Abstract Valuation Semantics · Representation of logics in algebraic setting (restricted to propositional based logics) Many-sorted and non-truth-functional logics Propositional

Motivation Algebraization Beh. algebraization Semantics Examples Conclusions

Abstract Valuation Semantics

Carlos Caleiro1,2

1Dept. Matematica, IST, TU Lisbon2SQIG - Instituto de Telecomunicacoes

Portugal

XVI EBL, May 2011Petropolis, Brasil


Outline

1 Motivation

2 AlgebraizationLogicAlgebraizationAALLimitations

3 Behavioral algebraizationMany-sorted signaturesBehavioral equivalenceBehavioral algebraizationBehavioral AAL

4 Semantical considerationsAbstract valuation semanticsBridge results

5 ExamplesK/2C1

6 Conclusions and future work


Motivation

Theory of algebraization of logics (AAL)[Lindenbaum & Tarski, Blok & Pigozzi, Czelakowski, Nemeti et al]

Representation of logics in algebraic setting (restricted topropositional based logics)

Many-sorted and non-truth-functional logics

Propositional based logics are often not enough for reasoningabout complex systemsTruth-functionality is not ubiquitous

Generalization of the notion of algebraizable logic

Many-sorted behavioral logic


Motivation

AAL _? // bAAL

Matrices ???


Motivation

AAL _? // bAAL

Matrices ???


Motivation

AAL _? // bAAL

Matrices _? // ???


Logic

Logic

Definition

A structural propositional logic is a pair L = 〈Σ,`〉, where Σ is apropositional signature and `⊆ P(LΣ(X))× LΣ(X) is aconsequence relation satisfying the following conditions, for everyT1 ∪ T2 ∪ ϕ ⊆ LΣ(X):

Reflexivity: if ϕ ∈ T1 then T1 ` ϕCut: if T1 ` ϕ for all ϕ ∈ T2, and T2 ` ψ then T1 ` ψ

Weakening: if T1 ` ϕ and T1 ⊆ T2 then T2 ` ϕStructurality: if T1 ` ϕ then σ[T1] ` σ(ϕ)


Algebraization

Algebraization

L

unisorted

ksStrong representation +3Equational logic

unisorted


Algebraization

Algebraizable logic

L is algebraizable if there are translations θ and ∆, and a class ofalgebras K s.t.

LΘ(x)

.. EqnΣK

∆(x1,x2)

mm

T ` ϕ iff Θ[T ] EqnΣK

Θ(ϕ)

∆(δi, εi) : i ∈ I ` ∆(ϕ1, ϕ2) iff δi ≈ εi : i ∈ I EqnΣKϕ1 ≈ ϕ2

ϕ a` ∆[Θ(ϕ)] ϕ1 ≈ ϕ2 =||=EqnΣK

Θ[∆(ϕ1, ϕ2)]


Algebraization

Examples

Example (I)

CPC Boolean algebras

Example (II)

IPC Heyting algebras

In both casesΘ(x) = x ≈ > and ∆(x, y) = x↔ y


AAL

Leibniz operator - The main tool

Definition (Leibniz operator)

Let L = 〈Σ,`〉 be a structural propositional logic. Then theLeibniz operator on the formula algebra can be given by:

Ω : ThL → Congr(LΣ(X))

T 7→ largest congruence compatible with T.

θ is compatible with T if 〈ϕ,ψ〉 ∈ θ implies (ϕ ∈ T ) iff (ψ ∈ T )


AAL

The Leibniz hierarchy

finitely algebraizablemonotone

injective

continuous

wwooooooooooooo

''OOOOOOOOOOO

finitely equivalentialmonotone

continuous

algebraizablemonotone

injective

commutes with inverse substitutions

ssgggggggggggggggggggggggggg

equivalentialmonotone


))RRRRRRRRRRRRRR

weakly algebraizablemonotone

injective

uullllllllllllll

protoalgebraicmonotone


AAL

Logical matrices

Consider fixed a propositional signature Σ.

