|S 4 / S 5, logic, philosophy: S 4 and S 5 are modal logical systems that differ in terms of what is expressible in them. The increase in expressiveness is achieved by adding axioms. S 5 results from S 4 by the added axiom Mp > NMp. "What is possible is necessarily possible". See also axioms, axiom systems, modal logic, modalities, stronger/weaker._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
Books on Amazon
System S4/Bigelow/Pargetter: contains T and in addition:
A10 Axiom S4 (Na > Nna)
Everyday language translation: if something has to be true, it must be true that it has to be true.
System B/Brouwer/Bigelow/Pargetter: contains T plus
A11. Axiom B: (a > NMa)
Everyday language translation: if something is true, it must be true that it is possible.
System S4/Bigelow/Pargetter: some of his theorems are not theorems of B, and some of B are not theorems of S4.
With an additional Axiom S5, we can prove both S4 and B as theorems:
A12. Axiom S5: (Ma > NMa)
System S5/Bigelow/Pargetter: contains all theorems of S4 and of B and nothing else.
Systems/Proveability/Bigelow/Pargetter: T plus S5 can prove S4 and B, but also T plus S4 and B together can prove S5.
Nevertheless: T plus S4 without B cannot prove S5
T plus B without S4 cannot prove S5.
Logical necessity/S5/Bigelow/Pargetter: the system S5 is a plausible characterization of the logical necessity.
System S4/Bigelow/Pargetter: when we interpret:
Rhomb/diamond/possibility/M: "cannot be proved by logic alone"
Box/Necessity/N: "can be proved by logic alone"
Then S4 becomes:
Everyday language translation: "If something can be proved by logic alone, then one can prove by logic alone that one can prove it by logic alone".
Bigelow/Pargetter: that is plausible.
Everyday language translation: "If something is true, one can show with logic alone that it cannot be refuted by logic alone.
Everyday language translation: If something cannot be refuted by logic alone, it can be proved by logic alone that it cannot be refuted by logic alone._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990