CC BY
8
Chelyabinsk Physical and Mathematical Journal. 2017. Vol. 2, iss. 3. P. 329-337.
УДК 510.644
CORRESPONDENCE ANALYSIS FOR LOGIC OF RATIONAL AGENT
Y. I. Petrukhin
Lomonosov Moscow State University, Moscow, Russia yaroslav.petrukhin@mail.ru
In this paper, we examine Kubyshkina & Zaitsev's Logic of Rational Agent (LRA) from a proof-theoretic point of view. We present three natural deduction systems for LRA which differ from Kubyshkina & Zaitsev's axiomatization of LRA. Moreover, we introduce a general method for axiomatizing LRA's unary and binary truth-functional extensions via natural deduction systems. This method is Kooi & Tamminga's correspondence analysis which we adapt for LRA.
Keywords: many-valued logics, generalized truth values, correspondence analisys, natural deduction systems.
Introduction
Kubyshkina & Zaitsev's Logic of Rational Agent (LRA) [1] is both one of so called logics of generalized classical truth values and one of epistemic logics. The generalization technique of classical truth values allows the authors "to introduce a system, where the epistemic operator for knowledge (ka-operator) does not appear, but the fact of knowing or not knowing some truths (or the falsity of some statement) can be defined truth-functionally" [1, p. 2]. The avoiding of the use of ka-operator allows to escape so called Church — Fitch's paradox (knowability paradox): p ^ kap, i.e. if it holds that p then agent a knows that p. This paradox arises in some epistemic systems. See [1] for the review of solutions of this paradox. Recall that one of solutions is reasoning with LRA which is based on generalized classical truth values.
In [2], Dunn suggested the idea of generalization of classical truth values. He considered the subsets of the set {T, F} of classical "true" and "false" as independent truth values. As a consequence, he obtained a very simple and intuitive four-valued semantics for FDE [3; 4] with the set {0, {T}, {F}, {t, f}} of truth values. Belnap [5; 6] interpreted these values as follows: 0 = "none", {T} = "true", {F} = "false", and {T,F} = "both". Later on the generalization technique of classical truth values have been developing by Shramko, Dunn & Takenaka [7], Shramko & Wansing [8], Zaitsev & Grigoriev [9; 10], Zaitsev & Shramko [11], Zaitsev [12], Grigoriev [13] and Wintein & Muskens [14].
In case of LRA, the generalization technique of classical truth values works as follows: consider the set {T, F} of truth values "true" and "false" and the set {1, 0} of truth values "known" and "unknown". These sets deal with ontological truth and falsehood and epistemic states of the agent, respectively. Then {T, F} and {1, 0} are multiplied. As a result, the set {T1, T0, F1, F0} of truth values arises. The set of designated values is {T1}.
The propositional formula of LRA (L-formula) is defined in a standard way from the propositional variables p, q, r,p\,..., unary operators —, ~ and binary operators Л, V.
A logical matrix for LRA is Mlra = <{T1, T0, F1, F0},/-,/„,/a,/v, {T1}). Functions /-,, /A, fv are defined as follows:
/- /a T1 T0 F1 F0 /v T1 T0 F1 F0
T1 F1 T0 T1 T1 T0 F1 F0 T1 T1 T1 T1 T1
T0 F0 T1 T0 T0 T0 F1 F0 T0 T1 T0 T0 T0
F1 T1 F0 F1 F1 F1 F1 F1 F1 T1 T0 F1 F0
F0 T0 F1 F0 F0 F0 F1 F0 F0 T1 T0 F0 F0
The notion of valuation of formula A in MLRA is defined in a standard way. Let r be an arbitrary set of L-formulas and A be an arbitrary L-formula. Then r =lra A iff for each valuation v if v(B) = T1 (for each B G r) then v(A) = T1. If r = 0 then A is said to be a tautology.