Definition

A Σ-matrix is a tuple m = 〈A, D〉 such that:

A is a Σ-algebra,

D ⊆ A (designated values).

m with an assignment h over A satisfies ϕ, denoted by m,h ϕ, if h(ϕ) ∈ D.

The class of Σ-matrix models of L will be denoted by Matr(L).


AAL

Canonical semantics

A congruence of a matrix m = 〈A, D〉 is a congruence of A compatible with D.The Leibniz congruence Ω(m) of m is simply the largest congruence of m.

Reduced matrix is m∗ = 〈A/Ω(m), [D]Ω(m)〉.

Definition

The canonical matrix semantics of L is Matr∗(L).


AAL

Bridge results

L is complete for Matr(L), Matr∗(L)(Bloom) L is finitary iff Matr(L) is closed under ultraproducts

(Blok& Pigozzi) L is protoalgebraic iff Matr∗(L) is closedunder subdirect products

(Czelakowski) L is equivalential iff Matr∗(L) is closed undersubmatrices and products

(Herrmann) L is finitely equivalential iff Matr∗(L) is aquasivariety


Limitations

Limitations

First-order logic

“Its” algebraization gives rise to cylindric algebras (Henkin,Monk & Tarski) or polyadic algebras (Halmos)

Limitation applicable to other many-sorted logics

Even at the propositional level there are some “relatively”well-behaved logics that are not algebraizable


Limitations

Paraconsistent logic C1 of da Costa

C1 (da Costa 1963) given by a structural Hilbert-style axiomatization

C1 is paraconsistent: the explosion principle ϕ,¬ϕ ` ψ is not valid in C1

Mortensen 1980: there are no non-trivial congruences on the formulaalgebra compatible with the set of theoremsExample:`C1 (p ∧ q)⇔ (q ∧ p) but 0C1 ¬(p ∧ q)⇔¬(q ∧ p)

Lewin, Mikenberg & Schwarze 1991: there exists a model in which theLeibniz operator is not injective

da Costa 1966, Carnielli & Alcantara 1984 have proposed an ”algebraiccounterpart” called da Costa algebras (paraconsistent algebras)

Caleiro et al 2003: alternative approach using two-sorted algebras


Limitations

A simpler counterexample

The logic K/2 of Beziau 1999 is not algebraizable!

Σ contains the connectives ¬, ⇒semantics defined by the set of all v : LΣ(X)→ 0, 1 st.

v(¬A) = 0 if v(A) = 1v(A⇒B) = 0 iff v(A) = 1 and v(B) = 0

Axiomatization (Humberstone)

` A⇒ (B⇒A)` (A⇒ (B⇒ C))⇒ ((A⇒B)⇒ (A⇒ C))

` ((A⇒B)⇒A)⇒A

` A⇒ ((¬A)⇒B)A,A⇒B ` B


Many-sorted signatures


Definition (Many-sorted signature)

A many-sorted signature is a pair Σ = 〈S, F 〉 where S is a set (of sorts)and F = Fwsw∈S∗,s∈S is an indexed family of sets (of operations).

Example (FOL)

ΣFOL = 〈S, F 〉 such that

S = φ, tFφφ = ∀x : x ∈ X ∪ ¬Fφφφ = ⇒,∧,∨Ftnφ = P : P n-ary predicate symbolFtnt = f : f n-ary function symbol




Definition (Many-sorted signature)

A many-sorted signature is a pair Σ = 〈S, F 〉 where S is a set (of sorts)and F = Fwsw∈S∗,s∈S is an indexed family of sets (of operations).

Example (FOL)

ΣFOL = 〈S, F 〉 such that

S = φ, tFφφ = ∀x : x ∈ X ∪ ¬Fφφφ = ⇒,∧,∨Ftnφ = P : P n-ary predicate symbolFtnt = f : f n-ary function symbol


Behavioral equivalence

Behavioral Equivalence: motivation

Dichotomy Visible vs Hidden - The OO paradigm

For security reasons and/or to simplify the process of updating andimproving program implementations, computational systems dealwith heterogeneous data and split it in two different kinds:

hidden data (internal data or states);

visible data (the real data).

