In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.

1. Point-free foundation of Mathematics
Dr M Benini
Università degli Studi dell’Insubria
Correctness by Construction project
visiting JAIST until 16th June 2014
marco.benini@uninsubria.it
April 22nd, 2014

2. Introduction
This seminar aims at introducing an alternative foundation of Mathematics.
Is it possible to define logical theories without assuming the existence
of elements?
This talk will positively answer to the above question by providing a sound
and complete semantics for multi-sorted, first-order, intuitionistic-based
logical theories.
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3. Introduction
Actually, there is more:
the semantics does not interpret terms as elements of some universe, but
rather as the glue which keeps together the meaning of formulae;
the semantics allows to directly interpret the Curry-Howard isomorphism
so that each theory is naturally equipped with a computational meaning;
the semantics allows for a classifying model, that is, a model such that
every other model of the theory can be obtained by applying a suitable
functor to it;
most other semantics for these logical systems can be mapped to the one
presented: Heyting categories, elementary toposes, Kripke models,
hyperdoctrines, and Grothendieck toposes.
In this talk, we will focus on the first aspect only.
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4. Introduction
Most of this talk is devoted to introduce a single definition, logically
distributive categories, which identifies the models of our logical systems.
These models are suitable categories, equipped with an interpretation of
formulae and a number of requirements on their structure.
Although the propositional part has already been studied by, e.g., Paul
Taylor, the first-order extension is novel.
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5. Logically distributive categories
Let Σ = 〈S,F,R〉 be a first-order signature, with
S the set of sort symbols,
F the set of function symbols, of the form f : s1 ×···×sn → s0, with si ∈ S
for all 0 ≤ i ≤ n,
R the set of relation symbols, of the form r : s1 ×···×sn, with si ∈ S for
all 1 ≤ i ≤ n.
Also, let T be a theory on Σ, i.e., a collection of axioms.
A logically distributive category is a pair 〈C,M〉 where C is a category and M
a map from formulae on Σ to ObjC, satisfying seven structural conditions,
indicated as (C1) to (C7).
Informally, objects of C will denote formulae while arrows will denote proofs
where the domain is the theorem and the co-domain is the assumption(s).
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6. Logically distributive categories
The first four conditions allows to interpret propositional intuitionistic logic,
as shown in P. Taylor, Practical foundations of mathematics, Cambridge
University Press, 1999.
(C1) C has finite products;
(C2) C has finite co-products;
(C3) C has exponentiation;
(C4) C is distributive, i.e., for every A,B,C ∈ ObjC the arrow
∆ ≡ [1A ×ι1,1A ×ι2] : (A×B)+(A×C) → A×(B +C) has an inverse. Here
[_,_] is the co-universal arrow of the (A×B)+(A×C) co-product,
1A is the identity on A,
and ι1 : B → B +C, ι2 : C → B +C are the injections of the B +C co-product,
_×_ is the product arrow.
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7. Logically distributive categories
To express the other conditions, we need additional notation.
For every s ∈ S, A formula, and x :s variable, let ΣA(x :s) be the functor from
the discrete category of terms of sort s to C, defined by t :s → M(A[t/x]).
Also, let C∀x :s.A be the subcategory of C whose objects are
the vertexes of the cones on ΣA(x :s)
such that each vertex is of the form MB for some formula B with
x :s ∈ FVB.
The arrows of C∀x :s.A, apart identities, are all the arrows in the category of
cones over ΣA(x :s) whose co-domain lies in C∀x :s.A and whose domain is
M(∀x :s.A).
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8. Logically distributive categories
Dually, C∃x :s.A is the subcategory of C whose objects are
the vertexes of the co-cones on ΣA(x :s)
such that each vertex is of the form MB for some formula B with
x :s ∈ FVB.
The arrows of C∃x :s.A, apart identities, are all the arrows in the category of
cones over ΣA(x :s) whose domain lies in C∃x :s.A and whose co-domain is
M(∃x :s.A).
We require that
(C5) All the subcategories C∀x :s.A have a terminal object, and all the
subcategories C∃x :s.A have an initial object.
Evidently, M(∀x :s.A) is the terminal object in C∀x :s.A, and M(∃x :s.A) is
the initial object in C∃x :s.A. More important for us, from each object MB in
C∀x :s.A there is a unique arrow to M(∀x :s.A), and dually, to each object
MB in C∃x :s.A there is a unique arrow from M(∃x :s.A).
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9. Logically distributive categories
(C6) We constrain the map M to be as follows:
M( ) = 1C, the terminal object of C,
M(⊥) = 0C, the initial object of C,
M(A∧B) = MA×MB, the binary product in C,
M(A∨B) = MA+MB, the binary co-product in C,
M(A ⊃ B) = MBMA, the exponential object in C,
M(∀x : s.A) = 1C∀x : s.A
, the terminal object in C∀x : s.A,
M(∃x : s.A) = 0C∃x : s.A
, the initial object in C∃x : s.A.
