A paraconsistent approach to da Costas deontic logic: beyond - - PowerPoint PPT Presentation
A paraconsistent approach to da Costas deontic logic: beyond contradictions and triviality Gregory Carneiro Department of Philosophy, University of Bras lia, Brazil Vichy-France, June, 2018 New Systems of Predicate Deontic Logic Newton
Gregory Carneiro
Department of Philosophy, University of Bras´ ılia, Brazil
Vichy-France, June, 2018
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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Lr 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 Lr 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
Formulas of Ls, as propositions, are truth-value holders. A λ A P T N F Γ ⊢Lr A iff Γ ⊢Ls λA
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
Formulas of Ls, as propositions, are truth-value holders. A λ A P T N F Γ ⊢Lr A iff Γ ⊢Ls λA
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
Formulas of Ls, as propositions, are truth-value holders. A λ A P T N F Γ ⊢Lr A iff Γ ⊢Ls λA
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
Formulas of Ls, as propositions, are truth-value holders. A λ A P T N F Γ ⊢Lr A iff Γ ⊢Ls λA
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 4 / 22
The “⇒” dyadic deontic operator is added to the language of Lr and Ls. α ⇒ A means that “If α is the case, then A is obrigatory”. Let the Ls formulas be called P-formulas, represented by α, β, γ... Let the Lr 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 “⇒” dyadic deontic operator is added to the language of Lr and Ls. α ⇒ A means that “If α is the case, then A is obrigatory”. Let the Ls formulas be called P-formulas, represented by α, β, γ... Let the Lr 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 “⇒” dyadic deontic operator is added to the language of Lr and Ls. α ⇒ A means that “If α is the case, then A is obrigatory”. Let the Ls formulas be called P-formulas, represented by α, β, γ... Let the Lr 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
Rules: R1. α ↔ β (α ⇒ A) ↔ (β ⇒ A) R2. (A ↔ B) (α ⇒ A) ↔ (α ⇒ B) Definitions:
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 6 / 22
“Note. The author is indebted to Professor G. H. von Wright and F. Mir´
have urged, on numerous occasions, the necessity of constructing more realistic logics of action, as well as paraconsistent and paracomplete deontic logics (see, for instance, von Wright [7] and Mir´
[4]F. Mir´
de Lima, 1986. [7]G. von Wright. Philosophical Logic, Volume III of Philosophical Papers, Basil Blackwell, 1983.
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 7 / 22
In a very simple and naive sense, we have two basic kinds of actions considered in the ethical and legal field: Basic Actions x Complex Actions
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 8 / 22
Basic actions can be defined by only its objective, external
same time strong and intuitive) face. For instance: To help a blind man to cross the road. It’s important note that the goal of the agent or the intention part does not play any immediate role in this kind of action. In this sense they are simple (basic).
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 9 / 22
In this kind of actions goals of the agents or their intentions does play a strong role. In this sense they are complex - with a external part and a internal one. ◮ To lie in order to deceive ◮ To stab in order to kill In a broad sense any action is can be said complex (!), but in the
and to deceive is to deceive no matter how its done. ◮ To Murder: by stabbing, shooting, poisoning... ◮ To deceive: by telling the truth, by lying, by remaining in silence...
