Bulletin of the Section of Logic
Volume 2/2 (1973), pp. 144–147
reedition 2013 [original edition, pp. 144–148]
Czeslaw Wojtkiewicz
AXIOMATIZATION OF FINITE N -ALGEBRAS BY THE
METHOD OF T. HOSOI
In [1] a method has been presented of axiomatizing finite and (so called)
regular models of intuitionistic propositional logic. The aim of the present
abstract is to apply the method in order to obtain an analogous result for
the propositional part of a constructive logic formulated by D. Nelson in
[2].
The logical and mathematical notation I shall use in the sequel does not
differ from the standard one. Especially, p1 , p2 , . . . (or, informally, p, q, . . .)
denote propositional variables and α, β, . . . stand for arbitrary formulas of
a given language. For any sets X, Y of formulas SbY X denotes the set of
all substitution-instances of formulas from X by means of formulas from Y .
1. The axiomatic system of propositional logic of Nelson, call it LN ,
can be described briefly as follows. Let LP be an axiomatic system of
(intuitionistic) positive propositional logic with denumerable set {p, q, . . .}
of propositional variables, with conjunction (∧), disjunction (∨) and implication (→) as propositional connectives and with substitution and detachment as the only rules of inference. Now, to obtain LN from LP introduce
to the language of LP a new unary propositional connective ¬ together with
the following new axioms (where (α ↔ β) abbreviates ((α → β) ∧ (β → α))
(1)
(2)
(3)
(4)
(5)
p → (¬p → q)
¬(p ∧ q) ↔ (¬p ∨ ¬q)
¬(p ∨ q) ↔ (¬p ∧ ¬q)
¬(p → q) ↔ (p ∧ ¬q)
¬¬p ↔ p
Axiomatization of Finite N -Algebras by the Method of T. Hosoi
145
Denote by Ax the axioms for LP together with axioms (1) – (5). Relative
to LN , the consequence operation CN can be defined in a standard way,
CN (X) being the smallest set containing X ∪ Ax and closed under substitution and detachment. For any formulas α, β of LN let (α ⇔ β) be an
abbreviation of ((α ↔ β)∧(¬α ↔ ¬β)). One can easily verify the following
(6) (α ⇔ α) ∈ CN (0) for every formula α of LN ,
(7) γ(α//β) ∈ CN (α ⇔ β, γ), where the formula γ(α//β) results from
the formula γ via replacing an occurrence of α in γ by the formula β.
2. By finite n-element N -algebra we mean any system N = hN, 1, ⇔,
∧, ∨, →, ¬i where
(i) hN, ⇔, ∧, ∨, →, ¬i is an algebra similar to the algebra of formulas of
LN ,
(ii) card N = n < ℵ0 ,
(iii) 1 ∈ N is a distinguished element from N ,
(iv) all the axioms of LN are true in N (i.e. v(α) = 1 for every axiom α
of LN and every valuation v in N of the formulas of LN ),
(v) 1 → x = 1 implies x = 1, for all x ∈ N ,
(vi) x ⇔ y = 1 implies x = y, for all x, y ∈ N .
Denote by E(N ) the set of all formulas of LN which are true in N .
3. Given the set Pm = {p1 , . . . , pm } of variables, denote by Lm
N the
sublanguage of LN generated by Pm . Given an n-element N -algebra N ,
let V (m) be the set of all valuations v : Lm
N → N . Of course, V (m) is finite.
,
let
[α]
be
the equivalence class {β : α ∼ β,
For v ∈ V (m) and α from Lm
v
N
v
df
v(α) = v(β). Put Lm
β formula of Lm
N /v = {[α]v : α
N }, where α ∼ β =
v
formula of Lm
N }. Finally, define in an arbitrary way a selector s, i.e. a
m
mapping s : Lm
N /v → LN such that s[α]v ∈ [α]v .
4. Now, given n, m, define (using generalized disjunctions and conjunctions as abbreviations) the following formulas
(a) γn =
W
(pi ⇔ pj )
16i<n+1
146
Czeslaw Wojtkiewicz
(b) εm
v =
V
(pi ⇔ s[pi ]v )
16i6m
V
(c) ηvm = {(s[o(α1 , . . .)]v ⇔ o(s[α1 ]v , . . .))}o∈{¬,∧,∨,→}
m
(d) δvm = εm
v ∧ ηv .
Finally define
W
(b0 ) εm = {εm
: v ∈ V (m)}
W vm
0
m
(c ) η = {ηv : v ∈ V (m)}
W
(d0 ) δ m = {δvm : v ∈ V (m)}
5. For given n-element N -algebra N consider the following set of
formulas
Ax(n) = {δ n , γn }
One can versify that γn ∈ E(N ) and for m > 1 εm , η m , δ m ∈ E(N ).
Moreover, one can verify, by induction on complexity of a formula α, that
α ∈ CN (Ax(n) ) implies α ∈ E(N ). So we have the following
Lemma 1. CN (Ax(n) ) ⊆ E(N ).
To prove the converse inclusion first observe the two following facts
Fact 1. α ∈ CN (γn , SbLm
).
N
m
Fact 2. If α ∈ E(M ) ∩ Lm
N then α ∈ CN (δ ).
Fact 1 is obvious in the view of definition of the formula γn together
with the property (7) of CN and several well known tautologies of LP concerning disjunction and conjunction. To prove Fact 2 assume α ∈ E(N ) ∩
Lm
N . Without loss of generality we may assume that, given any v ∈ V (m),
s[p1 → p1 ]v = (p1 → p1 ). Then (α ⇔ (p1 → p1 )) ∈ E(N ) ∩ Lm
N and so
v(α) = v(p1 → p1 ). Hence [α]v = [(p1 → p1 )]v and thus s[α]v = s[p1 →
p1 ]v = (p1 → p1 ). Hence by (6) we have (s[α]v ⇔ (p1 → p1 )) ∈ CN (0) and
so s[(α ⇔ (p1 → p1 ))]v ∈ CN (ηvm ) ⊆ CN (δvm ). From this we may deduce
(α ⇔ (p1 → p1 )) ∈ CN (δVm ). Thus (α ⇔ (p1 → p1 )) ∈ CN (δ m ) and finally
α ∈ CN (δ m ). This gives us α ∈ CN (Ax(n) ), as required. By Facts 1 and 2
we obtain
Lemma 2. E(N ) ⊆ CN (Ax(n) ).
Axiomatization of Finite N -Algebras by the Method of T. Hosoi
147
Lemma 1 and 2 give us the required
Theorem. Every finite N -algebra is finitely axiomatizable.
Acknowledgement. I am indebted to Dr Jerzy Perzanowski for calling
my attention to the paper of T. Hosoi and for his helpful remarks concerning
the present adaptation of the method of T. Hosoi to finite N -algebras.
References
[1] Tsutomu Hosoi, On the axiomatic method and the algebraic method
for dealing with propositional logic, Journal of the Faculty of Science,
University of Tokyo, Section 1, vol. 14 (1967), pp. 131–169.
[2] D. Nelson, Constructible falsity, J. S. L. 14 (1949), pp. 16–26.
The Section of Logic
Institute of Philosophy and Sociology
Polish Academy of Sciences