Ideas from 'An Introduction to Modal Logic' by GE Hughes/M Cresswell [1968], by Theme Structure

[found in 'An Introduction to Modal Logic' by Hughes,G./Cresswell,M. [Methuen 1972,416-29470-7]].

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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL

9540	A 'value-assignment' (V) is when to each variable in the set V assigns either the value 1 or the value 0

			Full Idea: A 'value-assignment' (V) is when to each variable in the set V assigns either the value 1 or the value 0.
			From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1)
			A reaction: In the interpreted version of the logic, 1 and 0 would become T (true) and F (false). The procedure seems to be called nowadays a 'valuation'.

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL

9541	The Law of Transposition says (P→Q) → (¬Q→¬P)

			Full Idea: The Law of Transposition says that (P→Q) → (¬Q→¬P).
			From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1)
			A reaction: That is, if the consequent (Q) of a conditional is false, then the antecedent (P) must have been false.

4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL

9543	The rules preserve validity from the axioms, so no thesis negates any other thesis

			Full Idea: An axiomatic system is most naturally consistent iff no thesis is the negation of another thesis. It can be shown that every axiom is valid, that the transformation rules are validity-preserving, and if a wff α is valid, then ¬α is not valid.
			From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1)
			A reaction: [The labels 'soundness' and 'consistency' seem interchangeable here, with the former nowadays preferred]

5. Theory of Logic / K. Features of Logics / 4. Completeness

9544	A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised

			Full Idea: To say that an axiom system is 'weakly complete' is to say that every valid wff of the system is derivable as a thesis. ..The system is 'strongly complete' if it cannot have any more theses than it has without falling into inconsistency.
			From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1)
			A reaction: [They go on to say that Propositional Logic is strongly complete, but Modal Logic is not]