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 10 / 22
What if even when the stabbing or shooting is a perfectly concluded action the victim does not die? What about lying without achieving the desired deception? Franca D’Agostini - The Logic of Lying Tutorial ◮Reflexion: Without the related desire, goal, there is a substantial change in ethical/legal intuitions involved. In a legal or ethical scenario where to kill or to deceive is wrong, we can say that a complex action without the desired outcome is still wrong/immoral?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 11 / 22
What if even when the stabbing or shooting is a perfectly concluded action the victim does not die? What about lying without achieving the desired deception? Franca D’Agostini - The Logic of Lying Tutorial ◮Reflexion: Without the related desire, goal, there is a substantial change in ethical/legal intuitions involved. In a legal or ethical scenario where to kill or to deceive is wrong, we can say that a complex action without the desired outcome is still wrong/immoral?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 11 / 22
What if even when the stabbing or shooting is a perfectly concluded action the victim does not die? What about lying without achieving the desired deception? Franca D’Agostini - The Logic of Lying Tutorial ◮Reflexion: Without the related desire, goal, there is a substantial change in ethical/legal intuitions involved. In a legal or ethical scenario where to kill or to deceive is wrong, we can say that a complex action without the desired outcome is still wrong/immoral?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 11 / 22
What if even when the stabbing or shooting is a perfectly concluded action the victim does not die? What about lying without achieving the desired deception? Franca D’Agostini - The Logic of Lying Tutorial ◮Reflexion: Without the related desire, goal, there is a substantial change in ethical/legal intuitions involved. In a legal or ethical scenario where to kill or to deceive is wrong, we can say that a complex action without the desired outcome is still wrong/immoral?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 11 / 22
What if even when the stabbing or shooting is a perfectly concluded action the victim does not die? What about lying without achieving the desired deception? Franca D’Agostini - The Logic of Lying Tutorial ◮Reflexion: Without the related desire, goal, there is a substantial change in ethical/legal intuitions involved. In a legal or ethical scenario where to kill or to deceive is wrong, we can say that a complex action without the desired outcome is still wrong/immoral?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 11 / 22
Augustine: Then perhaps lust is the evil in adultery, and you will run into difficulties as long as you are looking for evil in the outward visible deed. Now to understand that lust is the evil in adultery, consider the following. If a man does not have the opportunity to sleep with someone else’s wife but it is plain somehow that he wants to do so, and that he is going to do so should the opportunity arise, he is no less guilty than if he were caught in the act. Evodius: Nothing could be more obvious. Now I see that there is no need for a long discussion to persuade me about murder, sacrilege, and in fact all other sins. It is clear now that nothing but lust dominates in every kind of evildoing. Augustine: You do know, do you not, that lust is also called “desire”?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 12 / 22
Let’s assume that the action of murdering is forbidden (M). In D, considering γ a tautology of Ls
Using the simple semantical property of Lr, we know that “¬M” is performed iff “M” is not-performed. Assuming that, it is intuitive to think that the attempting of a murder (formalized as ¬M) is also
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 13 / 22
γO((¬¬M → M) ∧ (M → ¬¬M)), by a classical negation property (γO(¬¬M → M)) ∧ (γO(M → ¬¬M)), by A7 γO(¬¬M → M), By (α ∧ β) → α and Modus Ponens (γO¬¬M) → (γOM), by A4 and Modus Ponens 2*. γOM, by Modus Ponens The classical double negation equivalence is proved in D!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 14 / 22
3.(γOM) → ((γO¬M) → ((γOM) ∧ (γO¬M)), by Axiom for A-Formulas and Rule 2
A-formulas, Rule 2, A4 and Modus Ponens)
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 15 / 22
Let’s use C1, the first paraconsistent logic in da Costa’s hierarchy C0 Cn Cω: A → (B → A) (A → B) → ((A → (B → C)) → (A → C) (A ∧ B) → A (A ∧ B) → B A → (B → (A ∧ B)) A → (A ∨ B) B → (A ∨ B) (A → C) → ((B → C) → (A ∨ B → C)
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 16 / 22
The axiomatic diference from Classical logics comes in the negation and “well behaved” axioms: A ∨ ¬A ¬¬A → A Ao → (¬A)o Bo → ((A → B) → ((A → ¬B) → ¬A) Ao ∧ Bo → ((A → B)o ∧ (A ∧ B)o ∧ (A ∨ B)o) Definition: Ao =def ¬(A ∧ ¬A) Modus Ponens: A, A → B B
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 17 / 22
In C1, the Ex Falso Quodlibet only happens in the presence of
Bo → ((A → B) → ((A → ¬B) → ¬A). The extra axioms for DC1 are the same as D with three exceptions: A1’. λ ∼ A ↔∼ λA A5’. (γ ⇒ A) →∼ (γ ⇒∼ A), γ a p-tautology. A8’. Ao → (α ⇒ A)o The classical negation “∼” is defined in C1: ∼ A =def Ao ∧ ¬A
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 18 / 22
Since (A ∧ ¬A) → B, from γO(M ∧ ¬M) one cannot derive γOA, γ p-tautology and A any action, in the context presented before. Do we have here a real case of Dialetheism?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 19 / 22
Since (A ∧ ¬A) → B, from γO(M ∧ ¬M) one cannot derive γOA, γ p-tautology and A any action, in the context presented before. Do we have here a real case of Dialetheism?
