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Single Idea 15430

[filed under theme 4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic ]

Full Idea

From one point of view intuitionistic logic is a part of classical logic, missing one axiom, from another classical logic is a part of intuitionistic logic, missing two connectives, intuitionistic v and →

Gist of Idea

Is classical logic a part of intuitionist logic, or vice versa?

Source

John P. Burgess (Philosophical Logic [2009], 6.4)

Book Ref

Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.129

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]