Hidden data can only be indirectly accessed by programs withvisible output taking them as input; they are only compared bybehavioral equivalence. Intuitively, two data elements are said tobe behaviorally equivalent if they cannot be distinguished by anyprogram with visible output. [Reichel 1984]


Behavioral algebraization

Many-sorted logics

We will only consider many-sorted signatures with a distinguished sort φ

(of formulas). We will call formulas to the elements of TΣ,φ(X).

Definition (Many-sorted logics)

A many-sorted logic is a pair L = 〈Σ,`〉 where Σ is a many-sortedsignature and `⊆ P(TΣ,φ(X))× (TΣ,φ(X)) is a structuralconsequence relation over the set of formulas.



How to generalize?

L

unsorted


unsorted

Lmany-sorted

ksStrong representation +3

Behavioral equational logicmany-sorted



How to generalize?

L

unsorted


unsorted

Lmany-sorted

ksStrong representation +3

Behavioral equational logicmany-sorted



How to generalize?

L

unsorted


unsorted

Lmany-sorted

ksStrong representation +3 Behavioral equational logicmany-sorted



Γ-Behavioral equivalence

Consider given a subsignature Γ of Σ.

Given a Σo-algebra A, two elements a1, a2 ∈ As are Γ-behavioralequivalent a1 ≡Γ a2, if for every formula ϕ(x : s, u : r) ∈ TΓ(X),called a context or an experiment, and every tuple〈b1, . . . , bn〉 ∈ Abr:

ϕA(a1, b1, . . . , bn) = ϕA(a2, b1, . . . , bn).



Behavioral logic

Definition (Many-sorted behavioral equational logic BEqnΣK,Γ)

ti ≈ t′i : i ∈ I BEqnΣK,Γ

t ≈ t′

ifffor every A ∈ K and homomorphism h : TΣ(X)→ A,h(t)≡Γh(t′) whenever h(ti)≡Γh(t′i) for every i ∈ I




LΘ(x)

.. BEqnΣK,Γ

∆(x1,x2)

mm

T ` ϕ iff Θ[T ] BEqnΣK,Γ

Θ(ϕ)

∆(δi, εi) : i ∈ I ` ∆(ϕ1, ϕ2) iff δi ≈ εi : i ∈ I BEqnΣK,Γ

ϕ1 ≈ ϕ2

ϕ a` ∆[Θ(ϕ)] ϕ1 ≈ ϕ2 =||=BEqnΣK,Γ

Θ[∆(ϕ1 ≈ ϕ2)]



Examples

Example (Standard algebraization)

Algebraizable logics are Γ-behaviorally algebraizable by taking Γ = Σ.

Example (da Costa’s paraconsistent logic C1)

C1 is Γ-behaviorally algebraizable with Γ = ΣC1 \ ¬.

Example (Lewis’s modal logic S5)

S5 is Γ-behaviorally algebraizable with Γ = ΣS5 \ .

Example (Nelson’s constructive logic with strong negation)

N is Γ-behaviorally algebraizable with Γ = ΣN \ ∼. Makes explicit the keyrole that Heyting algebras play in the algebraic counterpart of N .


Behavioral AAL

Behavioral Leibniz operator

Definition (Behavioral Leibniz operator)

Let L = 〈Σ,`〉 be a structural many-sorted logic and Γ a subsignature of Σ.The Γ-behavioral Leibniz operator on the term algebra,

ΩbhvΓ : ThL → CongΓ(TΣ(X))

T 7→ largest Γ-congruence compatible with T.

We have:

ΩbhvΣ = Ω.

The operator ΩbhvΓ is monotone iff Ω is.


