conditional is false when its antecedent is true and its consequent is false. The rules of Weakening (Wk) that have conditionals as conclusions reflect the fact that a conditional is true if its consequent is and also if its antecedent is false.

With these rules, the system of derivations for truth-functional logic is complete. It is shown in the table below.

Rules for developing gaps
logical form as a resource as a goal
atomic
sentence 		  IP
negation
¬ φ 		CR
(if φ is not atomic
and the goal is ⊥) 		RAA
conjunction
φ ∧ ψ 		Ext Cnj
disjunction
φ ∨ ψ 		PC PE
conditional
φ → ψ 		RC
(if the goal is ⊥) 		CP
Rules for closing gaps
when to close rule
the goal is also
a resource 		QED
sentences φ and ¬ φ are
resources & the goal is ⊥ 		Nc
⊤ is the goal ENV
⊥ is a resource EFQ
Basic system
Detachment rules (optional)
main resource auxiliary resource rule
φ → ψ φ MPP
¬± ψ MTT
φ ∨ ψ ¬± φ or ¬± ψ MTP
¬ (φ ∧ ψ) φ or ψ MPT
Attachment rules
added resource rule
φ ∧ ψ Adj
φ → ψ Wk
φ ∨ ψ Wk
¬ (φ ∧ ψ) Wk
Rule for lemmas
prerequisite rule
the goal is ⊥ LFR
Added rules
(optional)

At the top and left appears the basic system, all of whose rules are progressive. It consists of the fundamental rules for developing gaps by exploiting resources or planning for goals, two rules each for negations, conjunctions, disjunctions, and conditionals along with a rule to plan for atomic sentences. There are the same four rules for closing gaps we had as of 3.2, and we now also have a set of four detachment rules that provide alternative ways of exploiting weak truth-functional compounds. In addition to the basic system, there is a group of rules that are not necessarily progressive although they are sound and safe. These are the rules makred off at the lower right in the table—the attachment rules and the general rule LFR for introducing lemmas in reductio arguments. As in the earlier tables of this form, the names of the rules in the following are links to places where they are actually stated.

Glen Helman 01 Aug 2013