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26Логические исследования 2015. Т. 21. № 2. С. 9-14 УДК 164.07+510.635
Logical Investigations 2015, vol. 21, no 2, pp. 9-14
Символическая логика
Symbolic Logic
V.I. Shalack
On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus
Shalack Vladimir Ivanovich
Department of Logic, Institute of Philosophy, Russian Academy of Sciences. 14/5 Volkhonka St., Moscow, 119991, Russian Federation. E-mail: [email protected]
In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.
Keywords: combinatory logic, definitions, definitional embedding, predicate logic, computable functions
The signature £ is a set of functional and predicate symbols. The first order language L(£) is the set of formulas of the signature £. The models are pairs M = < D, I >, where D is a nonempty set of individuals and I is the function of interpretation of functional and predicate symbols.
In standard way we define relations M,g = A — «formula A is true in the model M with assignment of values to individual variables g» and M = A — «formula A is true in the model M».
The theory in the language L(£) is some set of logical and non-logical axioms closed under derivability.
The first-order predicate calculus in the language L(£) is the theory in the language L(£) with empty set of non-logical axioms.
© Shalack V.I.
1. Defining new predicate symbols
We can use definitions in order to extend the language L(E) of theories with new predicate symbols. The definitions have the following form [2, p. 15]:
Vxi. ..Xn(P (xi, ...,xn) = A (xi,.. .,Xn)).
The definition must satisfy the conditions:
1. P / E.
2. A (xi,...,xn) e L(E).
3. The variables x1,... ,xn are pairwise distinct.
4. The set of free variables of A (x1,..., xn) is included into {x1,..., xn}.
After defining of the new predicate symbol P, it must be added to the signature E. As a result, there is a transition from the language L(E) to the language L(E U {P}).
In the language of the first order predicate calculus we can define the universal n-ary predicate Un by the following definition:
(DU) Vxi ...xn(Unxi,...,xn = Pxi y-Pxi).
The definition allows us to prove DU h Vxi... xnUnxi,..., xn.
This example is interesting because in the right part of the definition we use an arbitrary predicate symbol of the language of the first order predicate calculus. As another example, we can give a definition of a symmetric relation. Let B be an arbitrary predicate symbol of the language. We accept the following definition:
(DS) Vxy(Sxy = Vuv(Buv D Bvu) D Bxy).
Let us show that DS h Vxy(Sxy D Syx).
1. Sxy - hyp
2. Vuv(Buv D Bvu) D Bxy - from 1 by DS
3. Vuv(Buv D Bvu) - hyp
4. Bxy - from 2, 3
5. Bxy D Byx - from 3
6. Byx - from 4, 5
7. Vuv(Buv D Bvu) D Byx - from 3-6
8. Syx - from 7 by DS
9. Sxy D Syx - from 1-8
It turns out that many well-known theories can be defined within the first order predicate calculus.
Definition 1. The first-order theory T in a language L(£) with finite set of non-logical axioms Ax is definitionally embeddable into predicate calculus, iff there are a signature £' and a set of definitions DT of symbols £ \ £' by formulas of L(£') which met the following condition:
If A € L(£), then Ax h A & DT h A.
This definition is some variant of the notion of definitional embeddability of theories which was proposed by V.A. Smirnov [3].
2. Combinatory logic as the first-order theory
We can formulate the first order theory of combinatory logic in the language with signature = {K, S, *, >}, where
• K, S — individual constants.
• * — 2-argument functional symbol.
• > — 2-place predicate symbol.
The next formulas are non-logical axioms of the theory:
Ax.1 Vxy((K * x) * y > x)
Ax.2 Vxyz(((S * x) * y) * z > (x * z) * (y * z))
Ax.3 Vx(x > x)
Ax.4 Vxyz(x > y D (x * z) > (y * z))
Ax.5 Vxyz(x > y D (z * x) > (z * y))
Ax.6 Vxyz(x > y&y > z D x > z)
The rules of inference are modus ponens and generalisation. It is easy to verify that all the axioms and the rules of inference of combinatory logic as calculus of reductions [1] are provable in this theory.
3. The theorem of embeddability
Theorem 1. The theory of combinatory logic is definitionally embeddable into first-order predicate calculus.
Proof. We assume that the signature £p of the first-order predicate calculus contains the symbols K, S, *, > but doesn't contain the symbol >.
• K, S — individual constants.
• * — 2-argument functional symbol.
• > — 2-place predicate symbol.
We write Ax&> for conjunctions of the axioms Ax. 1-6 in which the symbol > is renamed into >.
We accept the following definition:
(DC) Vxy(x > y = (Ax&> D x>y)).
Let's check DC h Ax. 1-6.