Behavioral AAL

The behavioral Leibniz hierarchy

beh. finitely algebraizablemonotone

injective

continuous

wwnnnnnnnnnnnnnn

((PPPPPPPPPPPP

beh. finitely equivalentialmonotone

continuous

beh. algebraizablemonotone

injective


ssffffffffffffffffffffffffffffff

beh. equivalentialmonotone


))SSSSSSSSSSSSSSS

beh. weakly algebraizablemonotone

injective

uukkkkkkkkkkkkkkk

protoalgebraicmonotone


Behavioral AAL

The two hierarchies

algebraizable

6=

uujjjjjjjjjjjjjjj

**UUUUUUUUUUUUUUUU

equivalential

6=

===

====

====

====

====

====

=== weakly algebraizable

?

~~

behaviorally algebraizable

uujjjjjjjjjjjjjjj

**UUUUUUUUUUUUUUUU

behaviorally equivalential

))TTTTTTTTTTTTTTT behaviorally weakly algebraizable

ttiiiiiiiiiiiiiiii

protoalgebraic


Abstract valuation semantics

The problem

AAL _? // bAAL

Matrices _? // ???



The problem

Γ-congruence is in general not a congruence

How to reduce a model?

=⇒ Use an algebraically well-behaved notion of valuationsemantics (da Costa & Beziau 1994)



The problem

Γ-congruence is in general not a congruence

How to reduce a model?

=⇒ Use an algebraically well-behaved notion of valuationsemantics (da Costa & Beziau 1994)




Consider fixed a signature Σ = 〈S, F 〉 and a subsignature Γ of Σ.

Definition

An abstract Γ-valuation is a tuple α = 〈A, ϑ, 〈B, D〉〉 such that:

A is a (concrete) Σ-algebra,

〈B, D〉 is a (abstract) Γ-matrix, and

ϑ : A|Γ → B is a surjective Γ-homomorphism.

An abstract Γ-valuation semantics over Σ is a collection of abstractΓ-valuations.

α with an assignment h over A satisfies ϕ, denoted by α, h ϕ, if ϑ(h(ϕ)) ∈ D.

The class of abstract Γ-valuation models of L will be denoted by AValΓ(L).

=⇒ isomorphism, direct product, subvaluation, subdirect product, ultraproduct, . . .



Completeness

For each set Φ ⊆ LΣ(X) of formulas, we can define the abstract Γ-valuationλΦ

Γ = 〈LΣ(X)|Γ , id, 〈LΣ(X),Φ〉〉.

Lindenbaum abstract Γ-bundle of L : LindΓ(L) = λΦΓ : Φ is a theory of L.

Proposition

For every logic L,

LindΓ(L) ⊆ AValΓ(L);

L is complete with respect to its Lindenbaum abstract Γ-bundleLindΓ(L);

L is complete with respect to AValΓ(L).



Canonical semantics

A congruence of an abstract Γ-valuation α = 〈A, ϑ, 〈B, D〉〉 is simply acongruence of the Γ-matrix 〈B, D〉.The Leibniz Γ-congruence ΩΓ(α) of α is simply the Leibniz congruence of〈B, D〉.

Reduced abstract Γ-valuation is α∗ = 〈A, [ ]ΩΓ(α) ϑ, 〈B/ΩΓ(α), [D]ΩΓ(α)〉〉.

Definition

The canonical abstract Γ-valuation semantics of L is AVal∗Γ(L).



Canonical completeness

If 〈A, D〉 is a Σ-matrix then αΓ(m) = 〈A, id, 〈A|Γ, D〉〉 is an abstract

Γ-valuation.

Proposition

AVal∗Γ(L) is the closure for isomorphisms of α∗Γ(m) | m ∈ Matr(L).

Proposition

For every logic L,

L is complete with respect to its reduced Lindenbaum Γ-bundleLind∗Γ(L);

L is complete with respect to AVal∗Γ(L).


Bridge results

Bridge results

Let L be a many-sorted logic, and Γ a subsignature of Σ.

Theorem (Bloom generalized)

L is finitary iff AValΓ(L) is closed under ultraproducts.

