Bulletin of the Section of Logic
Volume 7/3 (1978), pp. 137–140
reedition 2011 [original edition, pp. 137–142]
Tadeusz Prucnal
ON FRIEDMAN’S PROBLEM IN MATHEMATICAL
LOGIC∗
(Preliminary Report)
0. Let F = hF , ∧, ∼, i be the free algebra in the class of all algebras
of the type h2, 1, 1i free-generated by the set V = {p1 , p2 , . . .} = {pi : i ∈
N }. By he we denote the extension of the function e : V → F to the
endomorphism of the algebra F .
H. Friedman in [1] conjectured that there are sets M ⊆ F such that:
(F 1)
(F 2)
(F 3)
(F 4)
V ⊆ M,
∼ α ∈ M ⇔ α 6∈ M ,
α ∧ β ∈ M ⇔ αW∈ M ∧∧β ∈ M ,
α ∈ M ⇔
he (α) ∈ M ,
e:V →F
for every α, β ∈ F .
In this paper we will show that there exists a set M ⊆ F such that
the conditions (F 1) − (F 4) are satisfied.
We shall use the symbols: ⇔, ⇒, ∧∧, ∨∨ as the well-known propositional
connectives from metalanguage. The symbols ∀ and ∃ will also be used as
quantifiers from metalanguage.
1. Let now F = hF, ∨, ∧, →, ∼i be the free algebra in the class of all
algebras of the type h2, 2, 2, 1i free-generated by the set V . By T we denote
the well-known McKinsey-Tarski transformation (cf. [2]), which maps F
into F in the following way:
a. T (pi ) = pi ,
b. T (∼ α) = ∼ T (α),
∗ This paper was also presented at the seminar of the Department of Logic, Jagiellonian University, Cracow, March 6, 1978.
138
Tadeusz Prucnal
c. T (α ∧ β) = T (α) ∧ T (β),
d. T (α ∨ β) =∼ [∼ T (α)∧ ∼ T (β)],
e. T (α → β) = ∼ [T (α)∧ ∼ T (β)]
for every i ∈ N and α, β ∈ F .
By IN T we mean the set of all theorems of intuitionistic propositional
logic, and by S4 – the set of all theorems of modal logic. CnIN T (X) is
the smallest set containing IN T ∪ X ⊆ F and closed under the modus
ponens rule. Similarly: CnS4 (Y ) is the least set containing Y ∪ S4 ⊆ F
and closed under the modus ponens rule: ∼ (α∧ ∼ β), α/β.
We have:
Lemma 1. (Cf. [7]). For every γ ∈ F and X ⊆ F :
γ ∈ CnIN T (X) ⇔ T (γ) ∈ CnS4 (T (X)),
where T (X) is the image of the set X.
By FT we denote the least set containing the image T (F ) and closed
with respect to: ∧, ∼, and .
Lemma 2. (Cf. [6]). For every α ∈ FT there are γ1 , γ2 , . . . , γn , δ1 , δ2 , . . . , δn ∈
F such that:
α ≡∼ [T (γ1 )∧ ∼ T (δ1 )] ∧ . . . ∧ [T (γn )∧ ∼ T (δn )],
where α ≡ β ⇔df ∼ (α∧ ∼ β)∧ ∼ (β∧ ∼ α) ∈ S4.
Let B be the least set containing {∼ pi : i ∈ N } and closed under the
connectives: ∨, ∧, →, and ∼.
Putting
M L =df {γ ∈ F : ∀e:V →B he (γ) ∈ KP },
where KP is an intermediate logic obtained by adding to IN T the axioms:
(∼ γ → α ∨ β) → (∼ γ → α) ∨ (∼ γ → β), α, β ∈ F , we obtain an
intermediate logic such that KP 6⊆ M L.
We have:
Lemma 3. (Cf. [3]). For every α, β ∈ F :
α ∨ β ∈ M L ⇔ α ∈ M L ∨∨β ∈ M L.
