Point-free foundation of Mathematics

Dr M Benini

Università degli Studi dell’InsubriaCorrectness by Construction projectvisiting JAIST until 16th June 2014

marco.benini@uninsubria.it

April 22nd, 2014

Introduction

This seminar aims at introducing an alternative foundation of Mathematics.

Is it possible to define logical theories without assuming the existenceof elements?

This talk will positively answer to the above question by providing a soundand complete semantics for multi-sorted, first-order, intuitionistic-basedlogical theories.

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Introduction

Actually, there is more:■ the semantics does not interpret terms as elements of some universe, butrather as the glue which keeps together the meaning of formulae;

■ the semantics allows to directly interpret the Curry-Howard isomorphismso that each theory is naturally equipped with a computational meaning;

■ the semantics allows for a classifying model, that is, a model such thatevery other model of the theory can be obtained by applying a suitablefunctor to it;

■ most other semantics for these logical systems can be mapped to the onepresented: 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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Introduction

Most of this talk is devoted to introduce a single definition, logicallydistributive categories, which identifies the models of our logical systems.

These models are suitable categories, equipped with an interpretation offormulae and a number of requirements on their structure.

Although the propositional part has already been studied by, e.g., PaulTaylor, the first-order extension is novel.

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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 ∈ Sfor all 0≤ i ≤ n,

■ R the set of relation symbols, of the form r : s1 ×·· ·× sn, with si ∈ S forall 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 Ma 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 proofswhere the domain is the theorem and the co-domain is the assumption(s).

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Logically distributive categories

The first four conditions allows to interpret propositional intuitionistic logic,as shown in P. Taylor, Practical foundations of mathematics, CambridgeUniversity 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. Hereä [_,_] is the co-universal arrow of the (A×B)+ (A×C) co-product,ä 1A is the identity on A,ä and ι1 : B →B+C , ι2 : C →B+C are the injections of the B+C co-product,ä _×_ is the product arrow.

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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 fromthe discrete category of terms of sort s to C, defined by t :s 7→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 withx :s 6∈FVB.

The arrows of C∀x :s .A, apart identities, are all the arrows in the category ofcones over ΣA(x :s) whose co-domain lies in C∀x :s .A and whose domain isM(∀x :s .A).

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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 withx :s 6∈FVB.

The arrows of C∃x :s .A, apart identities, are all the arrows in the category ofcones over ΣA(x :s) whose domain lies in C∃x :s .A and whose co-domain isM(∃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) isthe initial object in C∃x :s .A. More important for us, from each object MB inC∀x :s .A there is a unique arrow to M(∀x :s .A), and dually, to each objectMB in C∃x :s .A there is a unique arrow from M(∃x :s .A).

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Logically distributive categories

(C6) We constrain the map M to be as follows:ä M(>)= 1C, the terminal object of C,ä M(⊥)= 0C, the initial object of C,ä M(A∧B)=MA×MB, the binary product in C,ä M(A∨B)=MA+MB, the binary co-product in C,ä M(A⊃B)=MBMA, the exponential object in C,ä M(∀x : s .A)= 1C∀x : s .A , the terminal object in C∀x : s .A,ä 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 isimpredicative.

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Logically distributive categories

For each variable x :s, A,B formulae with x :s 6∈FVA, it is easy to see thatMA×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)) inC∃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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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 arenamed 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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Semantics

A model for T is a logically distributive category together with a map MAxfrom 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 inductivelyinterpret 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 tothe 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 ofthe initial object and the universal arrow of the terminal object,respectively;

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Semantics

■ universal elimination becomes the projection M(∀x :s .C)→M(A[t/x ]) inthe unique cone over ΣC (x :s) having M(∀x :s .C) as vertex;

■ universal introduction becomes the universal arrow to the terminal objectin 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 initialobject of C∃x :s .A∧C .

Actually, the precise definition is a bit more complex, to take into accountthe context. Also, some additional properties of the existential and universalsubcategories are needed. But this is just technique. . .

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Semantics

A formula A is valid in the model ⟨C,M ,MAx⟩ when there exists an arrow1C→MA.

A formula A is a logical consequence of B1, . . . ,Bn in the model when thereexists an arrow M(B1 ∧ . . .∧Bn)→MA in C.

A formula A is a logical consequence of B1, . . . ,Bn when it is so in any modelfor the theory.

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Soundness and completeness

Theorem 1A formula A is a logical consequence of B1, . . . ,Bn in the theory T if andonly 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, IntuitionisticFirst-Order Logic: Categorical Semantics via the Curry-HowardIsomorphism, http://arxiv.org/abs/1307.0108, 2013.

As a side effect of the completeness proof, it follows that the syntacticcategory forms a classifying model with respect to the class of functorspreserving the logically distributive structure.

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The role of terms

How terms get interpreted?■ Variables are used to identify the required subcategories C∀x : s .A andC∃x : s .A;

■ variables are also used to construct the substitution functor ΣA(x :s);■ all terms contribute to the substitution process, which induces thestructure 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 constructthe 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 islimited to link together formulae in subcategories that control howquantifiers are interpreted.

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Inconsistent theories

A theory T is inconsistent when it allows to derive falsity. However, in oursemantics, 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 notultimately easier or different than finding an internal contradiction in thetheory.

Actually, it does make sense in a purely computational view that aninconsistent theory has a model: it means that, although the specification ofa program is ultimately wrong, there are pieces of code which are perfectlysound.

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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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