Theorem (Blok-Pigozzi generalized)

L is Γ-behaviorally protoalgebraic iff AVal∗Γ(L) is closed undersubdirect products.

Proposition

Let L be a Γ-behaviorally algebraizable logic with behaviorallyequivalent algebraic semantics K. Then,

AVal∗Γ(L) = AVal∗Γ,K(L).


Bridge results

Bridge results

Let L be a many-sorted logic, and Γ a subsignature of Σ.

Theorem (Czelakowski generalized)

L is Γ-behaviorally equivalential iff AVal∗Γ(L) is closed undersubvaluations and direct products.

Theorem (half-way)

If L is Γ-behaviorally finitely algebraizable with Θ(x) and ∆(x, y),and has a complete Hilbert-calculus with axioms Ax and rules Inf ,then α = 〈A, ϑ, 〈B, D〉〉 ∈ AVal∗Γ(L) iff α satisfies

Θ(ϕ) for each ϕ ∈ Ax ,

Θ(ψ1)& . . .&Θ(ψn)→ Θ(ϕ) for 〈ψ1, . . . , ψn, ϕ〉 ∈ Inf ,

Θ(∆(x, y))→ x ≈ y,

D = ϑ(a) | α Θ(a).


K/2

K/2

Theorem

K/2 is (Σ \ ¬)-behaviorally algebraizable withΘ(x) = x ≈ (x⇒ x), ∆(x1, x2) = x1⇒ x2, x2⇒ x1, and itscanonical semantics is the class of abstract valuations

〈A, ϑ, 〈B, >〉〉

st.

B is a Boolean algebra

ϑ(x) u ϑ(¬x) = ⊥


C1

Signature of C1

Signature ΣC1 = 〈φ, F 〉 such that:

Fεφ = t, f, Fφφ = ¬ and Fφφφ = ∧,∨,⇒.

ϕ is an abbreviation of ¬(ϕ ∧ (¬ϕ));

∼ ϕ is an abbreviation of ¬ϕ ∧ ϕ

Subsignature Γ = 〈φ, FΓ〉 of ΣC1 such that:

FΓφφ = and FΓ

ws = Fws for every ws 6= φφ.


C1

Hilbert-style axiomatization

Axioms:

ξ1⇒ (ξ2⇒ ξ1)

(ξ1⇒ (ξ2⇒ ξ3))⇒ ((ξ1⇒ ξ2)⇒ (ξ1⇒ ξ3))

(ξ1 ∧ ξ2)⇒ ξ1

(ξ1 ∧ ξ2)⇒ ξ2

ξ1⇒ (ξ2⇒ (ξ1 ∧ ξ2))

ξ1⇒ (ξ1 ∨ ξ2)

ξ2⇒ (ξ1 ∨ ξ2)

(ξ1⇒ ξ3)⇒ ((ξ2⇒ ξ3)⇒ ((ξ1 ∨ ξ2)⇒ ξ3))

¬¬ξ1⇒ ξ1

ξ1 ∨ ¬ξ1ξ1 ⇒ (ξ1⇒ (¬ξ1⇒ ξ2))

(ξ1 ∧ ξ2)⇒ (ξ1 ∧ ξ2)

(ξ1 ∧ ξ2)⇒ (ξ1 ∨ ξ2)

(ξ1 ∧ ξ2)⇒ (ξ1⇒ ξ2)

t⇔ (ξ1⇒ ξ1)

f⇔ (ξ1 ∧ (ξ1 ∧ ¬ξ1))

Rule:

ξ1 ξ1⇒ξ2ξ2


C1

Extended signature

Signature ΣC1 :

φ¬,∨,∧,⇒ 99

Extended signature ΣoC1 :

v

φ¬,∨,∧,⇒ 99

o

OO


C1


Theorem

C1 is Γ-behaviorally algebraizable with Θ(x) = x ≈ t,∆(x1, x2) = x1⇒ x2, x2⇒ x1 and the equivalent Γ-hiddenquasivariety semantics, KC1 , is the class of two-sorted algebrasdescribed below.


