A paraconsistent approach to da Costa’s deontic logic: beyond contradictions and triviality Gregory Carneiro Department of Philosophy, University of Bras´ ılia, Brazil Vichy-France, June, 2018
New Systems of Predicate Deontic Logic Newton Carneiro da Costa, New Systems of Predicate Deontic Logic , The Journal of Non-Classical Logic, vol. 5(2), 1988, pp. 75–80. Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 2 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L r System L r is a propositional-like calculus (PC) with the related propositional-like language ( p , q , z , ... A , B , C , ... ∨ , → , ¬ , ... ) as a logic of action. p = “Mary to drive drunk”. p ∨ q = “Mary to drive drunk or Peter to sleep on the floor”. (...) The L r formulas do not hold truth-value, but performance-value: an action is performed (P) or not-performed (N). The semantical aspect is, like the language, also isomorphic in a natural way to the everyday PC. For instance: ¬ A is performed iff A is not-performed. A ∨ B is performed iff A is performed or B is performed. (...) Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 3 / 22
L s System A1. λ ¬ A ↔ ¬ λ A A2. λ ( A ∨ B ) ↔ ( λ A ∨ λ B ) A3. λ ∃ xA ↔ ∃ x λ A Formulas of L s , as propositions, are truth-value holders. A λ A P T N F Γ ⊢ Lr A iff Γ ⊢ Ls λ A Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
L s System A1. λ ¬ A ↔ ¬ λ A A2. λ ( A ∨ B ) ↔ ( λ A ∨ λ B ) A3. λ ∃ xA ↔ ∃ x λ A Formulas of L s , as propositions, are truth-value holders. A λ A P T N F Γ ⊢ Lr A iff Γ ⊢ Ls λ A Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
L s System A1. λ ¬ A ↔ ¬ λ A A2. λ ( A ∨ B ) ↔ ( λ A ∨ λ B ) A3. λ ∃ xA ↔ ∃ x λ A Formulas of L s , as propositions, are truth-value holders. A λ A P T N F Γ ⊢ Lr A iff Γ ⊢ Ls λ A Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
L s System A1. λ ¬ A ↔ ¬ λ A A2. λ ( A ∨ B ) ↔ ( λ A ∨ λ B ) A3. λ ∃ xA ↔ ∃ x λ A Formulas of L s , as propositions, are truth-value holders. A λ A P T N F Γ ⊢ Lr A iff Γ ⊢ Ls λ A Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
The System D The “ ⇒ ” dyadic deontic operator is added to the language of L r and L s . α ⇒ A means that “If α is the case, then A is obrigatory”. Let the L s formulas be called P-formulas, represented by α, β, γ... Let the L r formulas be called A-formulas, represented by A, B, C... Note that A ⇒ α is not a wff in D. Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 5 / 22
The System D The “ ⇒ ” dyadic deontic operator is added to the language of L r and L s . α ⇒ A means that “If α is the case, then A is obrigatory”. Let the L s formulas be called P-formulas, represented by α, β, γ... Let the L r formulas be called A-formulas, represented by A, B, C... Note that A ⇒ α is not a wff in D. Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 5 / 22
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