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Single Idea 7795

[filed under theme 4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5 ]

Full Idea

In S5 there are six modalities: no-operator; necessarily; and possibly (and their negations). In any sequence of operators we may delete all but the last to gain an equivalent formula.

Gist of Idea

S5 has just six modalities, and all strings can be reduced to those

Source

Rod Girle (Modal Logics and Philosophy [2000], 3.5)

Book Ref

Girle,Rod: 'Modal Logics and Philosophy' [Acumen 2000], p.46

A Reaction

Such drastic simplification seems attractive. Is there really no difference, though, between 'necessarily-possibly', 'possibly-possibly' and just 'possibly'? Could p be contingently possible in this world, and necessarily possible in another?

The 15 ideas from 'Modal Logics and Philosophy'

7786 Propositional logic handles negation, disjunction, conjunction; predicate logic adds quantifiers, predicates, relations [Girle]
7787 Possible worlds logics use true-in-a-world rather than true [Girle]
7788 Modal logic has four basic modal negation equivalences [Girle]
7789 Necessary implication is called 'strict implication'; if successful, it is called 'entailment' [Girle]
7790 If an argument is invalid, a truth tree will indicate a counter-example [Girle]
7792 A world has 'access' to a world it generates, which is important in possible worlds semantics [Girle]
7793 ◊p → □◊p is the hallmark of S5 [Girle]
7795 S5 has just six modalities, and all strings can be reduced to those [Girle]
7794 There are seven modalities in S4, each with its negation [Girle]
7796 Modal logics were studied in terms of axioms, but now possible worlds semantics is added [Girle]
7798 There are three axiom schemas for propositional logic [Girle]
7799 Proposition logic has definitions for its three operators: or, and, and identical [Girle]
7797 Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems [Girle]
7800 Analytic truths are divided into logically and conceptually necessary [Girle]
7801 Possibilities can be logical, theoretical, physical, economic or human [Girle]