C1

The class KC1

The class KC1 can be presented as the class of all ΣoC1 -algebras

A = 〈Aφ, Av, oA,∧A,∨A,⇒A, tA, fA,∼A,uA,tA,AA,−A,>A,⊥A〉

such that Av = 〈Av,uA,tA,AA,−A,>A,⊥A〉 is a Boolean algebra

A also Γ-behaviorally satisfies the following set of hidden equations:

ξ1⇒ (ξ2⇒ ξ1) ≈ t

(ξ1⇒ (ξ2⇒ ξ3))⇒ ((ξ1⇒ ξ2)⇒ (ξ1⇒ ξ3)) ≈ t

(ξ1 ∧ ξ2)⇒ ξ1 ≈ t

(ξ1 ∧ ξ2)⇒ ξ2 ≈ t

ξ1⇒ (ξ2⇒ (ξ1 ∧ ξ2)) ≈ t

ξ1⇒ (ξ1 ∨ ξ2) ≈ t

ξ2⇒ (ξ1 ∨ ξ2) ≈ t

(ξ1⇒ ξ3)⇒ ((ξ2⇒ ξ3)⇒ ((ξ1 ∨ ξ2)⇒ ξ3)) ≈ t

¬¬ξ1⇒ ξ1 ≈ t

ξ1 ∨ ¬ξ1 ≈ t

ξ1 ⇒ (ξ1⇒ (¬ξ1⇒ ξ2)) ≈ t

(ξ1 ∧ ξ2)⇒ (ξ1 ∧ ξ2) ≈ t

(ξ1 ∧ ξ2)⇒ (ξ1 ∨ ξ2) ≈ t

(ξ1 ∧ ξ2)⇒ (ξ1⇒ ξ2) ≈ t

t ≈ (ξ1⇒ ξ1)

f ≈ (ξ1 ∧ (ξ1 ∧ ¬ξ1))


C1

da Costa algebras

By a da Costa algebra we mean a structure

U = 〈S, 0, 1,≤,∧,∨,⇒,∼〉,

such that 0, 1 ∈ S and, for every a, b, c ∈ S, the following conditions hold:

≤ ⊆ S × S is a quasi-order;

a ∧ b ≤ a and a ∧ b ≤ b ;

a ∧ a ' a and a ∨ a ' a, where a ' b iff a ≤ b andb ≤ a;

a ∧ (b ∨ c) ' (a ∧ b) ∨ (a ∧ c);

a ≤ a ∨ b, b ≤ a ∨ b;if a ≤ c and b ≤ c then a ∨ b ≤ c;a ≤ (∼ a), where a =∼ (a∧ ∼ a);

a ≤ (b⇒ a)⇒ ((b⇒ ∼ a)⇒ ∼ b);

0 ≤ a, a ≤ 1;

a ∧ (a⇒ b) ≤ b;a ∧ c ≤ b then c ≤ (a⇒ b);

a∨ ∼ a ' 1;

∼ (∼ x) ≤ x;

a∧ ∼ (a) ' 0;

a ∧ b ≤ (a ∧ b);a ∧ b ≤ (a ∨ b);a ∧ b ≤ (a⇒ b).


C1

KC1 and da Costa algebras

Let A = 〈Aφ, Av, oA,∧A,∨A,⇒A,∼A, tA, fA〉 ∈ KC1 and consider thefollowing structure,

UA = 〈SA, 0, 1,≤A,∧A,∨A,⇒A,∼A〉,

obtained from A in the following way:

SA = Aφ

1 = tA and 0 = fA

≤A⊆ SA × SA is such that a ≤A b iff (a⇒A b) ≡Γ tA

Consider 'UA on SA defined by: a 'UA b if a ≤UA b and b ≤UA a.

'UA coincides with behavioral equivalence ≡Γ.


