This document discusses the concept of an ecumenical logic system that allows both classical and intuitionistic reasoning to coexist. It summarizes Dag Prawitz's approach to defining such a system, which uses different symbols for logical constants that have different meanings classically versus intuitionistically. However, the document raises the question of why Prawitz's system only includes one symbol for negation rather than separate classical and intuitionistic negation symbols. Possible answers discussed include the interderivability of the two notions of negation and the view that negation asserts a contradiction from assuming the negated proposition. The document does not conclude there is a definitive answer and suggests this as an interesting open problem area.

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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Negation in the Ecumenical System
Valeria de Paiva
Joint work with
Elaine Pimentel and Luiz Carlos Pereira
Brasil-Colombia 2021
Dec 2021
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Many Thanks!
Elaine Pimentel and Luiz Carlos Pereira
Also Marcelo, Hugo, Ciro, Samuel, Andrés and Pedro, for organizing!
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3. 3/27
Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
An Accidental Intuitionist
would want to use classical logic too!
Intuitionism because pragmatism: More convincing to have a term
t such that ∃x.A(x) is true means A(t) holds, then to have to say
that it is not the case that (∀x.¬A(x))
Our work:
Duas Negações Ecumênicas, L. C. Pereira, Elaine Pimentel
and V. de Paiva, to appear
An ecumenical notion of entailment, E. Pimentel, L. C.
Pereira and V. de Paiva, Synthese, 198(22), 5391-5413, 2019.
A proof theoretical view of ecumenical systems, Elaine
Pimentel, Luiz Carlos Pereira and Valeria de Paiva, short
paper LSFA 2018.
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
An Accidental Intuitionist
Inspiration: ‘Classical versus Intuitionistic Logic’, Prawitz 2015
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Dummett, did you say?
I am, thus, not concerned with justifications of intuitionis-
tic mathematics from an eclectic point of view, that is,
from one which would admit intuitionistic mathematics as
a legitimate and interesting form of mathematics alongside
classical mathematics: I am concerned only with the stand-
point of the intuitionists themselves, namely that classical
mathematics employs forms of reasoning which are not
valid on any legitimate construal of mathematical state-
ments (save, occasionally, by accident, as it were, under a
quite unintended reinterpretation).
Dummett, ‘The Philosophical Basis of Intuitionistic Logic’, 1973
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Michael Dummet and Dag Prawitz
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7. 7/27
Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz
My interest in logic is first of all an interest in deductive
reasoning. I see so-called classical first order predicate
logic as an attempted codification of inferences occurring
in actual deductive practice.[...]
Only if it can be shown that the classical ideas about mea-
ning are mistaken and that no intelligible meaning can be
attached to the logical constants agreeing with the classi-
cal use of them, is the intuitionistic codification of mathe-
matical reasoning of interest, according to Dummett.
[Dummett] dismisses an ecletic attitute that finds an in-
terest in both the classical and the intuitionistic codifica-
tion.
Prawitz, ‘Classical versus Intuitionistic Logic’ 2015
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz on classical constants
Gentzen’s introduction rules are of course accepted also in
classical reasoning, but some of them can clearly not serve
as explanations of meaning.
[...]an existential sentence ∃xA(x) may be rightly asserted
classically without knowing how to find a proof of some
instance A(t). Hence, Gentzen’s introduction rule for the
existential quantifier, which allows one to infer ∃xA(x)
from A(t), does not determine what is to count classi-
cally as a canonical proof of ∃xA(x) and therefore does
not either determine the classical meaning of the existen-
tial quantifier.
Prawitz, ‘Classical versus Intuitionistic Logic’ 2015
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz conclusion
Comparing the two codifications, it is clearly wrong to ar-
gue that classical logic is stronger than intuitionistic. What
can be said is instead that the intuitionistic language is
more expressive than the classical one, having access to
stronger existence statements that cannot be expressed in
the classical language. However, there is no need to cho-
ose between the two codifications because we can have a
more comprehensive one that codifies both classical and
intuitionistic reasoning based on a uniform pattern of me-
aning explanations.
