Intensionalisation of Logical Operators
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1 Intensionalisation of Logical Operators Vít Punčochář Institute of Philosophy Academy of Sciences Czech Republic Vít Punčochář (AS CR) Intensionalisation / 29
2 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
3 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
4 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
5 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
6 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
7 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
8 A nonstandard representation of classical logic Semantics based on an assertibility relation C is a set of possible worlds (a context). C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff for some D, E, D E = C, D ϕ and E ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
9 A nonstandard representation of classical logic Consequence relation Definition ψ iff for all C, if C, then C ψ. Vít Punčochář (AS CR) Intensionalisation / 29
10 A nonstandard representation of classical logic Consequence relation Definition ψ iff for all C, if C, then C ψ. Fact is identical with the consequence relation of classical logic. Vít Punčochář (AS CR) Intensionalisation / 29
11 Strict disjunction Extensional principle for disjunction If C ϕ and D ψ, then C D ϕ ψ. Vít Punčochář (AS CR) Intensionalisation / 29
12 Strict disjunction Factual sentences C John is in Germany. C r s t = John is in Berlin = John is in Hamburg = John is in Munich Vít Punčochář (AS CR) Intensionalisation / 29
13 Strict disjunction Factual sentences C John is in Germany. C r s t = John is in Berlin = John is in Hamburg = John is in Munich Vít Punčochář (AS CR) Intensionalisation / 29
14 Strict disjunction Factual sentences D John is in France. D u v w = John is in Paris = John is in Toulouse = John is in Strasbourg Vít Punčochář (AS CR) Intensionalisation / 29
15 Strict disjunction Factual sentences D John is in France. D u v w = John is in Paris = John is in Toulouse = John is in Strasbourg Vít Punčochář (AS CR) Intensionalisation / 29
16 Strict disjunction Factual sentences C D John is in Germany or he is in France. CuD r s t u v w = John is in Berlin = John is in Hamburg = John is in Munich = John is in Paris = John is in Toulouse = John is in Strasbourg Vít Punčochář (AS CR) Intensionalisation / 29
17 Strict disjunction Factual sentences C D John is in Germany or he is in France. CuD r s t u v w = John is in Berlin = John is in Hamburg = John is in Munich = John is in Paris = John is in Toulouse = John is in Strasbourg Vít Punčochář (AS CR) Intensionalisation / 29
18 Contextual sentences Strict disjunction C All suspects are men. C r s t = John commited the crime = Robert commited the crime = Michael commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
19 Contextual sentences Strict disjunction C All suspects are men. C r s t = John commited the crime = Robert commited the crime = Michael commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
20 Contextual sentences Strict disjunction D All suspects are women. D u v w = Anna commited the crime = Natalie commited the crime = Molly commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
21 Contextual sentences Strict disjunction D All suspects are women. D u v w = Anna commited the crime = Natalie commited the crime = Molly commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
22 Contextual sentences Strict disjunction C D All suspects are men or all suspects are women. CuD r s t u v w = John commited the crime = Robert commited the crime = Michael commited the crime = Anna commited the crime = Natalie commited the crime = Molly commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
23 Contextual sentences Strict disjunction C D All suspects are men or all suspects are women. CuD r s t u v w = John commited the crime = Robert commited the crime = Michael commited the crime = Anna commited the crime = Natalie commited the crime = Molly commited the crime Vít Punčochář (AS CR) Intensionalisation / 29
24 Strict disjunction Strict disjunction C ϕ ψ iff C ϕ or C ψ. Vít Punčochář (AS CR) Intensionalisation / 29
25 Strict disjunction Inquisitive semantics (J. Groenendijk) C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff C ϕ or C ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
