THE OPPOSITIONAL GEOMETRY

OF THE MODAL SYSTEM K45

Alessio Moretti & Frédéric Sart

Vatican City — May 8, 2014

4th World Congress on the Square of Opposition

1

Introduction 2

In 1944, McKinsey and Tarski proved that the free cyclic

S4-algebra, i.e. the free S4-algebra on one generator,

is infinite

This result gives an idea of the complexity of the modal

system S4

Back to basics (1) 3

Free Boolean algebras are commonly used to study

classical propositional logic (hereafter CPL) from an

algebraic point of view

The free Boolean algebra on n generators, which

characterizes CPL restricted to n-variable formulas, has

22n elements

Back to basics (2) 4

The free Boolean algebra on two generators

Atoms p∧q ¬p∧¬q ¬p∧q¬ ¬p∧¬q

Free modal algebras (1) 5

Free modal algebras generalize free Boolean algebras

In contrast to their Boolean counterparts, finitely

generated free modal algebras may be infinite

(McKinsey-Tarski result)

Free modal algebras (2) 6

In the next slides, we will focus on the case of one

generator, extensively studied in opposition theory

We will present several modal systems that are finitely

representable, i.e. have their free cyclic algebra finite

Finiteness result: S5 (= KT45) 7

Bass (1958)

The free cyclic S5-algebra has 24 = 16 elements

Finiteness result: KD45 8

Moretti (2009)

The free cyclic KD45-algebra has 26 = 64 elements

Finiteness result: K45 9

Sart (2009)

The free cyclic K45-algebra has 28 = 256 elements

K45 as a natural extension of CPL (1) 10

As in classical propositional logic, the atoms of K45 are

obtained in a purely combinatorial way

K45

One generator

1. ¬p∧¬p∧¬¬p

4. ¬p∧¬p∧¬¬p

5. ¬p∧¬p∧¬¬p

6. ¬p∧¬p∧¬¬p

7. ¬p∧¬p∧¬¬p

8. ¬p∧¬p∧¬¬p

2. ¬p∧¬p∧¬¬p

3. ¬p∧¬p∧¬¬p

As in classical propositional logic, the atoms of K45 are

obtained in a purely combinatorial way

K45

Two generators

K45 as a natural extension of CPL (2) 11

01. ¬p∧¬q∧¬(p∧q)∧¬(p∧¬q)∧¬(¬p∧q)∧¬(¬p∧¬q)

02. ¬p∧¬q∧¬(p∧q)∧¬(p∧¬q)∧¬(¬p∧q)∧¬(¬p∧¬q)

63. ¬p∧¬q∧¬(p∧q)∧¬(p∧¬q)∧¬(¬p∧q)∧¬(¬p∧¬q)

64. ¬p∧¬q∧¬(p∧q)∧¬(p∧¬q)∧¬(¬p∧q)∧¬(¬p∧¬q)

l

l

K45 as a natural extension of CPL (3) 12

In K45, as in CPL, every n-variable formula is equivalent

to one and only one disjunction of n-generated atoms

For example, in K45, p is equivalent to 3∨4∨7∨8

1. ¬p∧¬p∧¬¬p

4. ¬p∧¬p∧¬¬p

5. ¬p∧¬p∧¬¬p

6. ¬p∧¬p∧¬¬p

7. ¬p∧¬p∧¬¬p

8. ¬p∧¬p∧¬¬p

2. ¬p∧¬p∧¬¬p

3. ¬p∧¬p∧¬¬p

There are only ten distinct modal systems obtained by

adding one-variable axiom schemes to K45

K45-subvarieties (1) 13

Gärdenfors (1973)

KT!45 (= CPL)

KT45 (= S5)

KTc45

KD!45 KDc45

K45 KD45

KB45

K45+⊥ K45+⊥

K45-subvarieties (2) 14

The free cyclic algebras of the aforementioned ten

modal systems have respectively the following atoms

K45+⊥

KTc45

KD!45

KDc45

KT45 (=S5)

KB45

K45+⊥

KT!45 (= CPL)

KD45

K45

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}

K45-subvarieties (3) 15

The disjunctive normal form theorem still holds for these

ten systems, but with less atoms than in K45

For example, in KD45, p is equivalent to 3∨4

1. ¬p∧¬p∧¬¬p

4. ¬p∧¬p∧¬¬p

5. ¬p∧¬p∧¬¬p

6. ¬p∧¬p∧¬¬p

7. ¬p∧¬p∧¬¬p

8. ¬p∧¬p∧¬¬p

2. ¬p∧¬p∧¬¬p

3. ¬p∧¬p∧¬¬p

Conclusion (1) 16

In this presentation, we showed that the free cyclic K45-

algebra, characterizing K45 restricted to one-variable

formulas, is finite

This result contrasts with that obtained by McKinsey and

Tarski for S4 (=KT4)

Conclusion (2) 17

However, we are not yet able to give a necessary and

sufficient condition for finite representability

Infinitely representable

K45, KD45, S5, …

K, K4, S4, …

Finitely representable

References 18

Bass, H. (1958). Finite monadic algebras. Proceedings of the American

Mathematical Society, 9(2), 258-268.

Gärdenfors, P. (1973). On the extensions of S5. Notre Dame Journal of

Formal Logic, 14(2), 277-280.

Massey, G. J. (2001). Atomic Boolean Algebras and Classical

Propositional Logic. In Logic, Meaning and Computation (pp. 185-189).

Springer Netherlands.

McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals

of mathematics, 141-191.

Moretti, A. (2009). The geometry of standard deontic logic. Logica

Universalis, 3(1), 19-57.

Sart, F. (2009). A purely combinatorial Approach to Deontic Logic.

Logique et Analyse, 52(206), 131-138.