Ax.1 Vxy((K * x) * y > x)
1. h Ax&> D Vxy((K * x) * y > x) - by def. of Ax&>
2. h Vxy(Ax&> D (K * x) * y > x) - from 1
3. DC h Vxy((K * x) * y > x) - from 2 by DC
Ax.2 Vxyz(((S * x) * y) * z > (x * z) * (y * z))
1. h Ax&> D Vxyz(((S * x) * y) * z > (x * z) * (y * z)) - by def. of Ax&>
2. h Vxyz(Ax&> D ((S * x) * y) * z > (x * z) * (y * z)) - from 1
3. DC h Vxyz(((S * x) * y) * z > (x * z) * (y * z)) - from 2 by DC
Ax.3 Vx(x > x)
1. h Ax&> D Vx(x > x)
2. h Vx(Ax&> D x > x)
3. DC h Vx(x > x)
- by def. of Ax&>
- from 1
- from 2 by DC
Ax.4 "ixyz(x > y D (x * z) > (y * z))
1. h Ax&> D Vxyz(x > y D (x * z) > (y * z)) - by def. of Ax&>
2. h yxyz(Ax&> D (x > y D (x * z) > (y * z))) - from 1
3. h Vxyz((Ax&> D x > y) D (Ax&> D (x * z) >
(y * z))) - from 2 by self-distr. of D
4. DC h yxyz(x > y D (x * z) > (y * z)) - from 3 by DC
Ax.5 "ixyz(x > y D (z * x) > (z * y))
1. h Ax&> D Vxyz(x > y D (z * x) > (z * y)) - by def. of Ax&>
2. h yxyz(Ax&> D (x > y D (z * x) > (z * y))) - from 1
3. h Vxyz((Ax&> D x > y) D (Ax&> D (z * x) >
(z * y))) - from 2 by self-distr. of D
4. DC h Vxyz(x > y D (z * x) > (z * y)) - from 3 by DC
Ax.6 yxyz(x > y&y > z D x > z)
1. h Ax&> D Vxyz(x > y&y > z D x > z) - by def. of Ax&>
2. h Ax&> D Vxyz(x > y D (y > z D x > z)) - from 1
3. h yxyz(Ax&> D (x > y D (y > z D x > z))) - from 2
4. h yxyz((Ax&> D x > y) D ((Ax&> D y > z) D
(Ax&> D x > z))) - from 3 by self-distr. of D
5. h yxyz(x > y D (y > z D x > z)) - from 4 by DC
6. DC h yxyz(x > y&y > z D x > z) - from 5
Thus we have shown that all the axioms of the theory of combinatory logic are derivable from the DC. It follows that if A £ L(Eq) and Ax.1 — 6 h A, then DC h A.
Now we must show that if A £ L(^C) and DC h A, then Ax.1 — 6 h A.
Let us assume that DC h A but not Ax.1 — 6 h A. By the completeness theorem of the first-order logic it follows that DC \= A and there exists such a model M =< D, I > and an assignment of values to individual variables g that M = Ax.1 — 6 and M,g = -A.
We can extend the model M = < D, I > to the model M' = < D, I' > in which the formula DC is true. It is sufficient to expand the domain
of the function I so that the new function of interpretation I' ascribes value I'(>) = I(>) to predicate symbol >, and for all other functional and predicate symbols it retains the same value as I.
Since M = Ax.1-6, then in the model M' =< D, I' > by the definition of I' we will have M' = Ax&> and hence M' = x > y = (Ax&> D x > y). Therefore by our assumption DC = A it must be M',g = A. However the formula A doesn't contain the symbol >, while all other descriptive symbols are interpreted in the same way as in the model M, and by our assumption M,g = -A, it must be M\g |= -A. We have obtained the contradiction. Therefore the assumption that Ax.1 — 6 h A does not hold, is false. □
Corollary 1. All effectively computable functions can be represented in the first-order predicate logic.
Proof. This statement is true since all effectively computable functions can be represented in combinatory logic, which, as we have just shown, is definitionally embeddable in the first-order predicate logic. □
References
[1] Engeler, E. Metamathematik der Elementarmathematik. New York: Springer, 1983. 132 pp.
[2] Karpovich, V.N. Terminy v strukture teorii. Logicheskij analiz [Terms in the structure of the theory. Logical analysis]. Novosibirsk: Nauka, 1978. 128 pp. (In Russian)
[3] Smirnov, V.A. "Logical Relations between Theories", Synthese, 1986, 66(1), pp. 71-87.
[4] Shalack, V. "On Some Applied First-Order Theories which Can be Represented by Definitions", Bulletin of the Section of Logic, 2015, 44/1-2, pp. 19-24.
