1. Axioms of LB

Bulletin of the Section of Logic Volume 18/4 (1989), pp. 161–165 reedition 2006 [original edition, pp. 161–166]

Marek Tokarz

ON THE LOGIC OF CONSCIOUS BELIEF

This is an abstract of a paper winch has been submitted for a publication in Studia Logica. Here we are going to give a sketchy presentation of a logic of the phrase “x believes that p”, where x is a human being and p is a sentence, hereafter abbreviated into Bp. The system LB constructed here differs from Hintikka’s [Knowledge and Belief, 1962]: in LB conscious beliefs only are dealt with, that is, if x believes that p then x believes that he believes that p (cf. axiom Ax.2 below), and if x does not believe that p then x believes that he does not believe that p (cf. Ax.3). Traditionally x’s beliefs are also assumed to be “reasonable” in the following three respects: x believes in all logical laws (RB below); x’s beliefs are consistent (Ax.4); and are closed with respect to modus ponens (Ax.5). In section 3 some modifications of LB are considered.

1. Axioms of LB Let L denote the set of classical formulas in ∼, ∧, ∨ (→ and ≡ being defined as usual). P C is the set of classical tautologies; t denotes any fixed tautology, say p ∨ ∼ p; f denotes any fixed countertautology, say p ∧ ∼ p. S is the set of all formulas in ∼, ∧, ∨ and B (a unary connective). Taut is the set of all instances of P C in S; for example: p → p, Bp → (q → Bp), B(q ∧ B ∼ p) → B(q ∧ B ∼ p) all belong to Taut. Axioms of LB are the following (α, β ∈ S): Ax.1 α, where α ∈ T aut Ax.2 Bα ≡ BBα, Ax.3 ∼ Bα ≡ B ∼ Bα, Ax.4 B ∼ α → ∼ Bα,

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Ax.5 B(α → β) → (Bα → Bβ), Rules of inference are: M P : if ` α and ` α → β then ` β RB: if ` α then ` Bα LB is the set of all consequences of Ax.1 - Ax.5 that can be obtained by use of M P and RB. We shall write `LB α or simply ` α instead of α ∈ LB.

2. Interpretations Let T be a classical theory (in L), and let α ∈ S. By IT α we denote the interpretation of α in T , defined inductively as follows: 1o IT p = p if p is a variable 2o IT ∼ β = ∼ IT β 3o IT (β ∧ γ) = IT β ∧ IT γ 4o IT (β ∨ γ)= IT β ∨ IT γ β if IT β ∈ T 5o IT Bβ = f if IT β 6∈ T. Theorem 1. ` α iff for every consistent classical theory T, IT α ∈ P C. A structure U = < A, −, ∪, ∩, ∗, F > is a filter algebra, in symbols U ∈ F A, if < A, −, ∪, ∩ > is a Boolean algebra (with 1 and 0 understood as usual), F is a filter, and for every a ∈ A  ∗a =

1 0

if a ∈ F if a ∈ 6 F.

A formula α is true in U ∈ F A, in symbols U |= α, if hα = 1 for every valuation (homomorphism) of S into U. Where K ⊆ F A, K |= α stands for “U |= α for every U ∈ K”. A filter algebra U = < A, −, ∪, ∩, ∗, F > is a proper-filter algebra, in symbols U ∈ P F A, if F is a proper filter, that is, if 0 6∈ F . Theorem 2. ` α iff P F A |= α.

On the Logic of Conscious Belief

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3. Modifications A System LCB. The logic of complete belief, LCB, results by adding the following axiom Ax.6 to Ax.1 - Ax.5: Ax.6 Bα ∨ B ∼ α. (Ax.6 is equivalent to either of the following three formulas: ∼ Bα → B ∼ α, B(α ∨ β) ≡ (Bα ∨ Bβ), B(α → β) ≡ (Bα → Bβ).) A filter algebra U = < A, −, ∪, ∩, ∗, F > is an ultrafilter algebra, in symbols U ∈ M F A, if F is an ultrafilter (maximal filter). Theorem 3. `LCB α iff M F A |= α. B System LIB. By rejecting the axiom of consistency, Ax.4, we obtain an inconsistency-tolerating belief logic LIB. LIB is thus defined aa the set of all consequences of Ax.1, Ax.2, Ax.3, Ax.5. Theorem 4. `LIB α iff F A |= α. C System S5. The expression “x knows that p” differs from “x believes that p” in that the former presupposes that p is true, while the later does not. So, as to obtain the logic of the phrase “x knows that p” from LB, we should add to Ax.1 - Ax.5 a new axiom schema Ax.7: Ax.7 Bα → α. It is easily seen that Ax.7 is true in those filter algebras < A, −, ∪, ∩, ∗, F > only in which F is a unit filter, i.e., in which F = {1}. Let us denote the class of all such algebras by U F A. The logic based on Ax.1 - Ax.5 and Ax.7 happens to coincide with Lewis’ S5, which leads us to the following well-known completeness theorem, proved by Wajsberg in the thirties: Theorem 5. (Wajsberg, 1933) `S5 α iff U F A |= α.

4. Examples We are going to show how to apply our completeness theorems. We shall discuss three simple formulas: ∼ Bp → B ∼ p, Bp → p, and B(Bp → p). A. 6` ∼ Bp → B ∼ p.

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Since operation ∗ takes values from {0, 1} only we should look for a filter algebra with an element a such that neither a belongs to F (for we want − ∗ a to be 1), nor −a belong to F (for we want ∗ − a to be 0). We can thus take the following U F A-algebra: '$ F 1 • @ @ &% @ @ @ @• b

a • @ @ @ @

@ @• 0 and put hp = a to obtain h(∼ Bp → B ∼ p) = 0. Hence ∼ Bp → B ∼ p is not a thesis of S5, and consequently it is neither a thesis of LB nor of LIB. B. 6` Bp → p. We need an algebra one element b 6= 1 of which belongs to F (for then ∗b = 1). One such algebra is U4 below: ' @ @

U4 :

@ F 1 @ • @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @• b a • @ @ @ @ @ @ @ @ % @ @• 0

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We thus put hp = b (in U4 ) to obtain h(Bp → p) = ∗b ⇒ b = 1 ⇒ b = b 6= 1. Hence Bp → p is neither a thesis of LCB nor a thesis of LB or LIB. C. ` B(Bp → p). Suppose that hB(Bp → p) 6= 1 for some valuation h of S in an algebra with filter F . Since hBα ∈ {0, 1} for all α, we have hB(Bp → p) = 0 and consequently h(Bp → p) 6∈ F . Hence hp 6∈ F ; but then hBp = 0 and h(Bp → p) = 1 ∈ F − a contradiction.

5. Supplement A Decidability. The systems defined in Sections 1 and 3 are all decidable. B Cardinality of models. For LB, LIB, and clearly for S5, the following holds: there is no finite number k such that α is a thesis iff α is true in every algebra of cardinality less than or equal to k. There is, however, such a k for LCB; it is exactly 4; α is a thesis of LCB iff U4 |= α, where U4 is the algebra defined in example B, Section 4. C Classical interpretations. The following modifications of theorem 1 hold for system LCB and LIB: Theorem 6. a. `LCB α iff for every maximal classical theory T , IT α ∈ P C; b. `LIB α iff for every classical theory T , IT α ∈ P C..

Department of Logic Silesian University Katowice, Poland