On Prawitz Ecumenical system Luiz Carlos Pereira Elaine Pimentel - - PowerPoint PPT Presentation

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

1

What is Ecumenism?

2

Prawitz’ system

3

One digression

4

A bit of proof theory

5

New ecumenical systems

6

Related and future work

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

• n 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):

1

If the deviant/revisionist logician does not accept the general validity

• f a classical principle of reasoning, then he gives new meanings to

the concepts used in the formulation of the principle.

2

If the deviant logician gives new meanings to the concepts used in the formulation of the principle, then the deviant logician and the classical logician are not talking about the same thing (principle).

3

If they are are talking about different things, they cannot disagree!!!

4

The deviant logician does not accept the general validity of the 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 ¬cP) 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

1

Individual variables, individual parameters, predicate letters;

2

logical constants: ⊥, ∧, ¬, ∀, ∨i, ∨c, →i and →c, ∃i, ∃c;

3

Auxiliary signs: (, ) . 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:

1

The rules for ∧, ¬ and for the intuitionistic operators are the usual Gentzen-Prawitz introduction and elimination for these operators.

2

The intuitionistic absurd rule: ⊥ A

3

The rules for classical disjunction and classical implication are defined as follows:

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

[A] [¬B] Π1 ⊥ →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] Π1 ⊥ ∨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

∃cxA(x) ∃cxA(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

1

⊢NEc (A →i 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

⊢NEc ¬(A ∧ ¬B) ⇔i (A →c B) 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 Bc) Some more interesting theorems

1

⊢NEc (A →c Bc) →i (A →i Bc)

2

{A, (A →c Bc)} ⊢NEc Bc}

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:

1

We can prove that (A →i ⊥) and (A →c ⊥) are “interderivable” in 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.

2

We can argue that in fact there’s only one way to assert the 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

One could reply:

1

Interderivability is a weak form of equivalence. The fact that all theorems of classical propositional logic are “equivalent” clearly does not imply that we just have one theorem! Although it is not clear how to define a more robust notion of equivalence, it is clear that “material equivalence” alone is not sufficient to justify the use of a single negation

2

We may accept that there’s just one way to assert the negation of a proposition A, to wit, to produce a derivation of a contradiction from the assumption A. But we may also accept that there might be different ways to derive a contradiction from A, that there might be classical and intuitionistic derivations of ⊥ from A, and that this fact would establish two different ways we could use to negate A, and hence that we should have two negations, a classical one and an intuitionistic one.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

1

Question Can we find a derivation of ⊥ from A such that it is “essentially classic”, in the sense that it (differently from what happens in the example of Fact 1) “essentially” uses classical reasoning in the derivation of ⊥ from A? If such a derivation does exist, we would have a very good reason to defend the use of two negations, one classical, one intuitionistic.

2

In the case of propositional logic, the answer is no! Given any classical derivation of ⊥ from an assumption A, then there is also an intuitionistic derivation of ⊥ from the assumption A. This is a trivial consequence of Glivenko’s theorem and the normalization theorem.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The Joyal collapse and Natural Deduction

The collapse was formulated in categorial terms, but in a natural deduction setting it corresponds to the idea that there aren’t different intuitionistic derivations of a formula of the form ¬A, or equivalently, there’s just one intuitionistic derivation of ⊥ from the assumption A. Imteresting point: can we obtain the colapse without extra reductions?

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The Joyal collapse and Natural DEduction

Consider the following two derivations of ⊥ from (A ∧ ¬A): (A ∧ ¬A) A (A ∧ ¬A) ¬A ⊥ (A ∧ ¬A) A (A ∧ ¬A) ¬A ⊥ ⊥I A (A ∧ ¬A) ¬A ⊥ In what sense would these two derivations be equal?

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A new Reduction

Π1 ⊥ A Π2 (A → B) B Reduces to Π1 ⊥ B

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Lot’s of things to be done!

