Through Many-Valent Semantics Carolina Blasio IFCH/UNICAMP PhDs in - PowerPoint PPT Presentation

Through Many-Valent Semantics Carolina Blasio IFCH/UNICAMP PhD’s in Logic May 3 rd , BOCHUM Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 1 / 20 Through Many-Valent Semantics

Introduction Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 2 / 20 Through Many-Valent Semantics

Suszko’s Thesis "Obviously, any multiplication of logical values is a mad idea." (Roman Suszko, 1977) Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 3 / 20 Through Many-Valent Semantics

Suszko’s Thesis "Obviously, any multiplication of logical values is a mad idea." (Roman Suszko, 1977) Every logic can be characterized by bivalent semantics. (Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015) Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 3 / 20 Through Many-Valent Semantics

Suszko’s Thesis "Obviously, any multiplication of logical values is a mad idea." (Roman Suszko, 1977) Every logic can be characterized by bivalent semantics. (Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015) Many-valued semantics could be used as a tool. (Avron, 2009) Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 3 / 20 Through Many-Valent Semantics

Suszko’s Thesis "Obviously, any multiplication of logical values is a mad idea." (Roman Suszko, 1977) Every logic can be characterized by bivalent semantics. (Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015) Many-valued semantics could be used as a tool. (Avron, 2009) Two kinds of truth-values referential truth-values: inferential truth-values: make up many-valued consequence relation validity semantics Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 3 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Standard valuation matrix: M = �V , D , O� V V := Truth-values, D := D ⊆ V , the designated values, D O := Truth-functions for each connective of S . Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Standard valuation matrix: M = �V , D , O� V V := Truth-values, D := D ⊆ V , the designated values, D O := Truth-functions for each connective of S . Entailment relation based on M v (Φ) = { v ( φ ) | φ ∈ Φ } Γ | = ∆ iff there is no v such that v (Γ) ⊆ D and v (∆) ⊆ V − D Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Standard valuation matrix: M = �V , D , O� V V := Truth-values, D := D ⊆ V , the designated values, D O := Truth-functions for each connective of S . Entailment relation based on M v (Φ) = { v ( φ ) | φ ∈ Φ } Γ | = ∆ iff there is no v such that v (Γ) ⊆ D and v (∆) ⊆ V − D Proposition (The following holds in a standard logic:) Reflexivity α � α Monotonicity If Γ ′ | = ∆ ′ , then Γ ′ , Γ ′′ | = ∆ ′ , ∆ ′′ Transitivity If Σ , Γ | = ∆ , Π for every quasi-partition* � Σ , Π � of a Θ ⊆S , then Γ | = ∆ . * Σ ∪ Π = Θ and Σ ∩ Π = ∅ Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Standard valuation matrix: M = �V , Y , O� V V := Truth-values, Y := Y ⊆ V , the accepted values, Y O := Truth-functions for each connective of S . Entailment relation based on M v (Φ) = { v ( φ ) | φ ∈ Φ } Γ | = ∆ iff there is no v such that v (Γ) ⊆ Y and v (∆) ⊆ V − Y Proposition (The following holds in a standard logic:) Reflexivity α � α Monotonicity If Γ ′ | = ∆ ′ , then Γ ′ , Γ ′′ | = ∆ ′ , ∆ ′′ Transitivity If Σ , Γ | = ∆ , Π for every quasi-partition* � Σ , Π � of a Θ ⊆S , then Γ | = ∆ . * Σ ∪ Π = Θ and Σ ∩ Π = ∅ Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Many-valent Semantic, but Bivalent Logics Let S be a propositional language. Standard valuation matrix: M = �V , N , O� V V := Truth-values, N := N ⊆ V , the rejected values, N O := Truth-functions for each connective of S . Entailment relation based on M v (Φ) = { v ( φ ) | φ ∈ Φ } Γ | = ∆ iff there is no v such that v (Γ) ⊆ V − N and v (∆) ⊆ N Proposition (The following holds in a standard logic:) Reflexivity α � α Monotonicity If Γ ′ | = ∆ ′ , then Γ ′ , Γ ′′ | = ∆ ′ , ∆ ′′ Transitivity If Σ , Γ | = ∆ , Π for every quasi-partition* � Σ , Π � of a Θ ⊆S , then Γ | = ∆ . * Σ ∪ Π = Θ and Σ ∩ Π = ∅ Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 4 / 20 Through Many-Valent Semantics