On Friedman’s Problem in Mathematical Logic (Preliminary report)
139
This M L has also the following property1 :
Lemma 4. (Cf. [5]). For every α, β ∈ F :
α → β ∈ M L ⇔ ∀e:V →F [he (α) ∈ M L ⇒ he (β) ∈ M L].
Putting
M L(T ) =df CnS4 (T (M L)),
we obtain:
Lemma 5.
(i) γ ∈ M L ⇔ T (γ) ∈ M L(T ) , for every γ ∈ F .
(ii) α ∈ M L(T ) ⇔ α ∈ M L(T ) , for every α ∈ F .
(iii) ∼ (∼ α∧ ∼ β) ∈ M L(T ) ⇔ α ∈ M L(T ) ∨∨β ∈ M L(T ) ,
for every α, β ∈ FT .
Let now M L be a set defined as follows:
M L =df {α ∈ F : ∀e:V →FT he (α) ∈ M L(T ) }.
Lemma 6. For every γ ∈ F :
γ ∈ M L ⇔ T (γ) ∈ M L .
Lemma 7. For every α, β ∈ F :
(i)
(ii)
(iii)
(iv)
(v)
S4 M L ,
α, ∼ (α∧ ∼ β) ∈ M L ⇒ β ∈ M L ,
α ∈ M L ⇒ ∀e:V →F he (α) ∈ M L ,
α ∈ M L ⇔ α ∈ M L ,
α ∈ M L ∨∨β ∈ M L ⇔∼ (∼ α∧ ∼ β) ∈ M L .
We define now a set A ⊆ F in the following way:
β ∈ A ⇔ ∃e:V →F ∃α∈F −M L β =∼ (α ∧ he (α)),
for every β ∈ F .
1 Let us note that Lemma 4 states that the calculus M L is structurally complete
in the sense of W. A. Pogorzelski [4].
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Tadeusz Prucnal
Thus:
Lemma 8. CnS4 (M L ∪ A ∪ V ∪ {∼ T (γ) : γ ∈ F − M L}) 6= F .
Let M0 =df CnS4 (M L ∪ A ∪ V ∪ {∼ T (γ) : γ ∈ F − M L}) and let
M∗ be the maximal element in {M ⊆ F : M0 ⊆ M = CnS4 (M ) 6= F }.
Thus we have:
Theorem.2 The set M∗ satisfies the conditions (F 1) − (F 4).
References
[1] H. Friedman, One hundred and two problems in mathematical logic,
Journal of Symbolic Logic, Vol. 40 (1975), pp. 113–129.
[2] J. C. C. McKinsey and A. Tarski, Some theorems about the sentential calculi of Lewis and Heyting, Journal of Symbolic Logic, Vol. 13
(1948), pp. 1–15.
[3] Ú. T. Médvédev, Intérprétaciá logičeskih formul pośrédstvam finitnyh zadač i sváz éé s téorj́ réalizuémosti, Doklady Akadémii Nauk
SSSR, vol. 148 (1963), pp. 771–774.
[4] W. A. Pogorzelski, Structural completeness of the propositional
calculus, Bull. Acad. Polon. Sci., Ser. Math. Astr. et Phys., Vol. 19
(1971), pp. 349–351.
[5] T. Prucnal, Structural completeness of Medvedev’s propositional
calculus, Reports on Mathematical Logic, No. 6 (1976), pp. 103–105.
[6] H. Rasiowa, An algebraic approach to non-classical logics,
North-Holland Publishing Company, Amsterdam, PWN Polish Scientific
Publishers, Warszawa 1974.
[7] H. Rasiowa and R. Sikorski, The mathematics of metamathematics, PWN Warszawa 1963.
Institute of Mathematics
Pedagogical College, Kielce
2 Dr J. Perzanowski informed me that an analogous results had been obtained by Kit
Fine (unpublished).