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 19 / 22
Is the attempting of an action really so important? Avoiding the paradox by formalizing the method (i.e, the stabbing, the lie...)? Using the conjunction? O¬(S ∧ M)? O(¬S ∧ ¬M)? What about the intuition that follows the attempted action in most legal codes and moral theories? Attempting an action seems connected with some natural caracteristic of the negation
conjunction, disjunction or any other conective!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 20 / 22
Is the attempting of an action really so important? Avoiding the paradox by formalizing the method (i.e, the stabbing, the lie...)? Using the conjunction? O¬(S ∧ M)? O(¬S ∧ ¬M)? What about the intuition that follows the attempted action in most legal codes and moral theories? Attempting an action seems connected with some natural caracteristic of the negation
conjunction, disjunction or any other conective!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 20 / 22
Is the attempting of an action really so important? Avoiding the paradox by formalizing the method (i.e, the stabbing, the lie...)? Using the conjunction? O¬(S ∧ M)? O(¬S ∧ ¬M)? What about the intuition that follows the attempted action in most legal codes and moral theories? Attempting an action seems connected with some natural caracteristic of the negation
conjunction, disjunction or any other conective!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 20 / 22
Is the attempting of an action really so important? Avoiding the paradox by formalizing the method (i.e, the stabbing, the lie...)? Using the conjunction? O¬(S ∧ M)? O(¬S ∧ ¬M)? What about the intuition that follows the attempted action in most legal codes and moral theories? Attempting an action seems connected with some natural caracteristic of the negation
conjunction, disjunction or any other conective!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 20 / 22
Is the attempting of an action really so important? Avoiding the paradox by formalizing the method (i.e, the stabbing, the lie...)? Using the conjunction? O¬(S ∧ M)? O(¬S ∧ ¬M)? What about the intuition that follows the attempted action in most legal codes and moral theories? Attempting an action seems connected with some natural caracteristic of the negation
conjunction, disjunction or any other conective!
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 20 / 22
Paradoxes in Deontic Logic in general carry too many assumptions (i.e, the contrary-to-duty paradox). The incomplete actions paradoxes are very basic and just part of a common ethical and legal discourse. In fact, basic enough to make us question traditions of the normative logic community: Why use classical logic even when we do not want to talk about foundations or mathematical properties? Why use non-classical reasoning just as a back up plan (for classical reasoning) when we evidences shows we are better in the philosophical stand point with non-classical logic? The paradox of attempted action shows us that non-classical logic can be used to expand the expressive power of a logic too. Therefore, not just extensions has the power to do that.
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 21 / 22
Paradoxes in Deontic Logic in general carry too many assumptions (i.e, the contrary-to-duty paradox). The incomplete actions paradoxes are very basic and just part of a common ethical and legal discourse. In fact, basic enough to make us question traditions of the normative logic community: Why use classical logic even when we do not want to talk about foundations or mathematical properties? Why use non-classical reasoning just as a back up plan (for classical reasoning) when we evidences shows we are better in the philosophical stand point with non-classical logic? The paradox of attempted action shows us that non-classical logic can be used to expand the expressive power of a logic too. Therefore, not just extensions has the power to do that.
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 21 / 22
Paradoxes in Deontic Logic in general carry too many assumptions (i.e, the contrary-to-duty paradox). The incomplete actions paradoxes are very basic and just part of a common ethical and legal discourse. In fact, basic enough to make us question traditions of the normative logic community: Why use classical logic even when we do not want to talk about foundations or mathematical properties? Why use non-classical reasoning just as a back up plan (for classical reasoning) when we evidences shows we are better in the philosophical stand point with non-classical logic? The paradox of attempted action shows us that non-classical logic can be used to expand the expressive power of a logic too. Therefore, not just extensions has the power to do that.
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 21 / 22
Paradoxes in Deontic Logic in general carry too many assumptions (i.e, the contrary-to-duty paradox). The incomplete actions paradoxes are very basic and just part of a common ethical and legal discourse. In fact, basic enough to make us question traditions of the normative logic community: Why use classical logic even when we do not want to talk about foundations or mathematical properties? Why use non-classical reasoning just as a back up plan (for classical reasoning) when we evidences shows we are better in the philosophical stand point with non-classical logic? The paradox of attempted action shows us that non-classical logic can be used to expand the expressive power of a logic too. Therefore, not just extensions has the power to do that.
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 21 / 22
Gregory Carneiro University of Bras´ ılia Departament of Philosophy gregorycarneiro@outlook.com
Gregory Carneiro (University of Bras´ ılia) Beyond Contradictions and Triviality Vichy-France, June, 2018 22 / 22