This document discusses linear logic and its relationship to constructive mathematics. It begins by providing background on intuitionistic logic, classical logic, and the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic connectives. It then introduces linear logic and discusses how linear logic can provide a better account of negation and proofs by contradiction for constructive mathematics. The document explores models of linear logic using Heyting algebras and the dialectica construction. It considers how linear logic may help explain traditionally non-constructive ideas in mathematics.

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Introduction
BHK
Algebra
Constructivism
Linear Logic and Constructive Mathematics
(Algebraic Dialectica for Logicians)
Valeria de Paiva
Topos Institute
21 de abril de 2021
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Algebra
Constructivism
Thanks Luiz Carlos!
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BHK
Algebra
Constructivism
The courage of our convictions
This discussion is mostly based on reading Mike Shulman’s ‘Linear
Logic for Constructive Mathematics’. I am grateful to Mike for
ideas and even slides. (the mistakes are my own, of course)
I want to talk about an algebraic version of the dialectica
construction, but to do that first
classical vs. intuitionistic logic
intuitionistic vs. linear logic
linear mathematics?
Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics

4. 3/35
Introduction
BHK
Algebra
Constructivism
The courage of our convictions
This discussion is mostly based on reading Mike Shulman’s ‘Linear
Logic for Constructive Mathematics’. I am grateful to Mike for
ideas and even slides. (the mistakes are my own, of course)
I want to talk about an algebraic version of the dialectica
construction, but to do that first
classical vs. intuitionistic logic
intuitionistic vs. linear logic
linear mathematics?
Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics

5. 3/35
Introduction
BHK
Algebra
Constructivism
The courage of our convictions
This discussion is mostly based on reading Mike Shulman’s ‘Linear
Logic for Constructive Mathematics’. I am grateful to Mike for
ideas and even slides. (the mistakes are my own, of course)
I want to talk about an algebraic version of the dialectica
construction, but to do that first
classical vs. intuitionistic logic
intuitionistic vs. linear logic
linear mathematics?
Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics

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Introduction
BHK
Algebra
Constructivism
A Hundred Years Ago
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Introduction
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Constructivism
A Hundred Years Ago
Hilbert (1927) ”To prohibit existence statements and the principle
of excluded middle is tantamount to relinquishing the science of
mathematics altogether.”Brouwer-Hilbert controversy (from
wikipedia)
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Almost Fifty Years Ago
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More than Thirty Years Ago
Girard shook the basis of logic several times
“Broccoli logic”is still one of enduring jokes in the internet
Linear Logic has been very influential
Out of fashion now?
Linear thinking and variations permeated logic and theoretical
computing
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Introduction
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Categorical Proof Theory
Types are formulae/objects in appropriate category,
Terms/programs are proofs/morphisms in the category,
Logical constructors are appropriate categorical constructions.
Most important: Reduction is proof normalization (Tait)
Outcome: Transfer results/tools from logic to CT to CSci
How far can we push it?
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Algebra
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papers: Term calculus for intuitionistic linear logic (BBdePH1993),
Term assignment for ILL (TR1992) and Linear λ-calculus and
categorical models revisited (CSL1992)
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Intuitionistic Logic
Brouwer wanted to eliminate non-constructive proofs. Heyting
formulated intuitionistic logic where all valid proofs are necessarily
constructive. Kolmogorov, Glivenko, Weyl, Bishop, and many
others developed constructive maths
http://dx.doi.org/10.1090/bull/1556 FIVE
STAGES OF ACCEPTING CONSTRUCTIVE MATHEMATICS
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Introduction
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Intuitionistic Logic
Proof by contradiction is not allowed
a statement can be ’not false’ without being true: ¬¬P does
not imply P
De Morgan’s laws hold* except ¬(P ∧ Q) → (¬P ∨ ¬Q)
Similarly,¬∀x.P(x) does not imply ∃x.¬P(x)
The law of excluded middle P ∨ ¬P does not hold in general
The three connectives ∧, ∨, → are independent: neither can
be defined in terms of the others
Negation is a defined connective ¬A := A → ⊥
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Brouwer-Heyting-Kolmogorov (BHK) interpretation
This an informal description of the meanings of intuitionistic
connectives in terms of what counts as a proof of them
A proof of P ∧ Q is a proof of P and a proof of Q
A proof of P ∨ Q is a proof of P or a proof of Q (plus a
marker of which one it is)
A proof of P → Q is a construction transforming any proof of
P into a proof of Q
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BHK interpretation of Negation
Intuitionism defines ¬P to be P → ⊥,
A proof of ¬P is a construction transforming any proof of P
into a proof of a contradiction.
