On Prawitz’ Ecumenical system Luiz Carlos Pereira Elaine Pimentel Valeria de Paiva PUC-Rio/UERJ/CNPQ UFRN University of Birmingham Proof-theoretical semantics, T¨ ubingen, 2019 Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
With a very big help from my friends! (Dag Prawitz, Alberto Naibo, Luca Tranchini, Victor Nascimento, Wagner Sanz, Hermann Haeusler..) Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Plan for the talk What is Ecumenism? 1 Prawitz’ system 2 One digression 3 A bit of proof theory 4 New ecumenical systems 5 Related and future work 6 Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Ecumenism Ecumenical systems Main idea: a codification where two or more logics can coexist in peace. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Ecumenism Coexisting in peace: the different (maybe rival!) logics accept and reject the same things (principles, rules,...) Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
The ecumenical view Prawitz 2015 Dowek 2015 Krauss 1992 Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
A possible problem for revisionism in Logic An standard form of disqualifying the conflict between two logics is based on the somewhat reasonable idea that the litigants are talking about distinct things (or speaking different things), and that if they are talking about different things, there is not “the same thing” - a rule or a principle - on which they diverge and dispute. According to this position, it is as if the participants of the conflict spoke different languages and did not realize it. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
A possible problem for revisionism in Logic An easy argument (Quine, 1970): If the deviant/revisionist logician does not accept the general validity 1 of a classical principle of reasoning, then he gives new meanings to the concepts used in the formulation of the principle. If the deviant logician gives new meanings to the concepts used in 2 the formulation of the principle, then the deviant logician and the classical logician are not talking about the same thing (principle). If they are are talking about different things, they cannot disagree!!! 3 The deviant logician does not accept the general validity of the 4 principle. Thus, they do not disagree!!!! Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Dag Prawitz seems to agree with Quine when he says: ”When the classical and intuitionistic codifications attach different meanings to a constant, we need to use different symbols, and I shall use a subscript c for the classical meaning and i for the intuitionistic. The classical and intuitionistic constants can then have a peaceful coexistence in a language that contains both.” (Prawitz [2015]) Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
An alternative is to use the idea of Hilbert and Poincar´ e that axioms and deduction rules define the meaning of the symbols of the language and it is then possible to explain that some judge the proposition ( P ∨ ¬ P ) true and others do not because they do not assign the same meaning to the symbols ∨ , ¬ , etc. (Dowek [2015]) Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Taking this idea seriously, we should not say that the proposition ( P ∨ ¬ P ) has a classical proof but no constructive proof, but we should say that the proposition ( P ∨ c ¬ c P ) has a proof and the proposition ( P ∨ ¬ P ) does not, that is we should introduce two symbols for each connective and quantifier, for instance a symbol ∨ for the constructive disjunction and a symbol ∨ c for the classical one, instead of introducing two judgments: “has a classical proof” and “has a constructive proof’ (Dowek [2015]) Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
What’s Prawitz’ main idea? The same meaning explanation for classical logic and intuitionistic logic. But this does not seem possible! Gentzen’s introduction rule for disjunction (and for implication and the existential quantifier) is too strong! It cannot give the meaning of classical disjunction. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Solution: different introduction rules for classical disjunction Interesting: two disjunctions, but the same idea of meaning explanation. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
The Natural Deduction Ecumenical system NEc Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
The language of NEc is defined as follow: Alphabet Individual variables, individual parameters, predicate letters; 1 logical constants: ⊥ , ∧ , ¬ , ∀ , ∨ i , ∨ c , → i and → c , ∃ i , ∃ c ; 2 Auxiliary signs: (, ) . 3 The grammar of Ec is the usual. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
The Natural Deduction system NEc defined by Prawitz has the following rules of inference: The rules for ∧ , ¬ and for the intuitionistic operators are the usual 1 Gentzen-Prawitz introduction and elimination for these operators. The intuitionistic absurd rule: 2 ⊥ A The rules for classical disjunction and classical implication are 3 defined as follows: Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
[ A ] [ ¬ B ] Π 1 ⊥ → c -Int A → c B A → c B A ¬ B → c -Elim ⊥ Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
[ ¬ A ] [ ¬ B ] Π 1 ⊥ ∨ c -Int A ∨ c B A ∨ c B ¬ A ¬ B ∨ c -Elim ⊥ Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
[ ∀ x ¬ A ( x )] Π ⊥ ∃ c − I ∃ c xA ( x ) ∃ c xA ( x ) ∀ x ¬ A ( x ) ∃ c − E ⊥ Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Classical implication and modus ponens Classical implication: Contrary to what we could expect from any reasonable concept of conditional judgements (hypothetical judgement), the operator → c does not satisfy modus ponens . This is due to the fact that the introduction rule for → c is weaker than the introduction for → i , since the classical logician is allowed to assert ( A → c B ) in cases where the intuitionistic logician is not. It is interesting to observe that the general validity of modus ponens for → c would not depend solely on the meaning of → c , but would also depend on a concept of negation that is not determined by the introduction rule for negation. The classical implication → c clearly satisfies a weak form of modus ponens : { A, (A → c B) } ⊢ ¬¬ B . Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Some interesting theorems ⊢ NEc ( A → i B ) ⇒ i ( A → c B ) 1 ⊢ NEc ( A ∧ B ) ⇔ i ¬ ( ¬ A ∨ c ¬ B ) 2 ⊢ NEc ( A ∧ B ) ⇔ i ¬ ( A → c ¬ B ) 3 ⊢ NEc ¬ ( ¬ A ∧ ¬ B ) ⇔ i ( A ∨ c B ) 4 ⊢ NEc ¬ ( A ∧ ¬ B ) ⇔ i ( A → c B ) 5 Definition A formula B is called classical if and only if its main operator is classical (we sometimes indicate that B is classical with the notation B c ) Some more interesting theorems ⊢ NEc ( A → c B c ) → i ( A → i B c ) 1 { A, ( A → c B c ) } ⊢ NEc B c } 2 Interesting remark: The system NEc does not satisfy the deduction theorem! Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
One negation or Two negations In the propositional part of the ecumenical system defined by Prawitz, we have the following logical constants: ∧ , ⊥ , ¬ , → i , → c , ∨ i , and ∨ c . The problem now is: why do we have just one negation, given that we have two implications and the negation of A could be understood as “A implies ⊥ ”? (The “ecumenical” system defined by Peter Krauss uses a single negation, and although the system defined by Gilles Dowek begins with two negations, at some point in the paper (p.232), Dowek concludes that the system can work with just one negation). Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
Two possible answers: We can prove that ( A → i ⊥ ) and ( A → c ⊥ ) are “interderivable” in 1 the ecumenical system, in the sense that the equivalences (( A → i ⊥ ) ↔ i ( A → c ⊥ )) and (( A → i ⊥ ) ↔ c ( A → c ⊥ )) are provable in the ecumenical system. We can argue that in fact there’s only one way to assert the 2 negation of a proposition A: in order to assert ¬ A we have to derive a contradiction from A. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system
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