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22Логические исследования 2017. Т. 23. № 1. С. 83-104 УДК 510.662
Logical Investigations 2017, Vol. 23, No. 1, pp. 83-104 DOI: 10.21146/2074-1472-2017-23-1-83-104
A Precise Definition of an Inference (by the Example of Natural Deduction Systems for
Shangin Vasilyi Olegovich
Department of Logic, Faculty of Philosophy, Lomonosov Moscow State University. Lomonosovsky prospect, 27-4, GSP-1, Moscow, 119991, Russian Federation. E-mail: [email protected]
In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics I(a,p) ,a, 3 £ {0,1, 2, 3,... w}, such that I(o,o) is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbertstyle calculus HI(a,p) ,a,3 £ {0,1, 2, 3,... w}, for each logic in question, in order to construct examples of effects of Glivenko theorem's generalization. Here we, first, propose a subordinated natural deduction system NI(a,p), a, 3 £ {0,1, 2, 3,... w}, for each logic in question, with a precise definition of a N1^,^)-inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each a, 3 £ {0,1, 2, 3,... w}, a Hilbert-style calculus HI(a,p) and a natural deduction system NI(a,p) are equipollent, that is, a formula A is provable in HI(a,p) iff A is provable in NI(a,p).
Keywords: precise definition of inference, indirect rule, implication introduction rule, natural deduction, quasi-elemental formula, subordinated sequence
Introduction
In [10], V. Popov presents logics I(a>^) and Hilbert-style calculi HI{a>p),a,P € 0,1,2,3,...w, for these logics, such that is
propositional classical logic. He uses them in order to construct examples
xThe author is supported by Russian Foundation for Humanities, grant 16-0300749 "Logical-epistemic problems of knowledge representation".
Logics I{ae)
1
© Shangin V.O.
of effects of a generalization of Glivenko theorem. So, the purpose of the present paper is to present, within the framework of [8, 9], a subordinated natural deduction (abbreviated passim as 'ND') calculus NI(a,p), for each logic in question, with the precise definition of an N1^p)-inference, following the works of V. Smirnov [12, 14]. We, also, show the equipollentness between a Hilbert-style calculus HI(a,p) and a ND system NI^p), for each a, / € {0,1,2,3,...w}, that is, a formula A is provable in HI^a,p) iff A is provable in NI^a,p).
Following [10], we fix a standard propositional language L over an alphabet {p,p\,p2, ■ ■ ■, (,), &, v, d, -}. A notion of a formula of language L is defined as usual. (Passim by 'a formula' we mean 'a formula of language L'.) A formula is said to be quasi-elemental iff no logical connective &, v, d occurs in it ([10]). A length of a formula A is said to be the number of all occurrences of the logical connectives in L in A. Letters A, B, C, D, E with lower indexes run over arbitrary formulae. Letters r, A with upper and lower indexes run over arbitrary finite sets of formulae. Letters a and / run over {0,1,2,3,... w} passim.
In [10], V. Popov presents a Hilbert-style calculus HI^a,p). The language of the calculus is the language L mentioned above. We follow (and, for more details, refer the reader to) [10] in describing a Hilbertstyle calculus HI a p p). A formula is an axiom of HI^a, p) iff it is one of the following forms: (I) (A d B) d ((B d C) d (A d C)), (II) A d (A v B), (III) B d (A V B), (IV) (A D C) d ((B d C) d ((A v B) d C)), (V) (A&B) d A, (VI) (A&B) d B, (VII) (C d A) d ((C d B) d (C d (A&B))), (VIII) (A D (B d C)) d ((A&B) d C), (IX) ((A&B) d C) d (A D (B d C)), (X) (((A d B) d A) d A), (XI,a) -D d (D d A), where D is a formula which is not a quasi-elemental formula of a length less than a, (XII,/) (E d -(A d A)) D -E, where E is a formula which is not a quasi-elemental formula of a length less than /. Modus ponens is the only inference rule of the calculus.
Definitions of an inference in HI^a,p) (abbreviated as HI^a,p)-inference) and a proof in HI^a,p) are given in the standard way for a Hilbert-style calculus. Notions of the length of an inference and the length of a proof as well as the notion of a theorem are defined as usual.
In [10], the following fact is particularly highlighted: I^q^) is propositional classical logic, where I(0,0) is the set of formulae provable in HI(p>0). This fact implies both schemata A d (B D A) and (A d (B d C)) d ((A d B) d (A d C)) are theorems of HI^0,0) and, therefore, of each Hilbert-style calculus HI(a,p),a, / € {0,1, 2, 3,... w}. So, we, non-constructively, point out the standard deduction theorem holds for each calculus in question.
The paper is organized as follows. Section 1 presents a ND system NI{a>@) with both precise and traditional definitions of an NI(a,p)-inference. In Section 2, the Hilbert-style calculus NI^a,@) and the ND system NIa, ^ are shown to be equipollent. The final section concludes the work and outlines the future research.
1. ND systems NI{a,p)
Let us set up a subordinated ND system NI(a,p) and give a precise definition of a NI^a,^-inference. The language of the system is, again, the language L mentioned above. There are two kinds of rules in the system. Here is the list of the rules of the first kind (sometimes called direct). The rules of the second kind (sometimes called indirect) are defined with the precise definition of an inference below.
A&B A&B A,B
A B A&B
A B A d B,A
A r. Vin1 ^ Del
A v B A v B B
D, -D
—A— -in1(a), where D is a formula which is not a quasi-elemental formula of a length less than a.
The necessity of a precise definition of NI(a,^-inference is illustrated with V. Smirnov's thesis: "... By natural deduction systems we shall refer to logistic systems with a special notion of an inference. In these systems, an inference is more complex object than just a sequence of formulae or a tree-like of formulae. Due to this property of natural deduction systems, a definite object entitled a formal inference corresponds to both direct
and indirect ways of argument" [12, p. 96, both the translation and the italics are ours].
In defining both a NI a, ^-inference h its length, we, with modifications, follow V. Smirnov [12, p. 116-118], [14, p. 245]. Letters n h i with indexes denote -inferences, a letter 7 with indexes
denotes parts of NI^a^-inferences, and a letter h denotes the length of an inference.2
An precise definition of NI (a,p)-inference and a definition a height of NI a , ^-inference
1. A is an inference 1 of A from a set of premises {A}, and h(i) = 1.
2. If n is an inference from r and A is a formula, then A is an inference 1 of A from {A} U r, and h(i) = h(n) + 1.
3. If n is an inference from r, n contains A1r.., Ak (k = 1,2) and B is inferred from A1,... ,Ak via one of the rules &el1, &el2, &in, vin1, Vin2, del and —in1(a), then B is an inference 1 of B from r, and h(i) = h(n) + 1.
4. If n is an inference of B from {A} U r and n is Y1, where 71 is a part of n, starting from the last premise A in n until B itself,3 then
Y
Iyi is an inference 1 of A d B from r, and h(i) = h(n) + 1.
adb
5. If n is an inference of B from {A} U r and n is 71, where B is
C, A is C d D, y1 is a part of n, starting from the last premise
7
C d D in n until C itself, then Iyi is an inference 1 of C from r,
C
and h(C) = h(n) + 1.
6. If n is an inference of B from {A} U r and n is Y1, where B is —(A d A), A is E, where E is a formula which is not a quasi-elemental formula of a length less than /3 and y1 is a part of n,
2In every case, the precise definition specifies which part of a N/^^)-inference is under consideration. The reason we introduce a special letter to run over parts of inference is that, in general, as we will see, a part of an inference is not an inference.
3It is the last occurrence of B in n that is under consideration. In what follows, we will omit this specification everywhere, except clause 7.
starting from the last premise E in n until —(A d A) itself, then
y
\n is an inference i of —E from r, and h(i) = h(n) + 1.
-E
y
7. If n is an inference of B from {D1} U {D2} U r and n is , where
B is C and 7 contains D1 v D2, 71 is a part of n, starting from
the last premise D1 in n until C, 72 is a part of n, is a part of n,
y
starting from the last premise D2 in n until C itself, then Y is an
C
inference i of C from r, and h(i) = h(n) + 1.4
The core of modifications is as follows. An essential modification deals with V. Smirnov's suggestion that any discarded part of a NI^a, inference is a NI a, ^-inference. (A discarded part of an inference is marked with a horizontal line from the left.) This is not the case if a part of a NI a, ^-inference contains a formula that is not a premise and is inferred from the formulae which this part of a NI a, ^-inference does not contain. For example, in the clause 4, a part 71 may contain a formula that is inferred from some formula contained in a part 7 (and 7 may be an inference, itself). So, 71 is not an inference while YJ1 is. Sometimes, V. Smirnov applies a notion of an auxiliary inference (or a subderivation) to such sequences of formulae as 71. The name of this notion obviously reflects the idea that such an inference plays a secondary role, and can be considered only with respect to the 'key' inference. However, we can't find it satisfactory that an auxiliary inference is shown not to be a kind of an inference. At last, minor modifications deal with evaluating a height of NI(a>p)-inference in dei and vei rules as well as with evaluating the height of a NI a, ^-inference that now cannot be equal to 0.
Clause 4 (5, 6, and 7, respectfully) of the above definition is a formulation of an indirect rule of din(dp,—in2(p), and vei, respectfully). We pay attention (and exemplify it below) to the fact that clauses 5-7 are special cases of clause 4. (In case of clause 7, the situation is a little bit
4Clause 7 may have alternative formulations: 1. 73 occurs between 71 and 72; 2. 71 reorders with y2; 3. D1 V D2 occurs below a part y1 etc. This analysis goes beyond the scope of the paper.
more complex than in the other cases because it allows simultaneously discarding two parts of an inference, not one part. It is the reason why we choose clause 7 in the example below.) By the fact that a rule, say, dp, is a special case of a rule din we mean that if one can apply dp in the inference then one can apply din, either, but not vice versa. To be sure, we don't mean dp is derivable via din.
There different notation formats for a subordinated inference in ND systems [12, p. 119-126]. We will use so called Jaskowski-Quine notation in [2].5
Let us consider the following sequence of formulae:
1. A d C — premise.
2. B d C — premise.
3. A v B — premise.
4. A — premise.
5. C — Dei: 1, 4
6. B — premise.
7. C — Dei: 2, 6
In accordance to clause 4, we have an inference of C from premises A d C, B d C, A v B, A, and B. Thus, we are legitimate to proceed with an inference of B d C from premises A d C, B d C, A v B, and A:
1. A d C — premise.
2. B d C — premise.
3. A v B — premise.
4. A — premise.
5. C — Dei: 1, 4
|6. B — premise. |7. C — Dei: 2, 6
8. B d C — din: 7
5In the literature, a subordinated inference is sometimes called a linear-type ND or a Fitch-style ND [11]. A subordinated inference differs from a tree-like inference presented by G. Gentzen [5], where, roughly, no formula is used more than once in the inference as a premise.
On the other hand, in accordance to clause 7, an inference of C from premises A d C, B d C, A v B, A, and B contains a part, starting from the last premise A until C (steps 4-5), and a part, starting from the last premise B until C (steps 6-7), as well as it contains A V B which contains in no parts mentioned above. Thus, we are legitimate to proceed with an inference of C from premises A d C, B d C, and A v B:
A d C — premise. B d C — premise. A v B — premise. A — premise. C — Dei: 1, 4 B — premise. C — Dei: 2, 6
C v in: 7
8
As a result, we see the complexity of a notion of an inference in ND systems leads to the fact that a sequence of formulae turns out to be different inferences of the same formula from different set of premises. Discussing this fact (which is impossible for the other conventional proof systems like Hilbert-style calculus, sequent-style calculus and tree-like ND system) and its consequences is not a topic of the paper. We are fully aware, however, that the fact that a precise definition of an inference leads to some ambivalence seems to be absurd. But we strongly believe that the reason of this fact is caused by the nature of indirect argument, itself, which have been being under suspicion in the development of logic.6
On the other hand, the difference between direct and indirect rules has become more evident. A direct rule is applicable provided an inference contains formula (formulae) which is (are) above the line in a formulation of this some rule. One can apply a direct rule to any formula; it is not necessary for the formula to be the last one in this inference. For example, in applying &ei1 or some other direct rule, A&B (the one
6It is well-known that intuitionists have been criticizing the general version of reductio ad absurdum, a type of indirect argument.
that is above the line) is not necessary the last formula of the inference, i.e., it is not necessary that this inference is an inference of A&B from (possibly, empty) r.
The situation is not the same in case of indirect rules. An indirect rule is applicable, too, provided there is an inference of the formula which is above the line in the formulation of this rule. The crucial difference is that it applies to the last formula in an inference only. (Note, at any moment, there is only one formula that is the last one in an inference.) For example, in applying din, a formula B (the one that is above the line) is, necessarily, the last one in an inference, i.e., it is a must that there is an inference of B from a non-empty set of premises r, where A is a member of r. In the example above, we see that one has different options in applying indirect rules in the same way one has different options in applying direct rules. However, it is impossible for a sequence of formulae to be an inference of two formulae: such a possibility is allowed by so called traditional formulation of some indirect rules (in the next subsection it holds for the traditional formulation of vei). Sometimes negation introduction rule is formulated, roughly, as follows: if there is an inference of both formulae A and —A from the last premise C then there is an inference of a formula —C [1, p. 140], or 'to be applied, an indirect rule —in requires two auxiliary inferences r, A I B and r, A I—B' [13, p. 66]. In general, one can't determine another inference (so called 'auxiliary inference') 'inside' a given inference. Let us, again, say that discussing it goes beyond the scope of this paper.
In the end of this section, let us present so called the traditional formulations of both indirect rules and of an inference.7 In the rules below, a formula A (A d B or E) is the last premise. In —in2(p), a formula E is, additionally, a formula which is not a quasi-elemental formula of a length less than /. In this subsection, by 'inference' we mean 'NI(a>py inference'.
7For the sake of simplicity and without loss of generality, we don't present a traditional formulation of Vel and refer the reader to, for example, [6]. Note, sometimes, the traditional formulation of the indirect rules includes the derivability symbol 'h' [1].
[A]B [A D B]A [E]—(B d B)
A d B din A dp —E —in2(f)
An inference is said to be a non-empty finite linearly ordered sequence of formulae C1, C2,..., Ck, satisfying the following conditions:8
• Each Ci is either a premise or is inferred from the previous formulae via a rule;
• In applying din, each formula, starting from the last premise A until A d B, the result of this application, exclusively, is discarded from an inference;
• In applying dp, each formula, starting from the last premise A d B until A, the result of this application, exclusively, is discarded from an inference;
• In applying —in2(f3), each formula, starting from the last premise E until —E, the result of this application, exclusively, is discarded from an inference.
Given an inference C1,C2,... ,Ck with A1,A2,... ,An being non-discarded premises and with the last formula Ck being graphically identical to B, we say this is an inference of B from premises A1,A2,..., An. If a set of formulae r contains A1,A2,...,An and there is an inference of B from premises A1, A2, . . . , An then we say there is an inference of B from a set of formulae r [2, p. 129-130].
2. Metatheory of a ND system
We proof the following
Theorem 1. r hm(a,fi) A r hni(a,fi) A, for each a,P € {0,1,2,3,... u}.
8Here is (of course, incomplete) a list of (text)books reproducing the traditional formulation one way or another: [1]-[4], [6], [15], [16]. On the other hand, we are fully aware that textbooks' authors are, mostly, driven by pedagogy trying to 'not go deep into theoretical subtleties of all kinds' and following the principle 'to tell the truth and only the truth, but not all the truth' [2, p. 11, 12].
Proof Proof is by the method of complete induction on a height s of an arbitrary HI^a^-inference of A from r.9
The scheme of complete induction is as follows: (P(1)&vx(vy((y < x) d P(y)) d P(x))) d vxP(x).
Let P(s) denote a sentence "if there is a HI^a,^-inference of a height s of A from r then there is a NI^a,p)-inference of A from r".
Then the scheme looks as follows: ((if there is a HI^a,p)-inference of a height 1 of A from r then there is a NI^a,p)-inference of A from r) &vs(vt((t < s) d (if there is a HI^a,p)-inference of a height t of A from r then there is a NI(a,p)-inference of A from r)) D (if there is a HI^^y inference of a height s of A from r then there is a NI^a,p)-inference of A from r))) D Vs(if there is a HI^a, p)-inference of a height s of A from r then there is a NI^a,p)-inference of A from r).
The base case is trivial according to the definitions of inferences in both HI{app) and NI{a, p).
We prove the inductive step: vs(vt((t < s) d (if there is a HI^a,p)-inference of a height t of A from r then there is a NI^№pp)-inference of A from r)) D (if there is a HI^app)-inference of a height s of A from r then there is a NI^a,p)-inference of A from r)).
For modus ponens is an inference rule in both HI^a,p) and NI^a,p), it is enough to show that every HI^a,p)-axiom is provable in NI^a,p). We confine ourselves to proving two specific HI(ap)-axioms: axiom (XI,a) —D d (D d A), where D is a formula which is not a quasi-elemental formula of a length less than a, and axiom (XII,/) (E d —(A d A)) d —E, where E is a formula which is not a quasi-elemental formula of a length less than / .
ini^ —D d (D d A) |1.—D — premise ||2. D — premise
||3. A —in1(a): 1 2
|4. D d A — Din: 3 5.— D d (D d A) — Din: 4
9We recall the standard definition of a length of an inference in a Hilbert-style calculus.
\-m<a, « (E D —(A D A)) D —E |1. E d —(A D A) — premise ||2. E — premise 113.—(A D A) — Dei: 1, 2
|4. —in2(f): 3
5. (E D —(A D A)) d—E — Din: 4
Proof <<. Proof is by the method of complete induction on a height n of an arbitrary NI^a,f)-inference of A from r.
The scheme of complete induction is as follows: (Q(1)&Vx(Vy((y < x) d Q(y)) d Q(x))) d VxQ(x).
Let Q(n) denote a sentence "if there is a NI^a, f)-inference of a height n of A from r then there is a HI^a, f)-inference of A from r".
Then the scheme looks as follows: ((if there is a NI(a,p)-inference of a height 1 of A from r then there is a HI(a,p)-inference of A from r) &vn(vq((q < n) d (if there is a NI^a, f)-inference of a height q of A from r then there is a HI a, f)-inference of A from r)) D (if there is a NI(a, f)-inference of a height n of A from r then there is a HI^a, f)-inference of A from r)))D vn(ifthereisaNI(af)-inference of a height n of A from r then there is a HI(a,p)-inference of A from r).
The base case: h(n) = 1. According to clause 1 of the definition of a NI(aff)-inference, a NI^a,f)-inference n of a height 1 of A from a set of premises r looks as follows: A is an inference from {A}:
1. A — premise.
This inference corresponds to the following HI^a, f)-inference of A from a set of premises {A}:
1. A — premise.
We prove the inductive step: vn(vq((q < n) d (if there is a NI(a>p)-inference of a height q of A from r then there is a HI a , f)-inference of A from r))D(if there is a NI^a,f)-inference of a height n of A from r then there is a HI^a,f)-inference of A from r)). According to clauses 2-7 of the definition of a NI^a>f)-inference, a NI^^-inference n of a height n of A from a set of premises r looks as one of the six following cases:
Case 1 (2nd clause of the definition of a NI^a, p)-inference): A, where n' is an inference from a set of premises r' and r is {A} U r'.
r'
n. A — premise.
For h(n') < h(n),w one can, by the inductive hypothesis, build up a HI(a,p)-inference from a set of premises r'. Then a HI^a p)-inference of A from a set of premises r looks as follows:
r'
n'. A — premise.
Case 2 (3rd clause of the definition of a NI^a,p)-inference): a ', where n' is an inference of C from a set of premises r', n' contains A1,... ,Ak; A is inferred from A1,... ,Ak via one of the rules
&el1, &el2, &in, vinl, ^in2, del, and —
inl(a).
Subcase 2.1.: n' contains —D and D; A is inferred from —D and D via —in1(a), where j < n — 1 and m <n — 1.
r
j .—D m. D n — 1. C
n. A ——in1(a): j,m
Let n' be an NI^a^-inference of C from r, n1 be an NI^a,p)-inference of D from r, and n2 be an NI^a,p)-inference of D from r, where h(n[) < h(n') and h(n'2) < h(n'), by the definition. The fact that h(n')<h(n), implies that h(n1) < h(n) and h(n'2) < (n), and, by the inductive hypothesis, one can build up the following HI^a p)-inferences:
0By the definition, hfy = h(n') + 1-
a HI a f f)-inference of D from r, a HI a, f)-inference of D from r, and a HI {a , f)-inference of C from r. Then a HI a f f-inference of A from a set of premises r looks as follows:
r
j '.—D m'. D n' - 1. C
n'.—D d (D d A) — HI/a,f)-axiom (XI,a) n'+1. A — modus ponens: j',m',n' (two times)
Subcase 2.2.:n' contains A&B; A is inferred from A&B via &ei1, where m < n - 1.
r
m. A&B
n - 1. C
n. A — &ei1 : m
Let n' be an NI/a,f)-inference of C from r and n1 be an NI/a,f)-inference of A&B from r, where h(n1) < h(n'), by the definition. The fact that h(n') < h(n), implies that h(n1) < h(n) and, by the inductive hypothesis, one can build up the following HI/af)-inferences: a HI/aff)-inference of A&B from r, a HI/a,f)-inference of C from r. Then a HI/a,f)-inference of A from a set of premises r looks as follows:
r
m'. A&B n' - 1. C
n'. (A&B) D A — HI/a, f)-axiom (V)
n' + 1. A — modus ponens: m', n'
Subcase 2.3., where n' contains B&A; A is inferred from B&A via &ei2, is treated analogously to subcase 2.2.
Subcase 2.4.: n' contains B and D; A is B&D and is inferred from B and D via &in, where f < m, j < n - 1, and m < n - 1.
r'
j. B m. D n - 1. C
n. B&D — &in: j, m
Let n' be an NI/a^-inference of C from r, n' 1 be an NI/af)-inference of B from r, and n'2 be an NI/aff-inference of D from r, where h(n1) < h(n') and h(n2) < h(n'), by the definition. The fact that h(n') < h(n), implies that h(n1) < h(n) and h(n2) < h(n), and, by the inductive hypothesis, one can build up the following HI^af)-inferences: a HI/aff)-inference of B from r, a HI/af f)-inference of D from r, and a HI/aff)-inference of C from r. Then a HI/aff)-inference of B&D from r looks as follows:
r
f. Ai — any HI/a,f)-axiom
f' + 1. B d (A1 D B) — HI/a,f)-theorem
f' + 2. D d (A1 d D) — HI{af)-theorem
j .B
j' + 1. A1 d B — modus ponens: f' + 1, j' m . D
m' + 1. A1 d D — modus ponens: f' + 2,m'
n' — 1. C
n'. (A1 D B) d ((A1 D D) d (A1 D (B&D))) — HI{a p)-axiom (VII)
n' + 1. B&D — modus ponens: j + 1', m + 1', f', n' (three times)
Subcase 2.5.: n' contains B; A is B v D and is inferred from B via vin1, where m < n — 1.
r
m. B n — 1. C
n. B v D — Vin1: m
Let n' be an NI^a,p)-inference of C from r and n1 be an NI^a,p)-inference of B from r, where h(n[) < h(n'), by the definition. The fact that h(n') < h(n), implies that h(n[) < h(n) and, by the inductive hypothesis, one can build up the following HI^a p)-inferences: a HI^a,p)-inference of B from r, a HI a p p)-inference of C from r. Then a HI^a , p)-inference of B v D from a set of premises r looks as follows:
r
m . B n' — 1. C
n'. B d (B v D) — HI{app)-axiom (II) n' + 1. B v D — modus ponens: m', n'
Subcase 2.6., where n' contains D; A is B v D and is inferred from D via vin2, is treated analogously to subcase 2.5.
Subcase 2.7.: n' contains B d A and B; A is inferred from B d A and B via del, where j < n — 1, and m < n — 1.
r
j. B d A
m. B
n - 1. C
n. A — dei: j,m
Let n' be an NI/af)-inference of C from r, n1 be an NI/af)-inference of B d A from r, and n2 be an NI^a,f-inference of B from r, where h(n1) < h(n') and h(n2) < h(n'), by the definition. The fact that h(n') < h(n), implies that h(n1) < h(n) and h(n2) < h(n), and, by the inductive hypothesis, one can build up the following HI/af)-inferences: a HI/a,f)-inference of B d A from r, a HI/a,f-inference of B from r, and a HI/a,f)-inference of C from r. Then a HI/a, f)-inference of A from r looks as follows:
r
j'. B d A m . B n' - 1. C
n'. A — modus ponens: j', m'
Case 3 (4th clause of the definition of a NI/a,f)-inference). A is B d C and a NI/a,f)-inference n of a height n of B d C from r looks
Y
as follows: Iyi , where Yj1 is a NI/a,f)-inference i of C from {B} U r, bdc
Y1 is a part of i, starting from the last premise B in i until C, itself, and m < n - 1.
r
|m. B — premise
|n — 1. C
n. B d C — din■ n — 1
For h(i) < h(n), one can, by the inductive hypothesis,11 build up a HI(a p)-inference of C from {B} d r. Then a HI^a,p)-inference of B d C from r looks as follows:
r
m'. B — premise n' — 1. C
n'. B d C — deduction theorem: m',n' — 1
Case 4 (5th clause of the definition of a NI^a,p)-inference). A
y
NI(a^-inference n of a height n of A from r looks as follows: Iyi , where
Y is a NI(a,p)-inference i of A from {A d B} U r, y1 is a part of i, starting from the last premise A d B in i until A, itself, and m < n — 1.
r
| m. A d B — premise
|n — 1. A
n. A — dp■ n — 1
For h(i) < h(n), one can, by the inductive hypothesis, build up a HI(a,p)-inference of A from {A d B} U r. Then a HI^ap)-inference of A from r looks as follows:
r
m'. A d B — premise
11Here and in the cases below, we stress the fact that we proceed from one inference to another inference, not from a part of an inference to another inference. So, the inductive hypothesis of the theorem is applicable.
n' - 1. A
n'. (A D B) D A — deduction theorem: m', n' - 1 n' + 1. ((A D B) D A) D A — HI a f f)-axiom (X) n' + 2. A — modus ponens: n', n' + 1
Case 5 (6th clause of the definition of a NI^a,f)-inference). A is —E, where E is a formula which is not a quasi-elemental formula of a length less than /, and a NI^a,f)-inference n of a height n of —E from r looks as
Y
follows: Iyi , where Yj1 is a NI^a^-inference i of —(A d A) from {E}Ur,
—E
Y1 is a part of i, starting from the last premise E in i until —(A d A), itself, and m < n - 1.
r
|m. E — premise |n — 1. —(A D A)
n—E — —in2(f) : n - 1
For h(i) < h(n), one can, by the inductive hypothesis, build up a HI^a,f)-inference of —(A d A) from {E}U r. Then a HI^a, f-inference of A from r looks as follows:
r
m'. E — premise n' - 1. —(A D A)
n'. E d —(A d A) — deduction theorem: m',n' - 1 n' + 1. (E D —(A D A)) d—E — HIa f f)-axiom (XII,/) n + 2. A — modus ponens: n , n + 1
Case 6 (7th clause of the definition of a NI^a, ^-inference). a NI^a , f)-
Y YY
inference n of a height n of A from r looks as follows: |yi , where Y2 is a
|Y2 a
NI(a,f)-inference i of A from {D}U{B}Ur, y contains D vB, Y1 is a part
of n, starting from a premise D in n until A, j2 is a part of n, starting from the last premise B in n until A, itself, and f < g,g < j,j < n — 1.12
r
f. D v B | g. D — premise |j. A
|j + 1. B — premise |n - 1. A
n. A — vei ■ f,j,n — 1 First, let us consider a NI^a,p)-inference i of A from {D} U {B} U r.
r
f. D v B
g. D — premise j. A
j + 1. B — premise n — 1. A
By the construction, i contains the following NI^a p)-inferences: i1 of D v B from r, i2 of A from {D} U r, and i3 of A from {B} U {D} U r.
For h(ii) < h(i),13 for each i from {1,2,3}, one can build up a HI{a^-inference of DvB from r, a HI^p)-inference of A from {D}Ur,
12On alternatives of this case see the footnote to the 7th clause of the definition of an NI/a,^-inference.
13Unlike the other cases, this case requires the inductive hypothesis holds true for a Nl/a,,^)-inference of any length less than the length of l, not only for a NI/a>py inference of a length h(C) — 1.
and a HI(a,p)-inference of A from {B}U{D}Ur. Then a HI^a,p)-inference of A from {D} U {B} U r looks as follows:
r
f .DvB g'. D — premise j .A
j + 1. B — premise n' — 1. A
So, a HI(a,p)-inference of A from r looks as follows:
r
f .DvB g'. D — premise j .A
j + 1. B — premise n' — 1. A
n'. B d A — deduction theorem: j' + 1,n' — 1
r
f''. D v B
g . D — premise j .A
j + 1. D d A — deduction theorem: g , j
j'' + 2. (D d A) D ((B d A) D ((D v B) d A)) — HIa,p)-axiom (IV)
j'' + 3. A — Dei: f'',j'' + 1,j'' + 2 (three times)
The Theorem implies a Corollary: for each a,/ € {0,1, 2, 3,...w}, a Hilbert-style calculus HI a, f) and a ND system NI{a , f are equipollent, i.e., A is a HI{a,f)-theorem iff A is a NI{a,f)-theorem.
Final remarks
In the paper, for each logic, I{af), a,/ € {0,1,2,3,...w}, such that I(o , o) is propositional classical logic [10], we, continuing the series of works [8, 9], present a subordinated ND system NI{a,f). Moreover, each ND system has a precise definition of an inference which is a modification of V. Smirnov's approach. Our approach highlights a view on the implication introduction rule as the genus for the other indirect rules. Using a Hilbert-style calculus HI{a,f), for each logic in question, presented by V. Popov [10], we show that a formula A is provable in HI {a , f) iff it is provable in NI{a, f). In the future, we point out studying consequences of the precise definition with an application to complexity problems [7]. Last, not least, we look forward to formulating proof searching procedures for these ND systems in the fashion of [3, 4].
Corrections
The paper "Natural deduction in a paracomplete setting" by A. Bolotov and V. Shangin to have been published in this Journal's 20th volume needs two corrections. First, the 23rd entry in the references list should be replaced with "Popov V. and Solotschenkov A. Semantics of propositional paracomplete Nelson logic // Integrated scientific journal. V. 8. 2012. P. 31-32. (In Russian)". Second, the truth-table definitions for the connectives of logic PComp in the 2nd section must be added with the following footnote: A. Avron had told V. Popov about these definitions at the World Congress on Paraconsistency (Ghent, 1997) and then V. Popov told one of the paper's authors about these definitions.
Acknowledgments
The author thanks the referees for commenting the previous draft of the paper.
References
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