Note that {—,A,V}-fragment of LRA was first studied in Kubyshkina's abstract [15] and Zaitsev's paper [16]. Connectives — and respectively, are an ontological and epistemic negations, i.e. — changes 'T' and 'F' (ontological parts of truth values) and ~ changes '1' and '0' (epistemic parts of truth values).
The purpose of this paper is to present correspondence analysis for LRA. Kooi & Tamminga have first described this framework in their paper [17] where they considered unary and binary extensions of three-valued paraconsistent logic LP (Logic of Paradox) [18-20]. Later on Tamminga [21] applied this technique to strong three-valued logic K3 [22]. In [23], correspondence analysis was extended to four-valued relevant logic FDE [2-6].
Consider a logic LRA*0 with L*0-formulas constructed in a standard way from the propositional variables and unary operators —, *i, ... and binary operators V, A, °1, . . . ,
Let Mlra,0 = <{T1, T0, F1, F0}, // /a, fv, ,...,/*„, fox,..., fom, {T1}) be a logical matrix for LRA*0. The notion of valuation of formula A in MLRa+0 is defined in a standard way. Let r be an arbitrary set of L*0-formulas and A be an arbitrary L*0-formula. Then r =lra*0 A iff for each valuation v if v(B) = T1 (for each B G r) then v(A) = T1. If r = 0 then A is said to be a tautology.
So the first step of correspondence analysis is a characterization of all 16 possible equalities of the form *a = b and all 64 possible equalities of the form a ° b = c by inference schemes. On the second step we consider these inference schemes as inference rules. Using them and a natural deduction system for LRA, we define a class of natural deduction systems for LRA's extensions. On the third step we show soundness and completeness of these calculi with respect to their semantics.
1. Correspondence analysis for LRA
In this section, we introduce inference schemes for operators ° and *. If p is L*0-formula or p G {T1, T0, F1, F0}, then we introduce the notations
pT1 = p, pT0 = ~ p, pF1 = —p, pF0 = ~ —(1) By (1) for every a G {T1, T0, F1, F0}
a" = T1 (2)
and for every L*0-formula ^ and for every valuation v
v(p°) = T1 ^^ v(p) = a.
(3)
Theorem 1. Let a,b,c G {T1, T0, F1, F0}. Then
a o b = c ^^ A —b = o —)c for any L^-formulas —]. (4)
Proof. Let a o b = c. Given any valuation v suppose v(^a) = T1, v(—b) = T1. By (3) v(p) = a, v(-) = b. Hence v((p o -)c) = (a o b)c = cc. By (2) v((p o -)c) = T1.
Let A —b = (<£ o —)c for any L*0-formulas . Put ^ = p, — = q (p, q are propositional variables). Choose v(p) = a, v(q) = b. Then by (2) v(pa) = v(qb) = T1. Applying pa A qb = (p o q)c gives v((p o q)c) = T1, hence by (3) a o b = c. □
Theorem 2. Let a,b G {T1, T0, F1, F0}. Then
*a = b ^^ = for any LM-formula (5)
Proof. Adapt the proof of Theorem 1. □
Let us substitute a o b to the place of c in (4). Also let us substitute *a to the place of b in (5). We get
Theorem 3. For any a,b G {T1, T0, F1, F0} and for any L^-formulas
A = o -)aob, (6)