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10. 10/27
Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Ecumenical system
Definition
ecumenical (adjective) promoting or relating to unity among the
world’s Christian Churches.
catholic (adjective) including a wide variety of things;
all-embracing.
Here: a codification where two or more logics can coexist in peace
If they are sufficiently ecumenical and can use the other’s
vocabulary in their own speech, a classical logician and an
intuitionist can both adopt the present mixed system, and
the intuitionist must then agree that A ∨c ¬A is trivially
provable for any sentence A, even when it contains intui-
tionistic constants, and the classical logician must admit
that he has no ground for universally asserting A ∨i ¬A,
even when A contains only classical constants. 10 / 27

11. 11/27
Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Ecumenical System
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 diffe-
rent symbols, and I shall use a subscript c for the classical
meaning and i for the intuitionistic. The classical and in-
tuitionistic constants can then have a peaceful coexistence
in a language that contains both.
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12. 12/27
Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz System
The ecumenical system defined by Prawitz has:
two disjunctions (∨c, ∨i ),
two implications (→c, →i ),
two existential quantifiers (∃c, ∃i ),
but only one conjunction, one negation, one constant for the
absurd and one universal quantifier.
Why? Is this optimal? Which criteria can we use?
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz Ecumenical Natural Deduction
1 Gentzen’s introduction and elimination rules for ⊥, ∧, ¬, and
∀: intro rule for ⊥ is vacant, elim rule allows arbitrary
sentence from ⊥;
2 Gentzen’s introduction and elimination rules for ∨, →, and ∃,
where now i is attached as a subscript to the logical constant;
3 New classical rules
4 Predicates
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz Ecumenical Natural Deduction
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz Ecumenical Natural Deduction
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Prawitz Ecumenical Natural Deduction
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
The Problem
Given that
we have two implications one classical and one intuitionistic,
the negation of a proposition A can be understood as A
implies ⊥
why do we have only one negation? and why do we have a single
constant for absurd?
Why don’t we have
a classical negation ¬cA, understood as (A →c ⊥)
an intuitionistic negation ¬i A, understood as (A →i ⊥)?
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Partial answer: Interderivability
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Interderivability
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 may is not
sufficient to justify the use of a single negation.
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
A different answer
There is just one way to assert the negation of a proposition A, to
wit, to produce a derivation of a contradiction from the
assumption A.
Support from Proof Theory? Seldin’s result:
Given a normal proof Π of ¬A in classical first order or in
intuitionistic first-order logic, then the last rule applied in Π is
¬-introduction, i.e. ¬-introduction is the only rule that allows us
to prove the negation of a proposition
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
A different answer
It is true, we may have different ways to produce a contradiction
from a given set of assumptions, a classical and an intuitionistic,
but in both the same assertability condition holds: in order to
assert ¬A, we should deduce a contradiction from A!
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
A different question
Question: Can we find a derivation of ⊥ from A such that it is
‘essentially classic’, in the sense that it ‘essentially’ uses classical
reasoning in the derivation of ⊥ from A?
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Glivenko Theorems
In the case of propositional logic, the answer is a strong no!
Given any classical derivation of ⊥ from an assumption A, there is
also an intuitionistic derivation of ⊥ from the assumption A.
This is a consequence of Glivenko’s theorems and the
normalization theorem.
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Glivenko’s theorems
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Glivenko’s theorems
But two problems:
1. what’s the categorification of Seldin’s normalization strategy?
2. what do we do in first-order?
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Summing up
Some good reasons for just one negation:
Interderivability
Assertability conditions
Computational Isomorphism
Some hope for purity: STOUP!
So we say, but plenty of interesting problems...
Anyways:
Many different ways of putting together classical and intuitionistic
reasoning.
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Introduction
Ecumenical Logic
One or Two negations?
Parting Thoughts
Related Work
Our paper mentions Dowek 2015, Krauss 1992 (unpublished),
Caleiro&Ramos 2007, Liang and Miller 2014, Girard LC and LU.
Only Kripke semantics for most
I don’t know of any system where the syntax and the (categorical)
semantics work as well as for IPC. Is there one already in
existence? Can we find one? What’s the best we can do?
Thanks!
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