This explains the properties of negation in intuitionistic logic:
If it would be contradictory to have a construction
transforming any proof of P into a contradiction, it does not
follow that we have a proof of P. Hence ¬¬P does not imply
P
For an arbitrary P, we can not claim to have either a proof of
P or a construction transforming any proof of P into a
contradiction. (E.g. P might be the Riemann hypothesis.) So
P ∨ ¬P does not hold.
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Is there a better Negation?
Girard’s idea: formal de Morgan dual, the negation in LL
¬(P ∨ Q) =def ¬P ∧ ¬Q
¬(P ∧ Q) =def ¬P ∨ ¬Q
¬∃x.P(x) =def ∀x.¬P(x)
¬∀x.P(x) =def ∃x.¬P(x)
A constructive proof of ∃x.P(x) must provide an example
A constructive disproofof ∀x.P(x) should provide a
counterexample
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Is there a better Negation?
Shulman’s bold idea: LL’s involutive negation solves many of
the issues of intuitionistic negation in mathematics
To prove ∃x.P(x) by contradiction, we assume its negation
∀x.¬P(x). But in order to use this hypothesis at all, we have
to apply it to some x! we are necessarily constructing
something.
Hence an involutive negation makes proofs by contradiction
less objectionable
Moreover, he produces examples showing that traditional uses
of non-constructivity are disallowed and that convoluted ideas
like ’apartness’ can be better explained in linear terms
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Linear Logic for Constructive Mathematics
We divide the hypotheses into linear and nonlinear ones. The
linear ones can only be used once in the course of a proof.
All ‘hypotheses for contradiction’ in a proof by contradiction
are linear hypotheses.
Similarly, P −◦ Q is a linear implication that uses P only once.
It is contraposable, ¬(P −◦ Q) = (¬Q −◦ ¬P) (here we’re
talking about bi-implications)
Linearity is the default status of assertions. We mark the
nonlinear hypotheses with a modality, !P.
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Not so classical disjunctions
In classical logic, (P ∨ Q) = (¬P → Q) = (¬Q → P).
This is no longer true in intuitionistic logic. (connectives are
independent)
It also fails in linear logic for the ‘constructive’ disjunction ∨.
classical Linear Logic does have (¬P −◦ Q) = (¬Q −◦ P),
defining another kind of disjunction that is weaker than ∨.
in classical linear logic P ` Q = (¬P −◦ Q) = (¬Q −◦ P).
in CLL ∨-excluded middle P ∨ ¬P fails. But par-excluded
middle (P ` ¬P) = (¬P −◦ ¬P) is a tautology.
∨ supports proof by cases; ` supports the disjunctive
syllogism
in FILL no excluded middle, 5 independent connectives
Still the case that ”For an arbitrary P, we can not claim to
have either a proof of P or a construction transforming any
proof of P into a contradiction.”
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Introduction
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Algebra
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BHK for Linear Logic
The BHK interpretation privileges proofs over refutations.
A proof of P ∧ Q is a proof of P and a proof of Q. A
refutation of P ∧ Q is a refutation of P or a refutation of Q
A proof of P ∨ Q is a proof of P or a proof of Q. A refutation
of P ∨ Q is a refutation of P and a refutation of Q.
A proof of P ` Q is a construction transforming any
refutation of P into a proof of Q, and any refutation of Q
into a proof of P. A refutation of P ` Q is a refutation of P
and a refutation of Q.
A proof of P −◦ Q is a construction transforming any proof of
P into a proof of Q, and a construction transforming any
refutation of Q into a refutation of P. A refutation of P −◦ Q
is a proof of P and a refutation of Q.
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Introduction
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Algebra
Constructivism
Making this BHK more formal?
Shulman’s sections: intuitionistic logic, linear logic, the
standard interpretation, the hidden linear nature of
constructive mathematics
I cannot judge how good the linear logic modifications are for
constructive mathematics, but I do have issues with what he
calls the ’standard interpretation’. The Dialecica version over
a Heyting algebra H and 0 is as good as the Chu construction
Mike says ‘If constructive logic is the logic of affirmative
propositions, then affine logic is the logic of propositions that
are subject to both affirmation and refutation, and the Chu
construction is the canonical embedding of the former in the
latter. Why canonical?
‘in the Dialectica interpretation the forwards and backwards
information is explicitly carried by functions, rather than
proofs as in the Chu construction.’ hmm?
