Intensionalisation of Logical Operators

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Intensionalisation of Logical Operators Vít Punčochář Institute of Philosophy Academy of Sciences Czech Republic Vít Punčochář (AS CR) Intensionalisation 2013 1 / 29

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 2013 2 / 29

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 2013 2 / 29

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 2013 2 / 29

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 2013 2 / 29

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 2013 2 / 29

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 2013 2 / 29

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 2013 2 / 29

A nonstandard representation of classical logic Consequence relation Definition ψ iff for all C, if C, then C ψ. Vít Punčochář (AS CR) Intensionalisation 2013 3 / 29

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 2013 3 / 29

Strict disjunction Extensional principle for disjunction If C ϕ and D ψ, then C D ϕ ψ. Vít Punčochář (AS CR) Intensionalisation 2013 4 / 29

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 2013 5 / 29

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 2013 5 / 29

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 2013 6 / 29

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 2013 6 / 29

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 2013 7 / 29

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 2013 7 / 29

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 2013 8 / 29

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 2013 8 / 29

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 2013 9 / 29

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 2013 9 / 29

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 2013 10 / 29

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 2013 10 / 29

Strict disjunction Strict disjunction C ϕ ψ iff C ϕ or C ψ. Vít Punčochář (AS CR) Intensionalisation 2013 11 / 29

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 2013 12 / 29

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 2013 13 / 29

Weak negation Negation and implication (p q) p q Vít Punčochář (AS CR) Intensionalisation 2013 14 / 29

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 2013 15 / 29

Weak negation Weak negation expressing a refusal to assert a sentence C ϕ iff C ϕ. Vít Punčochář (AS CR) Intensionalisation 2013 16 / 29

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 2013 17 / 29

Weak negation Two kinds of modal operators ϕ = Df ϕ, ϕ = Df ϕ, ϕ = Df ϕ. ϕ = Df ϕ. Vít Punčochář (AS CR) Intensionalisation 2013 18 / 29

Weak negation The relationships between the modalities Fact (i) ϕ ϕ, (ii) ϕ ϕ. Vít Punčochář (AS CR) Intensionalisation 2013 19 / 29

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 2013 19 / 29

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 2013 20 / 29

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 2013 21 / 29

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 2013 21 / 29

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 2013 22 / 29

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 2013 22 / 29

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 2013 23 / 29

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 2013 23 / 29

Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation 2013 24 / 29

Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation 2013 24 / 29

Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation 2013 24 / 29

Conditional proof A system of natural deduction p, p but p p i.e. p p. Vít Punčochář (AS CR) Intensionalisation 2013 24 / 29

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 2013 25 / 29

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 2013 26 / 29

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 2013 27 / 29

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 2013 28 / 29

(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 2013 29 / 29

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