6 ideas
9540 | A 'value-assignment' (V) is when to each variable in the set V assigns either the value 1 or the value 0 [Hughes/Cresswell] |
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'. |
9541 | The Law of Transposition says (P→Q) → (¬Q→¬P) [Hughes/Cresswell] |
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. |
9543 | The rules preserve validity from the axioms, so no thesis negates any other thesis [Hughes/Cresswell] |
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] |
9544 | A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised [Hughes/Cresswell] |
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] |
10855 | Actual infinities are not allowed in mathematics - only limits which may increase without bound [Gauss] |
Full Idea: I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking, in which one properly speaks of limits ...which are permitted to increase without bound. | |
From: Carl Friedrich Gauss (Letter to Shumacher [1831]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.7 |
20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |