Zcitschr. 1. M h . Lwik und Orund&gcn d. Math. Bd. 27, S. 371-374 (1981)

A LOGIC OF KNOWLEDGE

by W. RICHARD STARK, Tampa, Florida (U.S.A.)

Introduction

Given a propositional language L and a set of individuals 8, the language of know- ledge, L*, is the smallest extension of L such that 1) ( A * 8 ) EL* when A E 3 and 8 E L*, 2) L* is closed under the propositional connectives. Axioms for L* are:

0 ) 8 , if 8 is an axiom for L ; 1) 8 , if 8 is a propositional axiom in L*; 2) A * 8 , if A €3 and 8 is an axiom for L*; 3) ( A * 8 ) + 8 , all A and 8 ; 4) ( A * (v --f y ) ) --f ( ( A * y ) + ( A * w ) ) , all A , v and y ; 5) ( A * e ) 6) T ( A * e ) -+ ( A * T ( A * 8 ) ) ; 7) (0 * e ) -+ ( A * 8 ) .

( A * ( A * 8 ) ) ;

Modus ponens is the deduction rule.

The logics of knowledge generated by axioms 1)-4) with (0 * 8 ) -+ 0 * ( A * 8 ) for every A , 8, by axioms 1) - 5), and by axioms 1) - 6) (with axiom 2) interpreted relative to each set) are denoted LK3, LK4 and LK5, respectively. These logics have been studied by MCCARTHY [2], [3], [4], HINTIKKA [l] and SATO [5]. This paper describes a new deduction rule, the model theorist’s deduction rule which gives a natural ex- tension MK of LK5. The logics LK3, LK4 and LK5’describe models in which the individuals use propositional logic to derive their knowledge (axioms l), 2) and 4)). MK describes models in which the individuals also use elementary model theoretic techniques to derive knowledge of their ignorance. MK, the logic of model theoretic knowledge, is, consequently, a non-monotonic logic. For a more complete discussion of the intended models of MK see STARK [6], section 1.

I n case 3 consists of only one individual LK3, LK4 and LK5 are equivalent to T, 5 4 and 55, respectively. This does not imply that if 3 consists of more than one individual then the logics of knowledge are merely redundant versions of the classical Modal logics. However, it does indicate that the classical Modal logics embed into MK.

Deduction Rule

I n a situation where basic model theoretic techniques are used in the analysis of knowledge, the individuals will be aware of what it is that they don’t know. In other- words, if A doesn’t know 8 then A does know - I (A * 8 ) . I n this section, we develop a deduction rule which -when added to the system already described -yields a for- malization of this sort of reasoning. We work in LK5.

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372 w. RICHARD STARK

With respect to a theory T L*, this deduction looks something like: if (A * 8) is not provable from T then deduce i (A * 0). From this axiom 6) yields A * i (A * 8). However, there is a problem if 8 is a statement about A’s knowledge. For example, let T be { W } and identify 0 with (A * W ) and i ( A * W ) . Neither A * ( A * W ) nor A * l ( A * W ) are provable from T . Applying the deduction rule to these sentences yields i ( A * (A * W ) ) and 1 ( A * i (A * W ) ) . By axioms 5) and 6), conclude (A * W ) and i ( A * W ) . Inconsistencies will be avoided by applying the deduction rule only to (A * 8 ) where 0 is external to A’s knowledge.

Define *rank on L* by: 1) *rank@) = 0 if 8 is atomic 2) *rank( l8) = *rank(@, 3) *rank@ A y ) = *rank@ -+ y ) = max {*rank(@, *rank(y)}, and 4) *rank@ * 8) = = 1 + *rank(@.

The sentences external to A are those in the smallest set : 1) containing the sentences of *rank zero, 2) containing ( B * 8 ) whenever B Zt; A, 0 and 8 is in L*, 3) closed under the propositional connectives. The sentences internal to A are those of the smallest set: 1) containing.(A * 8) whenever 0 is in L*, 2) closed under the propositional con- nectives. All other sentences of L* are A-mixed.

The model theorist’s deduction rule (m.t.d.r.): if 8 is external to A and ( A * 8) is not provable in LK5 from T then deduce i ( A * 8) from T.

The superscript “*” will be used to indicate deduction in the expanded logic formed by adding the m.t.d.r. to LK5. All non-superscripted words and logical operators refer to the system LK5. Deduction* has several peculiar features. First,, for X T, S I-* 8 does not imply T t* 8. For example, let X be { W > and T be S u {(A * W ) ) , then X I-* i ( A * W ) while T t* (A * W ) . Second, consistent does not imply con- sistent*, { ( A * W ) v ( A * i W ) } is inconsistent* and consistent. Third, if T k 0 or T I- 1 8 whenever *rank(8) = 0, then T t* 8 or T t* 1 0 for every 8 (proved in the Consistency* and Completeness* Theorem).

I n n e r logic lemma. T t 0.impZies {(A * y ) I y E T} t (A * 0). Proof. Suppose yl, y z , . . ., yn = 8 is a proof of 0 from T . We will establish the

“only if” direction by showing that (A * yl), (A * y 2 ) , . . . , ( A * y,,) can be expanded into a proof of (A * 0) from {(A * y ) 1 y E T } . If y is an axiom then ( A * y ) is an axiom. If y E T then ( A * y ) E {(A * y ) 1 y E T } . If yk follows from yi and yj = (yi -+ y k ) then ( A * yk) follows by axiom 4 from (A * yi) and ( A * y j ) .

To see that the implication does not reverse observe that {(A * O ) } t A * ( A i 8 ) does not imply { O } t- (A * 8) . This example also shows that t can not be replaced by t*. (8 ) b* 1 (A * 8) does not imply {A * 8 } t* A * i ( A * 8).

Knowledge lemma. If T is a theory such that either T t ( A * 8 ) or T t. i ( A * 8) when 8 i s external to A, then T t (A * y) or T t ( A * i y ) when y i s internal to A .

Proof . In case y is (A * p) and cp is external to A , the conclusion follows imme- diately from the lemma’s hypothesis by axioms 5 ) and 6).

Let y be (A * v), where cp is internal and we assume for the inductive hypothesis that the lemma holds for all sentences of *rank less than *rank(y). By the i.h., either T t (A * Q)) or T 1 (A * i p ) . Two applications of axiom 3) ((A * 0) -+ 8 and 1~1 -+ i ( A * p)) give: T t (A * p) or T b i ( A *v) . Now by axioms 5) and 6): T t. (A * y ) or T t (A * y) .

A LOGIC OF KNOWLEDGE 373

Assume that y is ( A * p) where a, is mixed. Let a,’s conjunctive normal form be

vnf: (ti V . . . V tiz) A . . . A (pi V . . . V t:n) , where each 6; is either atomic, negated atomic, (B * 0) or i (B * 0) where B mag or may not be A .

(9 tt a,’”) , by propositional logic ; A * (a, c-, a ,n f ) , by the internal logic lemma.

Each E j is either internal to A or external to A . If internal, then by the i.h. either

(i) T I- ( A * 4) or (ii) T t ( A * it;). Using an atomic proposition W , pest is formed from pnf by replacing each internal t j by either (W v -I W ) in case (i), or (W A i W ) in case (ii). t (A * (W v i W ) ) and t ( A * i ( W A i W ) ) , so by the internal logic lemma we have

T t A * (4 +-) (W A i W ) ) in case (ii),

T t ( A * y ) t.t ( A * yext). peXt is external

T t A * (4 tt (W v i W ) ) in case (i),

and T k A * (y”‘ tt yext).

to A , so by the first paragraph in this proof either By the first equivalence and axiom 4),

T k A * ( A * a,ext) or T t A * i ( A * pext). Finally, T t ( A * y ) or T I- ( A * i y ) .

The next case in y is (p, A q~,) with the inductive hypothesis that the lemma holds for all proper subsentences of y. Since y, and p, are internal, either T t ( A * y,) A

A ( A * i q ~ , ) . I n the first case T 1 ( A * y ) otherwise T t ( A * y), by internal logic. The case y is i p 7 is trivial.

A theory T describing a natural distribution of knowledge among a group of model theorists has the following properties :

1) T is consistent, 2) if y,, y z , . . ., y, are sentences external to A, , A , , . . ., A,,, respectively and

A ( A * y,), T t ( A * pll) A ( A * 197,), T k ( A * 797,) A ( A * pJ or T t ( A * i y l ) A

T t (A , * y1) v (A , * 972) v * . . v (A,,, * YtfJ then

T t (A, * y,) or T t (A, * y,) or . . . or T t ( A , * y m ) . If T satisfies these conditions then T is said to be natural.

