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Single Idea 15419

[filed under theme 10. Modality / A. Necessity / 6. Logical Necessity ]

Full Idea

To the extent that there is any conventional wisdom about the question, it is that S5 is correct for alethic logical modality, and S4 correct for apodictic logical modality.

Clarification

'Alethic' concerns truth; 'apodictic' concerns proof

Gist of Idea

General consensus is S5 for logical modality of validity, and S4 for proof

Source

John P. Burgess (Philosophical Logic [2009], 3.8)

Book Ref

Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.65

A Reaction

In classical logic these coincide, so presumably one should use the minimum system to do the job, which is S4 (?).

Related Idea

Idea 15417 Logical necessity has two sides - validity and demonstrability - which coincide in classical logic [Burgess]

The 34 ideas from John P. Burgess

15404 Technical people see logic as any formal system that can be studied, not a study of argument validity [Burgess]
15403 Philosophical logic is a branch of logic, and is now centred in computer science [Burgess]
15405 Classical logic neglects the non-mathematical, such as temporality or modality [Burgess]
15407 Formalising arguments favours lots of connectives; proving things favours having very few [Burgess]
15406 'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components [Burgess]
15408 'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics [Burgess]
15409 All occurrences of variables in atomic formulas are free [Burgess]
15411 We only need to study mathematical models, since all other models are isomorphic to these [Burgess]
15412 Models leave out meaning, and just focus on truth values [Burgess]
15413 With four tense operators, all complex tenses reduce to fourteen basic cases [Burgess]
15415 The temporal Barcan formulas fix what exists, which seems absurd [Burgess]
15414 The denotation of a definite description is flexible, rather than rigid [Burgess]
15416 We aim to get the technical notion of truth in all models matching intuitive truth in all instances [Burgess]
15418 Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency [Burgess]
15417 Logical necessity has two sides - validity and demonstrability - which coincide in classical logic [Burgess]
15419 General consensus is S5 for logical modality of validity, and S4 for proof [Burgess]
15420 De re modality seems to apply to objects a concept intended for sentences [Burgess]
15421 Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them [Burgess]
15422 Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth) [Burgess]
15423 It is doubtful whether the negation of a conditional has any clear meaning [Burgess]
15424 Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths [Burgess]
15426 We can build one expanding sequence, instead of a chain of deductions [Burgess]
15425 The sequent calculus makes it possible to have proof without transitivity of entailment [Burgess]
15427 The Cut Rule expresses the classical idea that entailment is transitive [Burgess]
15428 The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut' [Burgess]
15429 Relevance logic's → is perhaps expressible by 'if A, then B, for that reason' [Burgess]
15430 Is classical logic a part of intuitionist logic, or vice versa? [Burgess]
15431 It is still unsettled whether standard intuitionist logic is complete [Burgess]
10185 Set theory is the standard background for modern mathematics [Burgess]
10184 Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess]
10186 If set theory is used to define 'structure', we can't define set theory structurally [Burgess]
10187 Abstract algebra concerns relations between models, not common features of all the models [Burgess]
10188 How can mathematical relations be either internal, or external, or intrinsic? [Burgess]
10189 There is no one relation for the real number 2, as relations differ in different models [Burgess]