Alleged Assassins Bjrn Jespersen & Giuseppe Primiero Department - PowerPoint PPT Presentation

Alleged Assassins Bjørn Jespersen & Giuseppe Primiero Department of Computer Science, Technical University of Ostrava & Department of Logic, Czech Academy of Sciences, Prague FWO & Centre for Logic and Philosophy of Science, Ghent University bjorn.jespersen@gmail.com Giuseppe.Primiero@Ugent.be TbiLLC 2011 – Kutaisi, Georgia

Outline The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 2 / 24

The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 3 / 24

a is an alleged assassin ? Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 3 / 24

a is an alleged assassin ? what is the logical structure of the premise? what follows as conclusion? Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 3 / 24

Property Modification Let M be a modifier and F a property. Then ( MF ) is the result of the procedure of applying the function M to the argument F . Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 4 / 24

Property Modification Let M be a modifier and F a property. Then ( MF ) is the result of the procedure of applying the function M to the argument F . A full semantic theory of modification must include the following variants: ◮ Subsective: ( M ′ F ) a ∴ Fa ◮ Privative: ( M ′′ F ) a ∴ ¬ Fa ◮ Intersective: ( M ′′′ F ) a ∴ M ∗ a ∧ Fa ◮ Modal: M ′′′′ oscillates between subsection and privation Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 4 / 24

3 Negative Characterizations of M ′′′′ ( MF ) c x M ∗ c x ∧ F c x Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 5 / 24

3 Negative Characterizations of M ′′′′ ( MF ) c x M ∗ c x ∧ F c x F c x ↔ G c x F c x → ( MF ) c x G c x → ( MG ) c x Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 5 / 24

3 Negative Characterizations of M ′′′′ ( MF ) c x M ∗ c x ∧ F c x F c x ↔ G c x F c x → ( MF ) c x G c x → ( MG ) c x Fails to validate either of Fa , ¬ Fa as conclusion. Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 5 / 24

Task A positive characterization of modal modification. Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 6 / 24

A solution to privative modification [Primiero and Jespersen, 2010] offers two analyses of privative modification using two variants of procedural semantics : Realism: Tichý’s Transparent Intensional Logic Idealism: Martin-Löf’s Constructive Type Theory Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 7 / 24

A solution to privative modification [Primiero and Jespersen, 2010] offers two analyses of privative modification using two variants of procedural semantics : Realism: Tichý’s Transparent Intensional Logic Idealism: Martin-Löf’s Constructive Type Theory Common basic idea is to analyze modal modification in terms of possibility/contingency : TIL: alethic CTT: epistemic Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 7 / 24

The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 8 / 24

The Commmon Core a notion of construction 1 a functional language 2 a typed universe 3 an interpreted syntax 4 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 8 / 24

What Distinguishes TIL from CTT TIL CTT Semantics model-theoretic proof-theoretic Modifier property to property set to set Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 9 / 24

The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 10 / 24

TIL [Duži et al., 2010] Basic and Functional Types Ground Types: o , ι, τ, ω Property: ( o ι ) τω Property modifier: (( o ι ) τω ( o ι ) τω ) Proposition: o τω Propositional modifier: ( o τω o τω ) Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 10 / 24

Sentential Meaning “ a is an alleged assassin” λ w λ t [[ Alleged Assassin ] wt a ] Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 11 / 24

The speech act of allegation λ w λ t [ Alleges wt b λ w ′ λ t ′ [ F w ′ t ′ a ]] EG λ w λ t [ ∃ x [ ∃ P [ Alleges wt x P ]]] “ b alleges that a is an F ” “somebody alleges something” Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 12 / 24

Introduction rule for Alleged λ f [[ Alleged f ] wt a ] = λ f [ ∃ x [ Alleges wt x λ w λ t [ f wt a ]]] “being a property that a is alleged to have equals being a property that somebody alleges a to have” Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 13 / 24

Elimination Rule for Alleged [[ Alleged Assassin ] wt a ] ∃ w ′ [ ∃ t ′ [ Assassin w ′ t ′ a ]] ∧ ∃ w ′′ [ ∃ t ′′ ¬ [ Assassin w ′′ t ′′ a ]] Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 14 / 24

Introduction rule for Allegedly λ P [ Allegedly P ] = λ P [ λ w λ t [ ∃ x [ Alleges wt x P ]]] “being an alleged proposition equals being a proposition that somebody alleges” Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 15 / 24

Elimination rule for Allegedly [ Allegedly P ] wt ∃ w ′ [ ∃ t [ P w ′ t ′ ]] ∧ ∃ w ′′ [ ∃ t ′′ [ ¬ P w ′′ t ′′ ]] Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 16 / 24

The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 17 / 24

Two initial comments Given the judgemental structure of formulas in CTT, we can 1 model only the propositional modifier: ◮ from ‘ a is an alleged assassin’ to ‘Allegedly, a is an assassin’ The standard constructive syntax does not allow to deal with the 2 contingency required by modal modifiers: ◮ an extended language is required Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 17 / 24

Language [Primiero, 2012],[Primiero, 2011] Definition (Alphabet) The syntax is defined by the following alphabet: K : { type , type inf } (verifiers, possibly terminating processes) Types := A | ⊥ | A ∧ B | A ∨ B | A → B | A ⊃ B . Terms := x i | a i | ( a i , b j ) | ( x i ( b j )) | a i ( b j ) . Contexts := Γ i | ∆ i | ✷ i Γ | ✸ i Γ Judgements := ∆ i ; Γ i ⊢ A type | ✷ i ( A true ) | ✸ i ( A true ) | ◦ i , j Γ ⊢ ◦ i , j ( A true ) . Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 18 / 24

Language [Primiero, 2012],[Primiero, 2011] Definition (Alphabet) The syntax is defined by the following alphabet: K : { type , type inf } (verifiers, possibly terminating processes) Types := A | ⊥ | A ∧ B | A ∨ B | A → B | A ⊃ B . Terms := x i | a i | ( a i , b j ) | ( x i ( b j )) | a i ( b j ) . Contexts := Γ i | ∆ i | ✷ i Γ | ✸ i Γ Judgements := ∆ i ; Γ i ⊢ A type | ✷ i ( A true ) | ✸ i ( A true ) | ◦ i , j Γ ⊢ ◦ i , j ( A true ) . Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 19 / 24

Modal Modification Rule: Introduction Allegedly [a is an assassin] Assassin type [Γ] Property i type inf ∈ Γ Alleged ( x )[ x : Assassin ] ✷ Γ , ✸ ( Property i ) ⊢ a : Assassin [ x i / p i : Property i ] Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 20 / 24

Modal Modification Rule: Elimination I It is proven that [a is an assassin] ✷ Γ , ✸ ( Property i ) ⊢ a : Assassin [ x i / p i : Property i ] p i : Property i ✷ (Γ , p i : Property i ) ⊢ a : Assassin A type inf x : A ⊢ B type inf a : A β -conversion ( x ( b ))( a ) = b [ a / x ]: B type [ a / x ] Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 21 / 24

Modal Modification Rule: Elimination II The allegation that [a is an assassin] is false. ✷ Γ , x i : Property i ⊢ a : Assassin [ x i / p i : Property i ] p i : Property i → ⊥ a : Assassin → ⊥ Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 22 / 24

The Problem of Modal Modification 1 Solutions in terms of Procedural Semantics 2 Transparent Intensional Logic 3 Modal Constructive Type Theory 4 Conclusions 5 Jespersen, Primiero (Ostrava – Ghent) Modal Modification TbiLLC 2011 23 / 24