1 Proof theory 2 Semantics. Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A bit of Proof Theory

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Reductions The reductions for the intuitionistic operators are the usual Prawitz’ reductions. The reductions for the classical operators are defined below:

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

[A] [¬B] Π1 ⊥ A →c B Π2 A Π3 ¬B ⊥ Reduces to: Π2 [A] Π3 [¬B] Π1 ⊥

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

[¬A] [¬B] Π1 ⊥ A ∨c B Π2 ¬A Π3 ¬B ⊥ Reduces to: Π2 [¬A] Π3 [¬B] Π1 ⊥

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Problem for normalization: inductive measures! Solution: new measures of complexity!

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

In the case of the new reductions we immediately see that through the elimination of a maximum formula, new maximum formulas of the same degree may be produced, and because of this the usual normalization strategy does not work anymore. An easy way to solve this difficult is through the modification of the usual definition of the degree of a formula as the number of occurrences logical operators in the formula. It is clear that in the case of classical disjunction and classical implication there are some hidden negations, and that any definition of the complexity of a formula must take this point in consideration. The new measure of complexity of a formula A will be called the ecumenical degree of A, ed(A), and is defined as follows:

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

E-degree

1

ed(φ) = 0

2

ed(¬A) = ed(A) + 1

3

ed(A ✷ B) = ed(A) + ed(B) + 1, if ✷ is ∧ or an intuitionistic

• perator.

4

ed(A ∨c B) = ed(A) + ed(B) + 4

5

ed(A →c B) = ed(A) + ed(B) + 3

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The Normalization Theorem

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The main definitions are standard. Definition A formula A in a derivation Π is a maximum formula if and only if:

1

A is the conclusion of an application of an α-introduction rule and at the same time the major premisse of an α-elimination rule in Π.

• r

2

A is the conclusion of an application of the ⊥-rule and at the same time the major premisse of an elimination rule in Π.

• r

3

A is the conclusion of an application of the ∨i-elimination rule and at the same time the major premisse of an elimination rule in Π. Definition The ecumenical degree of a derivation Π, ed[Π] is defined as the max{ ed[A] s. t. A is a maximum formula in Π }.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Definition A derivation Π is called critical iff

1

Π ends with an elimination rule α;

2

The major premiss of α is a maximum formula;

3

For every proper sub-derivation Π′ of Π, ed[Π′] ≤ ed[Π].

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Lemma Let Π1 / A and A/Π2 be two derivations in NEc such that d(Π1) = n1 and d(Π2) = n2. Then, d(Π1/[A]/Π2) = max(d[A], n1, n2). Lemma Let Π be a critical derivation of Γ ⊢NEc A. Then, Π reduces to a derivation Π′ of ∆ ⊆ Γ ⊢NEc A such that d(Π′) < d(Π). Lemma Let Π be a derivation of Γ ⊢NEc A. Then, Π reduces to a derivation Π′

• f ∆ ⊆ Γ ⊢NEc A such that d(Π′) < d(Π).

Proof. Directly from the previous lemma using induction on the length of Π.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Theorem Let Π be a derivation of Γ ⊢NEc A. Then, Π reduces to a normal derivation Π′ of ∆ ⊆ Γ ⊢NEc A.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

New Ecumenical systems

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Translations

Theorem Let S1 and S2 be two logics formulated in the languages L1 and L2 respectively, and let F be a translation from L2 into L1 that interprets S2 into S1. Let S3 be an intermediate logic between S1 and S2. Then, if F also satisfies the property that A −||− F(A) in S2, then F is a translation from L2 into the language L3 of S3 that interprets S2 into S3.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Translations

Theorem The translation F of the previous theorem cannot be a translation from S3 into S1.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A new ecumenical system - Classical Logic (CL) and the Logic of Constant (CD) Domains

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The system FIL

Definition A (decorated) sequent is an expression of the form A1(n1), . . . , Ak(nk) ⇒ B1/S1, . . . , Bm/Sm where

• Ai for (1 ≤ i ≤ k) and Bj for (1 ≤ j ≤ m) are formulae of

intuitionistic propositional logic;

• ni for (1 ≤ i ≤ k) are natural numbers. We say that ni is the index
• f the formula Ai;
• Sj for (1 ≤ j ≤ m) are sets of natural numbers. We call Sj the

dependency set of the formula Bj.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