C1


Let U = 〈S, 0, 1,≤U ,∧U ,∨U ,⇒U ,∼U 〉 be a da Costa algebra.

Consider the equivalence relation 'U on S defined by: a 'U b if a ≤U b and b ≤U a.

Consider now the two-sorted algebra,

AU = 〈Aφ, Av , oA,∧A,∨A,⇒A,∼A, tA, fA〉

obtained from U in the following way:

Aφ = S

Av = S/'U

oA(a) = [a]'U

∧A = ∧U ∨A = ∨U ⇒A =⇒U ∼A=∼U

tA = 1 and fA = 0.


C1


Theorem

Given A ∈ KC1 then UA is a da Costa algebra.

Given a da Costa algebra U then AU ∈ KC1 .

Given A ∈ KC1 then AUA∼= A.

Given a da Costa algebra U then UAU∼= U .


C1

Valuation semantics

The complete reduced abstract valuation semantics for C1 resulting from its behavioralalgebraization is the class of valuations 〈A, ϑ, 〈B, D〉〉 such that B is a Booleanalgebra, D = >B, and:

ϑ(t) ≈ >B and ϑ(f) ≈ ⊥B;

ϑ(x ∧ y) ≈ ϑ(x) uB ϑ(y);

ϑ(x ∨ y) ≈ ϑ(x) tB ϑ(y);

ϑ(x ⊃ y) ≈ ϑ(x)⇒B ϑ(y);

−Bϑ(x) ≤B ϑ(¬x);

ϑ(¬¬x) ≤B ϑ(x);

ϑ(x) uB ϑ(x) uB ϑ(¬x) ≈ ⊥B;

ϑ(x) uB ϑ(y) ≤B ϑ((x ∧ y));

ϑ(x) uB ϑ(y) ≤B ϑ((x ∨ y));

ϑ(x) uB ϑ(y) ≤B ϑ((x ⊃ y)).

The well-known bivaluation semantics of C1 can be easily recovered from these

valuations by looking at the underlying irreducible Boolean algebras.


Conclusions

Generalization of the notion of algebraizable logic

Characterization results using the Leibniz operator

Covering many-sorted logics as well as some non-algebraizablelogics (according to the old notion)

Abstract valuations as behavioral logical matrices andapplications

More on the Suszko operator ...


Future work

Connections between different behavioral algebraizations, including thestandard notion ...

... with applications to the theory of combined logics

Establish semantic characterizations for each of the classes on the Leibnizhierarchy

Development of the strong behavioral approach, towards acharacterization of quasivarieties

Examples separating classes in the behavioral Leibniz hierarchy

Other interesting examples, such as first-order logic with many-sortedterms and exogenous probabilistic and quantum logics

Connections to Avron’s non-deterministic matrices and dyadic semantics


References

C. Caleiro and R. Goncalves, On the algebraization of many-sorted logics.Recent Trends in Algebraic Development Techniques, LNCS 4409:21–36, 2007.

R. Goncalves, Behavioral algebraization of logics. PhD, IST - TU Lisbon, 2009.

C. Caleiro and M. Martins and R. Goncalves, Behavioral algebraization oflogics. Studia Logica, 91(1):63–111, 2009.

C. Caleiro and R. Goncalves, Behavioral algebraization of da Costa’sC-systems, Journal of Applied Non-Classical Logics, 19(2):127–148, 2009.

C. Caleiro and R. Goncalves, Algebraic Valuations as Behavioral LogicalMatrices. WoLLIC 2009, LNAI 5514:13–25, 2009.

C. Caleiro and R. Goncalves, Towards a behavioral algebraic theory of logicalvaluations, Fundamenta Informaticae, 106(2-4):191-209, 2011.

C. Caleiro and R. Goncalves, Abstract valuation semantics, to appear in StudiaLogica.

Documents

Abstract Valuation Semantics · Representation of logics in algebraic setting (restricted to propositional based logics) Many-sorted and non-truth-functional logics Propositional