Since M is given, the definition is not circular. But, evidently, it is
impredicative.
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10. Logically distributive categories
For each variable x :s, A,B formulae with x :s ∈ FVA, it is easy to see that
MA×M(∃x :s.B) is an object of C∃x :s.A∧B.
Thus, there is a unique arrow δ: M(∃x :s.A∧B) → M(A∧(∃x :s.B)) in
C∃x :s.A∧B by (C5).
Our last condition is that
(C7) the δ arrow above has an inverse in C.
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11. Semantics
Given a theory T over a signature Σ and a logically distributive category
〈C,M〉, we interpret each formula on Σ as MA.
Given a proof π: Γ T B with Γ = {x1 :A1,...,xn :An}, where assumptions are
named x1,...,xn, π will become an arrow
x1 :A1,...,xn :An.π:B : A1 ×···×An → B .
To lighten notation, the context will be written as x, and A ≡ A1 ×···×An.
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12. Semantics
A model for T is a logically distributive category together with a map MAx
from T to ObjC such that each axiom A is mapped in an arrow a: 1C → MA.
Assuming the standard rules of natural deduction by Prawitz, we inductively
interpret proofs as follows:
a proof by assumption becomes a projection from the context;
a proof by axiom a:B becomes the universal arrow from the context to
the terminal object composed by the arrow given by MAx;
conjunction eliminations become the projections of the binary product,
while conjunction introduction becomes the universal arrow;
disjunction elimination become the injections of the binary co-product,
while disjunction introduction is reduced to the co-universal arrow;
false elimination and truth introduction become the co-universal arrow of
the initial object and the universal arrow of the terminal object,
respectively;
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13. Semantics
universal elimination becomes the projection M(∀x :s.C) → M(A[t/x]) in
the unique cone over ΣC (x :s) having M(∀x :s.C) as vertex;
universal introduction becomes the universal arrow to the terminal object
in the C∀x :s.C subcategory;
existential introduction becomes the injection M(A[t/x]) → M(∃x :s.C)
in the unique co-cone over ΣC (x :s) having M(∃x :s.C) as vertex;
existential elimination becomes the co-universal arrow from the initial
object of C∃x :s.A∧C .
Actually, the precise definition is a bit more complex, to take into account
the context. Also, some additional properties of the existential and universal
subcategories are needed. But this is just technique. . .
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14. Semantics
A formula A is valid in the model 〈C,M,MAx〉 when there exists an arrow
1C → MA.
A formula A is a logical consequence of B1,...,Bn in the model when there
exists an arrow M(B1 ∧...∧Bn) → MA in C.
A formula A is a logical consequence of B1,...,Bn when it is so in any model
for the theory.
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15. Soundness and completeness
Theorem 1
A formula A is a logical consequence of B1,...,Bn in the theory T if and
only if there is a proof of A from the hypotheses B1,...,Bn.
The proof is long and complex: it can be found in M. Benini, Intuitionistic
First-Order Logic: Categorical Semantics via the Curry-Howard
Isomorphism, http://arxiv.org/abs/1307.0108, 2013.
As a side effect of the completeness proof, it follows that the syntactic
category forms a classifying model with respect to the class of functors
preserving the logically distributive structure.
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16. The role of terms
How terms get interpreted?
Variables are used to identify the required subcategories C∀x : s.A and
C∃x : s.A;
variables are also used to construct the substitution functor ΣA(x :s);
all terms contribute to the substitution process, which induces the
structure used by the semantics.
Thus, it is really the substitution process, formalised in the ΣA(x :s)
functors, that matters: terms are just the glue that enable us to construct
the C∀x : s.A and C∃x : s.A subcategories.
It is clear the topological inspiration of the whole construction. In particular,
it is evident that terms are not interpreted in some universe, and their role is
limited to link together formulae in subcategories that control how
quantifiers are interpreted.
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17. Inconsistent theories
A theory T is inconsistent when it allows to derive falsity. However, in our
semantics, T has a model as well.
A closer look to each model of T reveals that they are categorically “trivial”
in the sense that the initial and the terminal objects are isomorphic.
This provides a way to show that a theory is consistent. However, this is not
ultimately easier or different than finding an internal contradiction in the
theory.
Actually, it does make sense in a purely computational view that an
inconsistent theory has a model: it means that, although the specification of
a program is ultimately wrong, there are pieces of code which are perfectly
sound.
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18. Conclusion
Much more than this has been done on logically distributive categories. But,
still, I am in the beginning of the exploration of this foundation setting.
So, any question, comment, suggestion is welcome!
Questions?
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