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Heyting algebras
Definition
A Heyting algebra is a cartesian closed lattice, i.e. a poset (H, ≤)
with
A top element > and bottom element ⊥,
Meets P ∧ Q and joins P ∨ Q,
An ’implication’ with (P ∧ Q) ≤ R iff P ≤ (Q → R)
Heyting algebras are the algebraic semantics of intuitionistic
propositional logic, just like Boolean algebras are for classical logic.
For algebraic semantics of CLL/ILL/FILL a bit more complicated. I
talked about lineales, which are simply posetal symmetric
monoidal closed categories. Shulman wants to force units of tensor
and product to be the same
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Dejà vu?
Theorem (de Paiva CTCS1989)
For any Heyting algebra H, consider the algebra Dial⊥(H):
Elements are pairs P = (P+, P−) where P+, P− ∈ H and
P+ ∧ P− = ⊥. (Think P+ = proofs, P− = refutations)
Define P ≤ Q to mean (P+ ≤ Q+ and Q− ≤ P−)
P ∧ Q = (P+ ∧ Q+, P− ∨ Q−) and
P ∨ Q = (P+ ∨ Q+, P− ∧ Q−) and > = (>, ⊥) and
⊥ = (⊥, >)
P ⊗ Q = (P+ ∧ Q+, (P+ → Q−) ∧ (Q+ → P−))
P ` Q = ((P+ → Q+) ∧ (Q+ → P+), P− ∧ Q−)
P −◦ Q = ((P+ → Q+) ∧ (Q− → P−), P+ ∧ Q−)
Then Dial⊥(H) is a model of Linear Logic (without exponentials).
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Bang Modality
Digression: The other theorem of CTCS1989...
For any Heyting algebra H which has free co-commutative monoids
we can define a !-comonad that makes Dial⊥(H) is a model of IL.
Too complicated?
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Back to constructivism: Mike says
Girard was interested in Proof nets, Geometry of Interactions,
Games, Ludics, etc Linear logicians were interested in having both
LL and IL, constructivists use DTT
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Back to constructivism
6 and 7 get my money!
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Introduction
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Algebra
Constructivism
Back to constructivism
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Algebra
Constructivism
Back to constructivism
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Algebra
Constructivism
Back to constructivism
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Introduction
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Algebra
Constructivism
Back to constructivism
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Back to constructivism
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Intuition for Dialectica objects?
Blass makes the case for thinking of problems in computational
complexity. Intuitively an object of the dialectica construction
A = (U, X, α)
can be seen as representing a problem.
The elements of U are instances of the problem, while the
elements of X are possible answers to the problem instances.
The relation α says whether the answer is correct for that instance
of the problem or not.
LL4CM only considers objects of the form (P+, P−) of proofs and
refutations, the relation is always contradiction ⊥. Presumably
sometimes one wants to have different relations...
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Examples of objects in Dialectica
1. The object (N, N, =) where n is related to m iff n = m.
2. The object (NN, N, α) where f is α-related to n iff f (n) = n.
3. The object (R, R, ≤) where r1 and r2 are related iff r1 ≤ r2
4. The objects (2, 2, =) and (2, 2, 6=) with usual equality and
inequality.
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Conclusions
Introduced you to Shulman’s bold idea of doing constructive
mathematics with linear logic.
Don’t see the canonicity of Chu’s construction.
Believe FILL and Dial⊥(H) work just as well and have an
associated linear λ-calculus
Hinted at its importance for interdisciplinarity:
Category Theory, Proofs and Programs
Much more work needed for applications, LinearLean anyone? In
particular work needed on connecting LL+IL with classical logic.
Ecumenical logic ftw!
Thank you!
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Some References
N.Benton, A mixed linear and non-linear logic: Proofs, terms and models.
Computer Science Logic, CSL, (1994).
A.Blass, Questions and Answers: A Category Arising in Linear Logic,
Complexity Theory, and Set Theory, Advances in Linear Logic, London
Math. Soc. Lecture Notes 222 (1995).
de Paiva, The Dialectica Categories, Technical Report, Computer Lab,
University of Cambridge, number 213, (1991).
de Paiva, A dialectica-like model of linear logic, Category Theory and
Computer Science, Springer, (1989) 341–356.
de Paiva, The Dialectica Categories, In Proc of Categories in Computer
Science and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol
92, American Mathematical Society, 1989 (eds. J. Gray and A. Scedrov)
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