THE SYSTEM FIL

Axiom A(n) ⊢ A{n} Γ ⊢ A/S, ∆′ A(n), Γ′ ⊢ ∆ Cut Γ, Γ′ ⊢ ∆′, ∆∗ Γ, A(n), B(m), Γ′ ⊢ ∆ ExL Γ, B(m), A(n), Γ′ ⊢ ∆ Γ ⊢ A/S, B/S′, ∆ ExR Γ ⊢ B/S′, A/S, ∆ Γ ⊢ ∆ WL Γ, A(n) ⊢ ∆∗ Γ ⊢ ∆ WR Γ ⊢ A/{ }, ∆ Γ, A(n), A(m) ⊢ ∆ ConL Γ, A(k) ⊢ ∆∗ Γ ⊢ A/S, A/S′∆ ComR Γ ⊢ A/S ∪ S′, ∆ Γ ⊢ A/S, ∆ Γ′, B(n) ⊢ ∆′ →L Γ, Γ′, A → B(n) ⊢ ∆, ∆′ Γ, A(n) ⊢ B/S, ∆ →R Γ ⊢ (A → B)/S − {n}, ∆

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

THE SYSTEM FIL

Γ ⊢ A/S, ∆ Γ′ ⊢ B/S′∆′ ∧L Γ, Γ′ ⊢ ∆, ∆′, (A ∧ B)/S ∪ S′ Γ, A(n), B(m) ⊢ ∆ ∧R Γ, (A ∧ B)(k) ⊢ ∆∗ Γ ⊢ ∆, A/S, B/S′ ∨R Γ ⊢ ∆, (A ∨ B)/S ∪ S′ Γ, A(n) ⊢ ∆ Γ′, B(m) ⊢ ∆′ ∧L Γ, Γ′, (A ∨ B)(k) ⊢ ∆∗, ∆′∗

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The System CD The system CD is obtained from the system FIL through the addition of the usual classical rules (now with decorations) for quantifiers:

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

THE SYSTEM CD The system CD is obtained from the system FIL through the addition of the usual classical rules (now with decorated) for quantifiers:

Γ, A(t)(n) ⊢ ∆ ∀L Γ, ∀xA(x)(n) ⊢ ∆ Γ ⊢ A(a)/S, ∆ ∀R Γ ⊢ ∀xA(x)/S, ∆ Γ, A(a) ⊢ ∆ ∃L Γ, ∃xA(x) ⊢ ∆ Γ ⊢ A(t)/S, ∆ ∃R Γ ⊢ ∃xA(x)/S, ∆

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

THE ECUMENICAL SYSTEM ECD The ecumenical system ECD is obtained from the system CD through the addition of the new rules (now with decoratations) for →c, ∨c and ∃c:

Γ, A(n), ¬B(m) ⊢ ⊥/S →c −L Γ ⊢ (A →c B)/S∗ Γ ⊢ A/S Γ ⊢ ¬B/S′ →c −R Γ, (A →c B)(m) ⊢ ⊥/S ∪ S′ ∪ {m} Γ, ¬A(n), ¬B(m) ⊢ ⊥/S ∨c − R Γ ⊢ (A ∨c B)/S Γ ⊢ ¬A/S Γ′ ⊢ ¬B/S′ ∨c − L Γ, (A ∨c B)(m) ⊢ ⊥/S ∪ S′ ∪ {m} Γ, ∀x¬A(x)(m) ⊢ ⊥/S ∃c − R Γ ⊢ ∃cxA(x)/S∗ Γ ⊢ ∀x¬A(x)/S ∃c − L Γ, ∃cxA(x)(m) ⊢ ⊥/S ∪ {m}

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A new ecumenical system NEm - Intuitionistic Logic (IL) and Minimal Logic (ML) Domains (Victor Nascimento)

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The system NEm

• Two implications: →i, →m
• Two universal quantifiers: ∀i, ∀m

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The ecumenical system NEm The rules for ∧, ⊥, ∨ and ∃ are the usual ones. The new rules for the new operators are as follows:

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

[A] . . . B ∨ ⊥

I →i

A →i B A →i B A

E →i

B ∨ ⊥

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

(A ∨ ⊥)

∀i − I

∀ixA(x) ∀ixA(x)

∀i − E

(A ∨ ⊥)

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

[A]2 [¬A]1 ⊥ (B ∨ ⊥)

→i I 1

(¬A →i B) ((¬A →i B) ∨ ⊥)

→i I 2

A →i (¬A →i B))

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Related work

The idea of using different signs for the different meanings attached to intuitionistic and classical operators is not new, it was used by P. Krauss in 1992. The same idea was used again in 2015 by Gilles Dowek. Both Krauss and Dowek have classical versions for ∧ and ∀. It is interesting to

• bserve that [1] ∧c does not satisfy (in general) projections and is not

idempotent and that [2] ∀c does not (in general) satisfy universal instantiation.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Related work

The main motivation of both Krauss and Dowek was to explore the possibility of hybrid readings of axioms of mathematical theories. The example discussed by Krauss is the axiom of choice and the example discussed by Dowek is also taken from set theory. The whole point is, in Dowek’s own words, to consider that ”which mathematical results have a classical formulation that can be proved from the axioms of constructive set theory or constructive type theory and which require a classical formulation of these axioms and a classical notion of entailment remais to be investigated”.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Future work

1

We have just indicated the way to obtain a normalization for NEc. Clearly there are lots of things to be done with respect to the proof theory of NEc. We know that we do not have as a corollary of normalization the sub-formula principle in its usual form. But can we have a weak sub-formula principle based on the intended meaning of the classical operators? Can we have confluence? Strong Normalization?

2

It would be interesting to explore the intended meaning of the classical operators in order to obtain a Curry-Howard type of result.

3

As we mentioned above, an interesting application of ecumenical systems is related to the analysis of mathematical results that depend on ecumenical readings of axioms (see Krauss and Dowek). It would certainly be interesting to pursue the investigation of other axiomatic theories. .

4

We are also planning to define a sequent calculus and a tableaux system for the Ecumenical modal logic S4.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

The system CILP

5 The system CILP - Caleiro & Ramos

Caleiro C., Ramos J. (2007) Combining Classical and Intuitionistic

• Implications. In: Konev B., Wolter F. (eds) Frontiers of Combining
• Systems. FroCoS 2007. Lecture Notes in Computer Science, vol 4720.

Springer, Berlin, Heidelberg

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A labelled ecumenical system

[ωi : A] [ωiRωj] Π ωj : B

→ −I

ωi : (A → B) ωi : (A → B) ωiRωj ωj : A

→ −E

ωj : B

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

A labelled ecumenical system

[ω0 : A] Π ωj : B

⇒ −I

ωi : (A ⇒ B) ωi : (A ⇒ B) ω0 : A

⇒ −E

ωj : B

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

An alternative Johansson/Heyting Ecumenical system

6

Still another system [A] . . . B

I →i1

A →i B [A] . . . ⊥

I →i2

A →i B A →i B A [B] . . . C [⊥] . . . C

E →i

C

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Semantics

7 The problem of impurity/separability - Too

many negations!

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

• J. Murzi, 2018

[A/⊥]i [B/⊥]i Π ⊥

I∨c

(A ∨c B) (A ∨c B) A/⊥ B/⊥

E∨c

⊥

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system

Further Reading I

• G. Dowek.

On the definitions of the classical connective and quantifiers. Why is this a proof? Edward Hermann Haeusler, Wagner Sanz and Bruno Lopes, editors College Books, 2015.

• P. Krauss.

A constructive interpretation of classical mathematics. Mathematische Schriften Kassel, preprint No. 5/92 (1992)

• D. Prawitz.

Classical versus intuitionistic logic. Why is this a proof? Edward Hermann Haeusler, Wagner Sanz and Bruno Lopes, editors College Books, 2015.

Luiz Carlos Pereira, Elaine Pimentel, Valeria de Paiva On Prawitz’ Ecumenical system