26 Strict disjunction Semantics of assertibility C iff C =. C p iff for all v C, v(p) = 1. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff C ϕ or C ψ. C ϕ ψ iff D ψ for all D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
27 Weak negation Negation and implication (p q) p q Vít Punčochář (AS CR) Intensionalisation / 29
28 Weak negation Paul Grice Denial of a conditional Sometimes a denial of a conditional has the effect of a refusal to assert the conditional in question, characteristically because the denier does not think that there are adequate non-truth-functional grounds for such an assertion. (Paul Grice, Indicative conditionals) Vít Punčochář (AS CR) Intensionalisation / 29
29 Weak negation Weak negation expressing a refusal to assert a sentence C ϕ iff C ϕ. Vít Punčochář (AS CR) Intensionalisation / 29
30 Weak negation Semantics of assertibility with weak negation For every C, C. C p iff for all v C, v(p) = 1. C ϕ iff C ϕ. C ϕ ψ iff C ϕ and C ψ. C ϕ ψ iff C ϕ or C ψ. C ϕ ψ iff D ψ for all nonempty D C such that D ϕ. ϕ = Df ϕ Vít Punčochář (AS CR) Intensionalisation / 29
31 Weak negation Two kinds of modal operators ϕ = Df ϕ, ϕ = Df ϕ, ϕ = Df ϕ. ϕ = Df ϕ. Vít Punčochář (AS CR) Intensionalisation / 29
32 Weak negation The relationships between the modalities Fact (i) ϕ ϕ, (ii) ϕ ϕ. Vít Punčochář (AS CR) Intensionalisation / 29
33 Weak negation The relationships between the modalities Fact (i) ϕ ϕ, (ii) ϕ ϕ. Proof. (i) ϕ = ϕ ϕ = ϕ. (ii) ϕ = ϕ ϕ = ϕ. Vít Punčochář (AS CR) Intensionalisation / 29
34 Weak negation Semantics of the modal operators Fact (i) ϕ is assertible in C iff ϕ is (classically) true in every world of C. (ii) ϕ is assertible in C iff ϕ is assertible in every subcontext of C. (iii) ϕ is assertible in C iff ϕ is (classically) true in some world of C. (iv) ϕ is assertible in C iff ϕ is assertible in some subcontext of C. Vít Punčochář (AS CR) Intensionalisation / 29
35 Weak negation Two dual operators: and ϕ 1... ϕ n = Df (ϕ 1... ϕ n ) ( ϕ 1... ϕ n ). ϕ 1... ϕ n = Df (ϕ 1... ϕ n ) ( ϕ 1... ϕ n ). Vít Punčochář (AS CR) Intensionalisation / 29
36 Weak negation Two dual operators: and ϕ 1... ϕ n = Df (ϕ 1... ϕ n ) ( ϕ 1... ϕ n ). ϕ 1... ϕ n = Df (ϕ 1... ϕ n ) ( ϕ 1... ϕ n ). Vít Punčochář (AS CR) Intensionalisation / 29
37 Weak negation Semantics of C ϕ 1... ϕ n. Every disjunct is true in at least one possible world and in every possible world at least one disjunct is true. Vít Punčochář (AS CR) Intensionalisation / 29
38 Weak negation Semantics of C ϕ 1... ϕ n. Every disjunct is true in at least one possible world and in every possible world at least one disjunct is true. Vít Punčochář (AS CR) Intensionalisation / 29
39 Weak negation Fact (i) ϕ 1... ϕ n (ϕ 1... ϕ n ), (ii) ϕ 1... ϕ n (ϕ 1... ϕ n ), (iii) ϕ 1... ϕ n (ϕ 1... ϕ n ), (iv) ϕ 1... ϕ n (ϕ 1... ϕ n ). Vít Punčochář (AS CR) Intensionalisation / 29
40 Weak negation Fact (i) ϕ 1... ϕ n (ϕ 1... ϕ n ), (ii) ϕ 1... ϕ n (ϕ 1... ϕ n ), (iii) ϕ 1... ϕ n (ϕ 1... ϕ n ), (iv) ϕ 1... ϕ n (ϕ 1... ϕ n ). Vít Punčochář (AS CR) Intensionalisation / 29
41 Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation / 29
42 Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation / 29
43 Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation / 29
44 Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation / 29
45 A system of natural deduction Restricted conditional proof (ϕ : ψ)/ϕ ψ In the scope of a hypotetical assumption, not all formulas from the outer proof are available. We can use only -free formulas and formulas of the form ϕ ψ. Vít Punčochář (AS CR) Intensionalisation / 29
46 A system of natural deduction A system of natural deduction ( I) ϕ, ψ/ϕ ψ ( E) (i) ϕ ψ/ϕ, (ii) ϕ ψ/ψ ( I) (i) ϕ/ϕ ψ, (ii) ψ/ϕ ψ ( E) ϕ ψ, [ϕ : χ], [ψ : χ]/χ ( I) (ϕ : ψ)/ϕ ψ ( E) ϕ, ϕ ψ/ψ ( I) ϕ, ϕ/ (IP) [ ϕ : ]/ϕ Vít Punčochář (AS CR) Intensionalisation / 29
47 A system of natural deduction A system of natural deduction (R1) p / p (R2) / (ϕ ϕ), (R3) ϕ ψ / (ϕ ψ) (R4) ϕ 1... ϕ n / (ϕ 1... ϕ n ). Vít Punčochář (AS CR) Intensionalisation / 29
48 A system of natural deduction Theorem The system of natural deduction is sound and complete with respect to the semantics of assertibility with weak negation. Vít Punčochář (AS CR) Intensionalisation / 29
49 (p q) p A system of natural deduction 1 (p q) premise 2 p hyp. assumption 3 p hyp. assumption 4 2,3 ( I) 5 q 4 Ex falso quodlibet (derivable rule) 6 p q 3-5 ( I)!!!!!!! 7 1,6 ( I) 8 p 2-7 (IP) Vít Punčochář (AS CR) Intensionalisation / 29
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