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Monadic Predicate ukasiewicz Logic. Standard versus General Tautologies Flix Bou University of Barcelona (UB) bou@ub.edu May 20th ASUV 2011 (Salerno) Flix Bou (UB) Monadic Predicate ukasiewicz Logic May 20th, ASUV2011, Salerno 1 /

  1. Monadic Predicate Łukasiewicz Logic. Standard versus General Tautologies Félix Bou University of Barcelona (UB) bou@ub.edu May 20th ASUV 2011 (Salerno) Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 1 / 23

  2. Preliminaries Classical Predicate Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ ) + vocabulary ϑ Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 2 / 23

  3. Preliminaries Classical Predicate Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ ) + vocabulary ϑ Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 2 / 23

  4. Preliminaries Classical Predicate Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ ) + vocabulary ϑ Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: 2-valued (for every structure, we get a mapping from sentences into { 0 , 1 } ). Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 2 / 23

  5. Preliminaries Classical Predicate Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ ) + vocabulary ϑ Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: 2-valued (for every structure, we get a mapping from sentences into { 0 , 1 } ). Full vocabulary Valid sentences are recursively enumerable (Gödel) Undecidability of valid sentences (Church, . . . ) FO 2 is decidable: “effective fmp” holds (Scott, Mortimer) FO 3 is undecidable (Surányi, . . . ) Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 2 / 23

  6. Preliminaries Classical Predicate Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ ) + vocabulary ϑ Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: 2-valued (for every structure, we get a mapping from sentences into { 0 , 1 } ). Full vocabulary Valid sentences are recursively enumerable (Gödel) Undecidability of valid sentences (Church, . . . ) FO 2 is decidable: “effective fmp” holds (Scott, Mortimer) FO 3 is undecidable (Surányi, . . . ) Monadic vocabulary: P 1 , P 2 , P 3 , . . . Decidability of valid sentences: filtration method provides an “effective fmp” (Löwenheim). Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 2 / 23

  7. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  8. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  9. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  10. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: [ 0 , 1 ] -valued (for every structure, we get a mapping from sentences into [ 0 , 1 ] ). Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  11. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: [ 0 , 1 ] -valued (for every structure, we get a mapping from sentences into [ 0 , 1 ] ). Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  12. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: [ 0 , 1 ] -valued (for every structure, we get a mapping from sentences into [ 0 , 1 ] ). ∀ xPx = ∃ x ( Px ∨ ¬ Px ) = Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  13. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: [ 0 , 1 ] -valued (for every structure, we get a mapping from sentences into [ 0 , 1 ] ). ∀ xPx = 0 . 2 ∃ x ( Px ∨ ¬ Px ) = Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  14. Preliminaries Predicate Łukasiewicz (standard) Logic Syntax: Primitive symbols ( ∀ , ∃ , ¬ , ∨ , ∧ , → ) + vocabulary ϑ Definable symbols: ⊙ , ⊕ , ⊖ , etc . Examples of Sentences: ∀ xPx , ∀ x ∃ yRxy , etc. Semantics: [ 0 , 1 ] -valued (for every structure, we get a mapping from sentences into [ 0 , 1 ] ). ∀ xPx = 0 . 2 ∃ x ( Px ∨ ¬ Px ) = 1 Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 3 / 23

  15. Preliminaries Three semantics using MV-chains Standard ( stL ∀ ): [ 0 , 1 ] -valued General ( genL ∀ ): A -valued (where A is an arbitrary MV-chain) structures requiring “safeness” condition (all formulas in ϑ have a truth value). Supersound ( spsL ∀ ): A -valued (where A is an arbitrary MV-chain) structures only requiring the existence of the value of your formula. Some Trivial Remarks Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 4 / 23

  16. Preliminaries Three semantics using MV-chains Standard ( stL ∀ ): [ 0 , 1 ] -valued General ( genL ∀ ): A -valued (where A is an arbitrary MV-chain) structures requiring “safeness” condition (all formulas in ϑ have a truth value). Supersound ( spsL ∀ ): A -valued (where A is an arbitrary MV-chain) structures only requiring the existence of the value of your formula. Some Trivial Remarks spsL ∀ ⊆ genL ∀ ⊆ stL ∀ Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 4 / 23

  17. Preliminaries Three semantics using MV-chains Standard ( stL ∀ ): [ 0 , 1 ] -valued General ( genL ∀ ): A -valued (where A is an arbitrary MV-chain) structures requiring “safeness” condition (all formulas in ϑ have a truth value). Supersound ( spsL ∀ ): A -valued (where A is an arbitrary MV-chain) structures only requiring the existence of the value of your formula. Some Trivial Remarks spsL ∀ ⊆ genL ∀ ⊆ stL ∀ Safeness holds when A is complete (e.g., A finite). Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 4 / 23

  18. Preliminaries Three semantics using MV-chains Standard ( stL ∀ ): [ 0 , 1 ] -valued General ( genL ∀ ): A -valued (where A is an arbitrary MV-chain) structures requiring “safeness” condition (all formulas in ϑ have a truth value). Supersound ( spsL ∀ ): A -valued (where A is an arbitrary MV-chain) structures only requiring the existence of the value of your formula. Some Trivial Remarks spsL ∀ ⊆ genL ∀ ⊆ stL ∀ Safeness holds when A is complete (e.g., A finite). Safeness holds in the following cases: finite structures, structures where the range of vocabulary symbols is finite (“secure”), witnessed structures. Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 4 / 23

  19. Preliminaries Full vocabulary genL ∀ is Σ 1 -complete (Chang, Belluce) stL ∀ is not in Σ 1 (Scarpellini) stL ∀ is Π 2 -complete (Ragaz) stL ∀ = � n ∈ ω Taut ( L n ) = Taut ([ 0 , 1 ] ∩ Q ) . (Rutledge) General and standard semantics are complete for witnessed structures (Hájek) Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 5 / 23

  20. Preliminaries Full vocabulary genL ∀ is Σ 1 -complete (Chang, Belluce) stL ∀ is not in Σ 1 (Scarpellini) stL ∀ is Π 2 -complete (Ragaz) stL ∀ = � n ∈ ω Taut ( L n ) = Taut ([ 0 , 1 ] ∩ Q ) . (Rutledge) General and standard semantics are complete for witnessed structures (Hájek) Monadic vocabulary: P 1 , P 2 , P 3 , . . . stL ∀ is in Π 1 . Filtration method shows fmp (i.e., if ϕ �∈ stL ∀ then it is not valid in some finite [ 0 , 1 ] -structure) (Hájek) standard and general semantics coincide for FO 1 , and it is decidable (Rutledge) standard and general semantics coincide for “classical formulas” (i.e., ∀ , ∃ , ¬ , ∧ , ∨ ), and this fragment is decidable. Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 5 / 23

  21. Preliminaries From Scarpellini’s (and Ragaz) result it follows that there are sentences which are standard tautologies while not general tautologies, but his proof do not provide us any explicit example. . Félix Bou (UB) Monadic Predicate Łukasiewicz Logic May 20th, ASUV2011, Salerno 6 / 23

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