Due to (4), (5) and (6) applying (1) we obtain characteristics in the spirit of Kooi & Tamminga [17; 21] for 80 equalities of the form a o b = c and *a = b. For example, combining a = T0, b = F1, c = F0 with (4), (1) yields
T0 o F1 = F0 ^^ [— A — — =- — (<£ o -0) for any L+0-formulas —]. (7)
It is easy to prove the following analog of (6):
A = A -)aAb, A = V -)aVb, = (—= (- (8)
2. Natural deduction systems
A natural deduction system for LRA is as follows:
• Axiom:
(EM) ^ V —^ V —^ V — —
• Rules for negations:
(EFQ.) ^, (EFQ2) ^, (EFQ3) ^,
—imA „ —icz? A ~ —icz? , „ ~(z?A~—I(/J (EFQ4) ^ ^, (EFQ5) ^ 0 ^, (EFQ6) -^,
(——, (-- , (-—I) —-^
r^j
( -1 r^
-I — rrr — r——
, (r—I—1/) -—— , (—I rrr I) -—-,
—— r-1-1— -| rr —
Positive fragment:
(A/) ^-4, (V/,) ——-, (V/a) -
— A — — V — — V — '
(VE )
ы . . . [Pn] pi V ... V pra X ... X
X
where n ^ 2 and [w] means that the assumption w is discharged. • Rules for negations of conjunction and disjunction:
(—V I) —f^, (—A I) —f^, (— V I) ~ (—f A—. v 7 — (f V -) — (f A -) --i(p V -)
(~ f A -) V (f A ~-) V (~ f A ~-)
(-ЛI )
(^ ^ л
(~V I)
~ (p Л -0) '
(^ ^ Л ~ 0) V Л ~0) V ^ Л 0) V (—Л ~ 0) V ^ Л —0)
л i)
~ (<£ V -0)
— ^ л 0) v — ^ л ~0) v — ^ л v ^ л v л
--,(<p л 0) ■
The notion of a derivation of f from r in NDlra and other natural deduction systems described in this paper is defined in a tree-format (Gentzen-style) in a standard way. We introduce an example of derivation in ND1LRA in Figure 1.
[~p] , r)
[—p] ——P/T\ ——P — p ( ——''1 ) — — p [~—p]
. -(a/) -^-— (a/) -" . 1 11 (a/)
—p a ——p ——p a p v ' ——p a p
p v—p v~p v—p [p] -p-(EFQ l) -p-(EFQ 2) -^-{EFQ 4)
—-----------p---— (EM), (vE)
Fig. 1. A derivation of p from ——p
Proposition 1. The following rules are derivable in NDlLRA :
(——E) --, (--E) --, (--—E) --, ( — ~~ E) -,
f f ~ f —f
(A Ei) f^, (A E2) fA-, (—V E) —(f V -, (—A E) —(f A -, v 1 f K 2 - —f A— K ' — f V—-
(—V E) (f V , (~ A E) ~(f A
(~V E )
(~ — л E)
(—p Л—■0)' (~ p Л -0) V (p Л ~ -0) V (~ p Л ~
~ (^ V 0)
(^^ Л ~0) V ^ Л ~0) V ^ Л 0) V (—^ Л ~0) V ^ Л —0);
Ч^ л 0)
^ л 0) v ^ л ~0) v л ^—0) v <£> л ^—0) v (^ л ^—0)
Proposition 2. Let Rlra2 := {(EM), (EFQi), (EFQ2), (EFQ3), (EFQ4), (EFQ5),
(EFQ6), (——E), (--E), (~—I), (—~ I), (~ ——E), (—--E), (Л/), (ЛE1), (ЛE2),
(vE), (— V E), (— Л E), (~— V E), (^E), (~ VE), (~ — Л E) } be a set of inference rules for natural deduction system Then for any set of L-formulas Г and for
any L-formula p
Г h p m NDlra ^ Г h p m ND2LRA.
The Priest's results [20] show that Rk3 := {(EFQi), (——/), (——E), (A/), (AEi), (AE2), (V/1), (V/2), (VE), (—V I), (—V E), (—A I), (—A E)} is a set of inference rules for strong Kleene's logic K3 [22]. Therefore K3 is a fragment of LRA.
Proposition 3. Let Rlra := {(EM), (EFQi), (EFQ2), (EFQ3), (EFQ4), (EFQ5),
(EFQ6), (——I), (——E), (--/), (--E), (--1/), (—-/), (--.—I), (----E), (—--1),
(—--E), (A/), (AEi), (AE2), (V/i), (V/2), (VE), (—V /), (—V E), (—A /), (—A E),
(-— V /), (-—V E), (-A/), (-AE), (-V/), (-VE), (-—A /), (-—A E)} be a set of inference rules for a natural deduction system NDLRA. Then r h p in NDlra ^^ r h p in NDlra ^ r h p in ND LRA.
Although all these natural deduction systems are deductively equivalent, hereafter we will work with NDlra, because it seems to be the most convenient one.
A natural deduction system N3LRAto is an extension of NDLRA by inference rules based on Theorem 3: for any a, b G {T1, T0, F1, F0} we add the rules (cf. (6))
Ro(a, b) ^ A f*., R*(a)—^—. (9)
For example, if T0 o F1 = F0 (see (7)) and *T1 = T0, then NDLRA is extended by the rules R(T0, F1) and R*(T1):
Ro(T0, F1) - ^ A—R*(T1)- P
i( — o — ) ' ~ (*—)
Proposition 4. For any a,b G {T1, T0, F1, F0} the inference rules (cf. (8))
Ra(a, &), — A — A,, Rv(a,&), — A -f,.,, (a)^^r—, («)t——a— (10) either are rules of NDLRAw, or derivable in NDLRAw.
3. Soundness and completeness of
Soundness follows by a simple routine check. Theorem 4. (Soundness). For any set of L+0-formulas r and for any L+0-formula —
r h — r = —.
Completeness proof proceeds by Henkin's method [24]. We follow the notational conventions of [17; 21].
Definition 1. For any set of L+0-formulas r and for any L+0-formulas — and — r is a nontrivial prime theory, if the following conditions hold :
(ri) r = Form*0 where Form*0 is a set of all L+0-formulas (non-triviality);
(r2) r h — ^^ — G r (closure property of r);
(r3) —, V ... V — n G r (—, G r or ... or — n G r) where n ^ 2 (primeness).
Definition 2. For any set of L+0-formulas r and for any L+0-formula — e(—, r) is a canonic valuation, if the following conditions hold :
T1 T0 F1 F0
01 02
e(f, r)= {
07
010 011 012
f G r,
f G r, f G r, f G r, f g r, f G r, f G r, f G r, f G r, f G r,
f G r,
f G r, f G r, f G r, f G r, f G r,
nf G r, nf G r,
nf G r,
nf G r,
nf G r, if G r, nf G r, nf G r,
nf G r, nf G r, nf G r,
nf G r, nf G r, nf G r,
nf G r, nf G r,
G r
G r
G r G r
G r G r
G r G r
G r G r
G r
G r G r
G r
G r
Gr
and and and and and and and and and and and and and and and and
nf g r nf g r nf g r
nf G r nf G r
nf G r
nf G r
nf G r
nf G r
nf G r
nf G r nf G r
nf G r
nf G r nf G r
nf G r
Lemma 1. For any set of L*0-form,ulas r and for any L*0-form,ulas f and —
(1) e(f, r) = 0i where 1 < i < 12;
(2) e(f, r) ° e(—, r) = e(f ° —, r);
(3) *e(f, r) = e(*f, r);
(4) —e(f, r) = e(—f, r);
(5) ~ e(f r) = e(~f r);
(6) e(f, r) V e(—, r) = e(f V —, r);
: Form*0, contrary (EFQ6) instead of
(7) e(f, r) A e(—, r) = e(f A —, r).
Proof. (1) Suppose f G r and —f G r. Then by (AI) and (EFQ1) r to (r1). Therefore, e(f, r) = 0i where 1 < i < 4.
Repeating the previous arguments with using the rules (EFQ2) (EFQi) leads to e(f, r) = 0i where 5 < i < 11.
It remains to prove that e(f, r) = 012. Suppose f G r, —f G r, ~ f G r, and f G r. However, by (r3) and (EM) f G r or —f G r or ~ f G r or ~ —f G r, a contradiction. Therefore, e(f, r) = 012.
(2) From the preceding part of the Lemma and Definition 2 we deduce for any a G {T1, T0, F1, F0} and for any L*0-formula f (see also (1))
e(f r) = a
f
^ f G r,
,e(^,r) ^ r.
11) 12)
By (12) and R0(e(f, r),e(—, r)) (see (9)) we obtain r h (f ° —)e(^,r)0eW,r). Hence, by the closure property of r we have (f ° — )e(^,r)0e(^,r) g r, and by (11) we conclude e(f, r) ° e(—, r) = e(f ° —, r).
(3) By (12) and R*(e(f, r)) (see (9)) we get r h (*f)*e(^r). Closure property of r gives (*f)*e(^,r) g r, and by (11) we obtain *e(f, r) = e(*f, r).
The proofs of (4) and (5) are similar to (3) with the rules R-,(a) and R-,(a) (see (10)) instead of R*(a). The proofs of (6) and (7) are similar to (2) with the rules Rv(a, b) and RA(a,b) (see (10)) instead of R0(a,b). □
Standard proofs show that the following Lemmas 2 and 3 hold. Notice that Lemma 1 is used in proof of Lemma 2.
Lemma 2. For any nontrivial prime theory r and for any valuation vr such that
vr(p) = e(p, r), for any propositional variable p: vr(p) = e(p, r), for any L^-formula p.
Lemma 3. (Lindenbaum). For any set of L^-formulas r and for any L+0-formula p: if r h P, then there is a set of L^-formulas r* such that: (1) r C r*, (2) r* h p, and (3) r* is a nontrivial prime theory.
Theorem 5. (Completeness). For any set of L^-formulas r and for any L^-formula
p: r = p r h p.
Proof. The proof proceeds by contraposition. Let r h p. Then, by Lemma 3, there is a set of L*0-formulas r* such that: (1) r C r*, (2) r* h p, and (3) r* is a nontrivial prime theory. By Lemma 2, there is a valuation vr such that vr(—) = T1, for any — G r, and vr(p) = T1. But then r = p. □
In the light of Theorems 4 and 5 the following Theorem 6 is obvious.
Theorem 6. (Adequacy). For any set of L*0-formulas r and for any L*0-formula p:
r = p ^^ r h p. Conclusion
In this paper, we have presented a general method (correspondence analysis) for axiomatizing LRA's unary and binary truth-functional extensions via natural deduction systems. The future work concerns an investigation of logics with LRA's connectives but with the other sets of designated values and constructing correspondence analysis for them.
Acknowledgments. I would like to express my sincere gratitude and appreciation to an anonymous referee for his formulation of Theorems 1-3, a generalization of their proofs, and a simplification of Lemma 1's proof as well as other helpful comments and suggestions.
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Accepted article received 03.05.2017 Corrections received 13.09.2017
Челябинский физико-математический журнал. 2017. Т. 2, вып. 3. С. 329-337.
КОРРЕСПОНДЕНТСКИЙ АНАЛИЗ
ДЛЯ ЛОГИКИ РАЦИОНАЛЬНОГО АГЕНТА
Я. И. Петрухин
Московский государственный университет имени М. В. Ломоносова,
Москва, Россия
yaroslav.petrukhin@mail.ru
Рассматривается с теоретико-доказательной точки зрения логика рационального агента (LRA) Кубышкиной и Зайцева. В работе построено три системы натурального вывода для LRA, отличающиеся от аксиоматизации LRA, осуществлённой Кубышкиной и Зайцевым. Кроме того, сформулирован общий метод аксиоматизации с помощью натуральных исчислений расширений LRA любыми истинностно-функциональными одноместными и двухместными операторами. Этот метод есть не что иное, как описанный Коем и Таммингой корресподентский анализ, адаптированный в данном случае для LRA.
Ключевые слова: многозначные логики, обобщённые истинностные значения, корреспондентский анализ, натуральное исчисление.
Поступила в 'редакцию 03.05.2017 После переработки 13.09.2017
Сведения об авторе
Петрухин Ярослав Игоревич, студент кафедры логики философского факультета, Московский государственный университет имени М. В. Ломоносова, Москва, Россия; e-mail: yaroslav.petrukhin@mail.ru.