Consis tency* a n d completeness* theorem. T is consistent* if and only if I’ i s natural. If T i s natural then either T t* ( A * 0 ) or T I-* i ( A * €I), for each ( A * 0). If T is natural and T I- y or T I- i l y for every atomic p then T is complete*.

Proof . a) T is natural. Let U , = (0 I T t 0>, U , be the closure of U , under m.t.d.r. and U, = (6 I U , t 131. By condition 1) of natural, U , is consistent, U , - U , consists of sentences (A , * el), (A, * O,), . . ., (A, * On), . . . where each O i is external to A i and (Ai * Oi) 4 U,. By 2) of natural, not T I- ( (A , * 0,) v . . . v (A, * On)), for each n. U , = U , u { ( A , * el), . . . , (A,,, * en,), . . .> is consistent. U, is consistent. To see that U , is closed under m.t.d.r., consider an immediate consequence ( A * 0) of U , by m.t.d.r. We must have (not U , 1 ( A * 0) ) and, since U, U, , (not U , !- ( A * 6 ) ) . Thus ( A * 0 ) is a consequence by m.t.d.r. of U , and 80 ( A * 0) E U,. U , = (0 I T I-* 01. Hence I is consistent*.

374 W. RICHARD STARK

b) T is consistent*. We need only show that T satisfies 2) of natural. Let y,, . . . , y m be external to A, , . . . , A, , respectively. If (not T t (A, * yl)) and . . . and (not T t (A,n * y,J) then T I-* i ( A , * wl) and . . . and T t* i ( A , , & * yn), therefore

T t* l ( ( A , * Yl) v . * . v (A,, , * y m ) ) ,

and not T I-* ( (A, * y,) v . . . v (A,,, * y m ) ) , by the consistency* of T , and not T t ( (A , * y,) v . . . v (Ana * ym)).

Hence T is natural. c) By the m.t.d.r., U , contains either ( A * cp) or i ( A * cp) whenever cp is external

to A . By the Knowledge Lemma and ( A * 19) + i ( A * y ) , U3 contains ( A * cp) or i(A * q ~ ) whenever cp is internal to A . By the argument given in the proof of the Knowledge Lemma, if cp is mixed then there is an external yext such that ( ( A * 9) t) ++ ( A * ye”‘)) E U3. Since U , must contain either ( A * veXt) or its negation, U , must contain either ( A * cp) or i ( A * y ) . We have shown that for all ( A * g ~ ) either T I-” ( A * cp) or T t* i ( A * 9).

The last remark in the theorem is immediate from the second remark.

Puzzle The model theorist’s deduction rule was found as a result of a problem mentioned

by MCCARTHY to the author. The problem was to formally derive the response of each individual in the famous puzzle of the wise men. The puzzle: “The,King, armed with a pot of red paint and a pot of white paint, paints a spot on the forhead of each of his wise men. Each wise man can not see his own spot but can see the spot on the head of each of the others. The King tells them that a t least one spot is white. In turn, each man is asked ‘Do you know the color of your spot?’ Each man’s answer is heard by each of the others.’’ In the most difficult case, when every spot is white, the last man is able to answer “I know that the color of my spot is white.”

For three wise men, let 3 = {L, M , N , 0}, where L, M , N are wise men and 0 is a pseudoindividual used to represent common knowledge. Use atomic propositions W L , WhI , W, for “L’s spot is white”, etc., etc. Add the axiom schema

1) (0 * 8 ) -+ (A * O ) , for each A and 8. In this system, the puzzle’s initial configuration can be described by finite theory T , . We leave it to the reader to construct the theories T I , T, and T,, and show that T , t* ( N * WAr).

Bibliography [l] HINTIKKA, J., Knowledge and belief, an introduction to the logic of the two notions. Cornell

[Z] MCCARTHY, JOHN, The puzzle of the wise men. (Unpublished.) 133 MCCARTHY, JOHN, First order theories of individual concepts. (Unpublished.) [41 MCCARTHY, JOHN, An axiomatization of knowledge and the example of the wise man puzzle.

[5] SATO, M., A study of Kripke-type models for some modal logics by Gentzen’s sequential method.

[6] STARK, RICHARD, W., Logics of knowledge and their decision procedures. Theoretical Computer

(Eingegangen rtm 30. August 1079)

University Press, Ithaca and London 1962.

(Unpublished 1977.)

Publications of the Research Institute for Math. Sciences Kyoto University 13 (1977).

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