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29Логические исследования 2016. Т. 22. № 1. С. 125-135 УДК 164.07+510.635
Logical Investigations 2016, vol. 22, no 1, pp. 125-135
Символическая логика
Symbolic Logic
V.I. Shalack
On First-order Theories Which Can Be Represented by Definitions
Shalack Vladimir Ivanovich
Department of Logic, Institute of Philosophy, Russian Academy of Sciences. 12/1 Goncharnaya St., Moscow, 109240, Russian Federation. E-mail: shalack@gmail.com
In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic.
Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others.
It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 = x'.
Keywords: definition, definability, predicate calculus, theory, logicism
1. Logicism
As we know the main idea of logicism was that mathematics was an extension of logic and was reducible to logic by appropriate definitions.
One of the explications of logicism might look like if you are given a theory T with the set of postulates Ax. It is required to find such a set of logical definitions DF of mathematical notions of the theory T that for every formula B € LT holds:
Axh B & DF\- B.
© Shalack V.I.
As we know the attempt to implement the program of classical logicism has failed. It needs the higher-order logic, and far from intuitively obvious axioms: reducibility, multiplicativity (choice) and infinity, which can hardly be called logical. This was a major rebuke to the logicism.
It is interesting to find an answer to the more specific question:
To what limits classical logicism program can be implemented in the first-order predicate logic?
2. Defining new predicate symbols
We assume that the language of first-order predicate calculus is defined in the standard way as the set of terms and formulas over the signature £, which consists of nonlogical relational and functional symbols. We write L(£) for the first-order language over signature £. Models are pairs M = (D, I}, where D is a non-empty set of individuals, and I is an interpretation of the function and predicate symbols in the domain D. The relations "formula A is true in the model M for value assignment to individual variables g" and "formula A is true in the model M" are defined as usual and are written as M, g N A and M N A.
A first-order theory in the language L(£) is a set of logical axioms and non-logical postulates closed by derivability. Predicate calculus is the first-order theory with the empty set of non-logical postulates. We consider equality axioms as non-logical postulates.
We can extend the language of a theory by definitions of new predicate symbols, which have the following form:
Vxi.. .Xn(P (xi,... ,Xn) = A).
The definition must satisfy the conditions:
1. P / £.
2. A e L(£).
3. The variables x1,...,xn are pairwise distinct.
4. The set of free variables of A is included in {x1,..., xn}.
The newly defined predicate symbol P must be added to the signature £. As the result, there is a transition from the language L(£) to the language L(£ 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
ÀU nx1,
= Px1 V -Px1).
The definition allows us to prove DU h Vxi... xnUx1,..., xn.
This example is interesting because in the right part of the definition we use an arbitrary predicate symbol of the signature of the first order predicate calculus, but with the help of it, we define the specific predicate symbol with the specific properties.
As another example, we can give a definition of a symmetric relation. Let B be an arbitrary predicate symbol of the signature. We accept the following definition:
(DS1) Vxy(S1xy = Vuv(Buv D Bvu) D Bxy)
Let us show that DS1 h Vxy(S1xy D S1yx).
1. S1xy
2. yuv(Buv D Bvu) D Bxy
3. Vuv(Buv D Bvu)
4. Bxy
5. Bxy D Byx
6. Byx
7. Vuv(Buv D Bvu) D Byx
8. S1yx
9. S1xy D S1 yx
- hyp
- from 1, DS1 by replacement
- hyp
- from 2, 3 by m.p.
- from 3 by yei
- from 4, 5 by m.p.
- from 3-6 by Din
- from 7, DS1 by replacement
- from 1-8 by Din
There is another way to define a symmetric relation: (DS2) S2xy = Vuv(Buv D Bvu)&Bxy. Let us show that DS2 h Vxy(S2xy D S2yx).
1. S2xy
2. yuv(Buv D Bvu)&Bxy
3. Vuv(Buv D Bvu)
4. Bxy
5. Bxy D Byx
6. Byx
7. Vuv(Buv D Bvu)&Byx
8. S2yx
9. S2xy D S2yx
- hyp
- from 1, DS2 by replacement
- from 2 by &ej
- from 2 by &ei
- from 3 by yei
- from 4, 5 by m.p.
- from 3, 6 by &in
- from 7, DS2 by replacement
- from 1-8 by D in
x
x
n
These examples motivate us to find the general criterion of definability of the specific predicates with the help of predicate logic.
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 if and only if there are a signature £' and a set of definitions DT of symbols £ \ £' by formulas of L(£') which met the following condition:
If Be L(£), then Ax h B & DT h B.
This definition is some variant of the notion of definitional embeddability of theories, which was proposed by V.A. Smirnov in [2], [3, p. 65].
3. Auxiliary lemmas
To formulate the main theorem, we need to define function n, which translates formulas of first-order theories into formulas of the propositional logic. This function simply "erases" all terms and quantifiers in formulas.
Definition 2.
1. n(P(ti,...,tn)) = P.
2. n(-A) = —n(A).
3. n(A ▽ B) = n(A) ▽ n(B), where ▽ e {&, V, D, =}.
4. n(£xA) = n(A), where £ e {V, 3}.
Lemma 1. Let v be some truth-value assignment to propositional variables that is in the standard way extended to all formulas of propositional logic, then the next statements are true:
(A) If for each atomic subformula Pi(t) of formula A holds Vg[M,g N Pi(t-) & v(n(Pi)) = True], then it holds Vg[M,g N A & v(n(A)) = True].
(B) If for each atomic subformula Pi(t) of formula A holds Vg[M,g N Pi(t) & v(n(Pi)) = True], then it holds [M N A & v(n(A)) = True].
Proof.
(A) We prove the statement by structural induction. The basis of induction is the condition of the lemma Vg[M,g \ Pi(t) & v(n(Pi)) = True]. So we have to prove the induction step.
Case 1. A = -B
1. M,g \ -B -hyp
2. Vh[M, h \ B & v(n(B)) = True] - inductive hyp
3. M,g ¥ B
4. M,g \ B & v(n(B)) = True
5. v(n(B)) = False
6. v(-n(B)) = True
7. v(n(-B)) = True
1. v(n(-B)) = True
2. Vh[M, h \ B & v(n(B)) = True]
3. v(-n(B)) = True
4. v(n(B)) = False
5. M,g \ B & v(n(B)) = True
6. M,g ¥ B
7. M,g \ -B
Case 2. A = B&C
1. M, g \ B&C
2. Vh[M, h \ B & v(n(B)) = True]
3. Vh[M, h \ C & v(n(C)) = True]
4. M, g \ B
5. M, g \ C
6. M,g \ B & v(n(B)) = True
7. M,g \ C & v(n(C)) = True
8. v(n(B)) = True
9. v(n(C)) = True
10. v(n(B)&n(C)) = True
11. v(n(B&C)) = True
1. v(n(B&C)) = True
2. Vh[M, h \ B & v(n(B)) = True]
3. Vh[M, h \ C & v(n(C)) = True]
- from 1 by definition
- from 2
- from 3, 4 and definition v
- from 5 by definition v
- from 6 by definition n
- hyp
- inductive hyp
- from 1 by definition n
- from 3 by definition v
- from 2
- from 4, 5
- from 6 by definition
hyp
inductive hyp inductive hyp from 1 by definition from 1 by definition from 2 from 3 from 4, 6 from 5, 7
from 8, 9 by definition v from 10 by definition n
- hyp
- inductive hyp
- inductive hyp
4. v(n(B)&n(C)) = True
5. v(n(B)) = True
6. v(n(C)) = True
7. M, g = B & v(n(B)) = True
8. M,g = C & v(n(C)) = True
9. M, g = B
10. M,g = C
11. M,g = B&C
Case 3. A = /xB
1. M,g = /xB
2. /h[M, h = B & v(n(B)) = True]
3. M,g' = B
4. M,g' = B & v(n(B)) = True
5. v(n(B)) = True
6. v(n(yxB)) = True
1. v(n(yxB)) = True
2. \/h[M, h = B & v(n(B)) = True]
3. v(n(B)) = True
4. M,g ¥ /xB
5. M, g' ¥ B
6. M,g' = B & v(n(B)) = True
7. M, g' = B
8. contradiction
9. M,g = /xB
- from 1 by definition n
- from 4 by definition v
- from 4 by definition v
- from 2
- from 3
- from 5, 7
- from 6, 8
- from 9, 10 by definition
- hyp
- inductive hyp
- from 1 for arbitrary g' g
- from 2
- from 3, 4
- from 5 by definition n
- hyp
- inductive hyp
- from 1 by definition n
- hyp
- from 4 for some g' g
- from 2
- from 3, 6 -5,7
- from 4, 8
Since all logical connectives and the existential quantifier are definable through {-i, V} , the part (A) of the lemma is proved.
(B) The metalanguage statement Vg[M,g N A & v(n(A)) = True] implies the statement [Vg(M,g N A) & v(n(A)) = True], but Vg(M,g N A), it means the same as M N A. So part (B) of the lemma follows trivially from the part (A).
□
If Ax is the set of formulas then n(Ax) will denote the set of formulas {n(A)\A € Ax}.
Lemma 2. If T is a theory with a set of axioms Ax then the set of formulas n(Ax) is consistent if and only if for every set D there exists such a function of interpretation I, that M = (D, I} and for each A € Ax holds M N A.
Proof.
Suppose, n(Ax) is consistent. It follows that there is the truth-value assignment v to propositional variables, at which all the formulas n(Ax) are true.
Suppose that D is a non-empty set of individuals. We define the function of interpretation I of nonlogical language symbols in the set D. Let us choose an element e of the set D.
(1) If c — individual constant then I(c) = e.
(2) If f is n-ary function symbol then I(f) : D x .. .n x D — {e}.
(3) For any n-ary predicate symbol Pi, if v(n(Pi)) = True then I (Pi) = D x ...n x D, else I(Pi) = 0.
Let us show that in the model M = (D, I} holds M N Ax.
According to the constructed model, Vg[M,g N Pi(t) & v(n(Pi))]. From the Lemma 1 we obtain M N A & v(n(A)). Because for all A € Ax holds v(n(A)) = True, so we have M N A.
(^) The proof is trivial, since the consistency of n(Ax) follows from the existence of a one-element model M = ({a} ,I}. □
4. The main theorem
The following theorem is a stronger form of the theorem proved in [1].
Theorem 1. Let T be a first-order theory in a language L(T,) with a finite set of closed non-logical postulates Ax = {A1,..., Ak}.
(A) T is definitionally embeddable into the first-order predicate calculus if and only if the set of formulas {n(A1),... ,n(Ak)} is logically consistent.
(B) T is definitionally embeddable into the first-order predicate calculus if and only if it does not impose any restrictions on the power of models.
Proof.
(A) We must prove that if the set of formulas {n(A1),..., n(Ak)} is logically consistent then the theory T is definitionally embeddable into the first order predicate calculus.
Let {P1,..., Pm} be the set of all predicate symbols of signature £, which occur in nonlogical postulates {A1,..., Ak}.
The logical consistency of {n(A1),...,n(Ak)} means that there exists at least one truth-value assignment v to propositional letters n(P1),... ,n(Pm) with property v(n(P1)) = True,... ,v(n(Pm)) = True. Let us fix some such assignment v.
Take the signature £' which satisfes the two conditions:
• £ \ £' = {P1,...,Pm}.
• For each predicate symbol Pi e {P1,..., Pm} there exists such a predicate symbol Ri of the corresponding arity, that Ri e £'.
We use Ax to denote the conjunction A1&... &Ak of all postulates A1,... ,Ak and Ax [R/P] to denote the result of renaming all occurrences of symbols P1,..., Pm into R1, ... , Rm.
We associate the definition with each predicate symbol Pi {P1,..., Pm} by the following rule:
1) If v(n(Pi)) = True, then
Vx(Pi(x) = Ax [R/P] D Ri(x))
2) If v(n(Pi)) = False, then
Vx(Pi(x) = Ax [R/P] &Ri(x))
Let DT = {D1,..., Dm} be the set of all definitions.
(A.1) We must show that if Be L(£) and Ax h B, then DT h B. By the properties of the deducibility relation it suffices to show DT h Ax. By the completeness theorem of the first-order predicate calculus it is equivalent to DT N Ax.
Let M = (D, I} be a model in which all formulas of DT are true.
Since the formula Ax [R/P] is closed we have either M N Ax [R/P] or M N -Ax [R/P].
Case 1. M N Ax [R/P]. For each Pi we have one of the following two subcases:
Subcase 1.1. v(n(Pi)) = True
M,g N Pi(t) &
M,g N Ax [R/P] D Ri(t) &
M,g N Ri(t)
Subcase 1.2. v(n(Pi)) = False
M,g N Pi(t) &
M,g N Ax [R/P] &Ri(t) &
M,g N Ri(t)
In each case Pi is interpreted as Ri and therefore M N Ax.
Case 2. M N -Ax [R/P]. For each Pi we have one of the following two subcases:
Subcase 2.1. v(n(Pi)) = True
M,g N Pi(T) &
M,g N Ax [R/P] &Ri(t) &
M,g N Ax [R/P] &Ri(t) V -Ax [R/P] &(Ri(t) V -Ri(t)) & M,g N -Ax [R/P] &(Ri(t) V -Ri(t)) & M,g N Ri(t) V-Ri(t) & v(n(Pi))
Subcase 2.2. v(n(Pi)) = False
M,g N Pi(f) &
M,g N Ax [R/P] &Ri(t) &
M,g N Ax [R/P] &Ri(t) V -Ax [R/P] &(Ri(t)&-Ri(t)) & M,g N -Ax [R/P] &(Ri(t)&-Ri(t)) & M,g N Ri(t)&-Ri(t) & v(n(Pi))
For all atomic formulas Pi (t) and all assignments g to individual variables we have M,g N Pi(t) & v(n(Pi)). The value of the atomic formula Pi(t) doesn't depend on the particular assignments of values to individual variables. As a result, according to Lemma 1, we obtain M N Ax & v(n(Ax)). But according to the properties of the function v it holds v(n(A1)) = True,... ,v(n(Ak)) = True, and Ax is the conjunction of A1,..., Ak. Hence v(n(Ax)) = True and M N Ax.
With the help of the completeness theorem of the first-order predicate calculus, we obtain DT h Ax.
(A.2) We must show that if B € L(E) and DT h B, then Ax h B. By the completeness theorem of the first-order predicate calculus it is equivalent to show that if DT N B, then Ax N B.
Let us assume that B e L(£) and DT N B but Ax ¥ B. Then there exists such a model M = (D, I} of the theory T that M N Ax and M ¥ B.
We can extend the model M = (D, I} to the model M' = (D, I'} in which all the formulas of DT will be true. It is sufficient to expand the domain of the function I so that the new function of interpretation I' ascribed value I'(Ri) = I(Pi) to a predicate symbol Ri, and for all other functional and predicate symbols retained the same values as I.
Since M N Ax, then in the model M' = (D, I'} by definition of I' we will have M' N Ax [R/P], and hence, M' N Pi(x) = Ax [R/P] &Ri((x)) for each Ri. It follows that all the formulas DT are true in the model M'. Therefore by our assumption DT N B it must be M' N B. However, the formula B doesn't contain symbols R1,...,Rm, while all the other descriptive symbols are interpreted in the same way as in the model M, and by assumption it must be M', g ¥ B. We have obtained a contradiction. Therefore, the assumption that Ax N B does not hold is false.
(A) We must prove that if a theory T is definitionally embeddable into first-order predicate calculus, then the set of formulas {n(A1),..., n(Ak)} is consistent.
Let us assume that Ax h B & DT h B.
Take an arbitrary one-element model M = ({a} ,I} for signature £'. For each predicate symbol Pi £ \ £ , if it was introduced by definition Pi(x1,..., xn) = D, we expand the domain of the interpretation function I as follows:
I'(Pi) = {(g(x1),...,g(xn)} : M,g N D}.
Note that since the domain of individuals consists of only one element, the function assigning values to individual variables, too, is the only one, and, consequently, predicate symbol Pi will be interpreted as either empty set 0, or singleton {(a,...., a}}.
Performing this operation with all the new predicate symbols, we obtain the model M' = ({a} ,I'}, in which all the definitions of the set DT will be true.
Since we assumed that Ax h B & DT h B, then every axiom Ai e {A1,..., Ak} is derivable from DT. With the help of the completeness theorem of first-order predicate calculus, we obtain DT N Ai. It means that there is at least one one-element model of the theory T, and hence, the set {n(A1),... ,n(Ak)} is logically consistent.
(B) The second part of the theorem follows from the part (A) and Lemma 2. □
5. Conclusion
The main theorem of this article can be considered as a solution of the classical logicism program for first-order theories. Those and only those theories which don't impose any restrictions on the power of their models can be reduced to pure logic.
Among of such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others.
References
[1] Shalack, V.I. "On Some Applied First-Order Theories which Can be Represented by Definitions", Bulletin of the Section of Logic, 44(1-2) (2015). pp. 19-24.
[2] Smirnov, V.A. "Logical Relations between Theories", Synthese, 1986, 66(1), pp. 71-87.
[3] Smirnov, V.A. Logicheskiye metody analiza nauchnogo znaniya [The logical methods of analysis of scientific knowledge], Moscow: Nauka, 1987. 256 pp. (In Russian)