Logical Bivalence into Question Before Suszko’s Thesis non-determinism, probability, predictions and uncertainty issues; 1920’s: Łukaziewicz’s Ł 3 ; referential truth-values; Suszko Reduction: bivalent. Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 5 / 20 Through Many-Valent Semantics

Logical Bivalence into Question Before Suszko’s Thesis After Suszko’s Thesis non-determinism, non-determinism, probability, predictions probability, predictions and uncertainty issues; and uncertainty again!; 1920’s: Łukaziewicz’s Ł 3 ; 1990’s: Malinowski q -entailment; referential truth-values; many-valent; Suszko Reduction: bivalent. Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 5 / 20 Through Many-Valent Semantics

Logical Bivalence into Question Before Suszko’s Thesis After Suszko’s Thesis non-determinism, non-determinism, probability, predictions probability, predictions and uncertainty issues; and uncertainty again!; 1920’s: Łukaziewicz’s Ł 3 ; 1990’s: Malinowski q -entailment; referential truth-values; many-valent; Suszko Reduction: bivalent. non-reflexive/ non-transitive entailments!!! Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 5 / 20 Through Many-Valent Semantics

Section 1 Trivalent Logics Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 6 / 20 Through Many-Valent Semantics

q -entailment (G. Malinowski, 1990) Related to the reasoning by hypotheses; If no statement of the conclusion is accepted then some of the premisses should be rejected. Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 7 / 20 Through Many-Valent Semantics

q -entailment (G. Malinowski, 1990) Related to the reasoning by hypotheses; If no statement of the conclusion is accepted then some of the premisses should be rejected. L q = �S , � q � q -matrix: Q = �V , Y , N , O� V V := Truth-values; Y := the accepted values; N:= the rejected Y N values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S . Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 7 / 20 Through Many-Valent Semantics

q -entailment (G. Malinowski, 1990) L q = �S , � q � q -matrix: Q = �V , Y , N , O� V V := Truth-values; Y := the accepted values; N:= the rejected Y N values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S . Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 7 / 20 Through Many-Valent Semantics

q -entailment (G. Malinowski, 1990) L q = �S , � q � q -matrix: Q = �V , Y , N , O� V V := Truth-values; Y := the accepted values; N:= the rejected Y N values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S . q -entailment relation based on Q = q ∆ iff there is no v such that v (Γ) ⊆ Y Γ | N and v (∆) ⊆ Y where := V − Y and N := V − N. Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 7 / 20 Through Many-Valent Semantics

q -entailment (G. Malinowski, 1990) L q = �S , � q � q -matrix: Q = �V , Y , N , O� V V := Truth-values; Y := the accepted values; N:= the rejected Y N values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S . q -entailment relation based on Q = q ∆ iff there is no v such that v (Γ) ⊆ Y Γ | N and v (∆) ⊆ Y where := V − Y and N := V − N. Proposition (The following holds in a q -logic:) Monotonicity If Γ ′ | = ∆ ′ , then Γ ′ , Γ ′′ | = ∆ ′ , ∆ ′′ Transitivity If Σ , Γ | = ∆ , Π for every q-partition* � Σ , Π � of a Θ ⊆S , then Γ | = ∆ . * Σ ∪ Π ⊆ Θ and Σ ∩ Π = ∅ Carolina Blasio (IFCH/UNICAMP) Logic and Epistemology 7 / 20 Through Many-Valent Semantics