Two Standard and Two Modal Squares of Opposition Studied in - PowerPoint PPT Presentation

Two Standard and Two Modal Squares of Opposition Studied in Transparent Intensional Logic Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK) doc. PhDr. Jiří Raclavský, Ph.D. ( raclavsky@phil.muni.cz ) Department of Philosophy, Masaryk University, Brno

1 1 1 1 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic Abstract Abstract Abstract Abstract In this paper, we examine modern reading of the Square of Opposition by means of intensional logic. Explicit use of possible world semantics helps us to sharply discriminate between the standard and modal (‘alethic’) readings of categorical statements. We get thus two basic versions of the Square. The Modal Square has not been introduced in the contemporary debate yet and so it is in the heart of interest. It seems that some properties ascribed by mediaeval logicians to the Square require a shift from its Standard to its Modal version. Not necessarily so, because for each of the two there is its mate which can be easily confused with it. The discrimination between the initial and modified versions of the Standard and Modal Square enable us to sharply separate findings about logical properties of the Square into four groups, which makes their proper comparison possible. Keywords: Square of Opposition; modal Square of Opposition; modality; intensional logic; Math. Subject Classification: 03A05 some terminology: - the Standard Square of Opposition = the Square with categorical statements - the Modal Square of Opposition = the Square with modal versions of categorical statements - classical reading etc. = what is held by classical logicians (followers of Aristotle) - modern reading etc. = what is based on modern logic or held in modern logic textbooks Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

2 2 2 2 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic Content Content Content Content I. A very brief introduction to Transparent Intensional Logic I. I. I. II. Modern reading of the Standard Square of Opposition II. II. II. III. Modified modern reading of the Standard Square of Opposition III. III. III. IV. Modified modern reading of the Modal Square of Opposition IV. IV. IV. V. Modal reading of categorical statements V. V. V. VI. Modern reading of the Modal Square of Opposition VI. VI. VI. VII. Modal Hexagon of Opposition VII. VII. VII. VIII. Conclusions VIII. VIII. VIII. IX. Some prospects IX. IX. IX. Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

3 3 3 3 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I. I. I. I. A very brief introduction to Transparent Intensional Logic A very brief introduction to Transparent Intensional Logic A very brief introduction to Transparent Intensional Logic A very brief introduction to Transparent Intensional Logic Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

4 4 4 4 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I. I.1 I. I. 1 1 1 A very brief i A very brief i A very brief i A very brief introduction ntroduction ntroduction ntroduction to Transparent Intensional Logic to Transparent Intensional Logic to Transparent Intensional Logic to Transparent Intensional Logic - Transparent Intensional Logic ( TIL ) developed by Pavel Tichý (1936 Brno - 1994 Dunedin, New Zealand) in the very beginning of 1970s - TIL can be seen as a typed λ -calculus , i.e. a higher-order logic (with careful formation of its terms) - till now, most important applications of TIL are in semantics of natural language (propositional attitudes, modalities, subjunctive conditionals, verb tenses, etc. are analysed in TIL; I will suppress temporal parameter), rivalling thus the system of Montague Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

5 5 5 5 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I.2 I.2 TIL semantic I.2 I.2 TIL semantic TIL semantic TIL semantic scheme scheme scheme scheme expression E | E expresses construction C (= meaning explicated as an hyperintension ) | E denotes , C means intension/extension (= an PWS-style of explication of denotation ) - constructions are structured abstract entities of algorithmic nature - they are written by λ -terms: constants | variables | compositions | λ - closures - ‘intensional principle’ of individuation: every object O is constructed by infinitely many congruent , but not identical constructions Cs - every construction C is thus specified by : i. the object O constructed by C , ii. the way how C constructs the object O Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

6 6 6 6 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I. I.3 I. I. 3 3 3 T T T Type theory ype theory ype theory ype theory of TIL of TIL of TIL of TIL - Tichý modified Church’s Simple Theory of Types (and ramified it in 1988, which is omitted here; the type of k -order constructions is ∗ k ) - Let base B be a non-empty class of pairwise disjoint collections of atomic objects, e.g. B TIL ={ ι , ο , ω , τ }: a) Any member of B is a type over B . b) If α 1 , …, α m , β are types over B , then ( βα 1 … α m ) – i.e. the collection of all total and partial m -ary functions from α 1 , …, α m to β – is a type over B . - (possible world) intensions ( propositions , properties , …) are functions from possible worlds Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

7 7 7 7 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I. I.4 I. I. 4 4 4 T T T Type ype ype ypes of some basic objects s of some basic objects s of some basic objects s of some basic objects - “/” abbreviates “ v -constructs an object of type” • x / ξ (a ξ -object, i.e. an object belonging to the type ξ ) • p / ( οω ) (a proposition); let P P and Q Q be concrete examples of P P Q Q constructions of propositions • f, g /(( ο ξ ) ω ) (a property of ξ -objects; its extension in W is of type ( ο ξ )); let F F F F and G G be concrete examples of constructions of properties G G • ∀ ∀ ξ ξ ξ ξ /( ο ( ο ξ )) (the class containing the only universal class of ξ -objects; ∀ = ∀ ∀ ={U}) = = • ∃ ∃ ξ ξ ξ /( ο ( ο ξ )) (the class containing all nonempty classes of ξ -objects) ξ ∃ ∃ • 1 0/ ο (True, False); o / ο (a truth value); ¬ ¬ /( οο ) (the classical negation); ∧ ∧ , ∨ ∨ , → → , ↔ ↔ /( οοο ) ¬ ¬ ∧ ∧ ∨ ∨ → → ↔ ↔ 1, 0 1 1 0 0 = ξ ξ ξ /( ο ξξ ) (a ξ (the classical conjunction, disjunction, material conditional, equivalence); = = = ≠ ξ ξ ξ ξ /( ο ξξ ); ‘ ξ ξ ξ ξ ’ will be usually suppressed even in the case familiar relation between ξ -objects); ≠ ≠ ≠ of other functions/relations Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)

8 8 8 8 Jiří Raclavský (2014): Two Standard and Two Modal Squares of Oppositions Studied in Transparent Intensional Logic I.5 I. I. I. 5 5 5 Definitions Definitions Definitions Definitions - Tichý’s system of deduction for his simple type theory (1976, 1982) - sequents are made from matches x : C C (“the variable or trivialization x v -constructs the C C same ξ - object as the compound construction C C ”, loosely: “ C = x ”) C C - definitions are certain deduction rules of form C ⇔ x : D |- x : C C C D D D D are different constructions of the same object as x ; ⇔ means where C C C C and D D D interderivability of sequents flanking the ⇔ sign C ⇔ D C ⇔ x : D - “ C C C D D D ” abbreviates “|- x : C C C D D ” D - example (where ∅ ∅ /( ο ξ ), the total empty ξ -class): ∅ ∅ ∅ ∅ ⇔ df λ x F ∅ ∅ F F F Logika: systémový rámec rozvoje oboru v ČR a koncepce logických propedeutik pro mezioborová studia (reg. č. CZ.1.07/2.2.00/28.0216, OPVK)