D. Perlis. Languages with self-reference ii: Knowledge, belief and modality. Arti...cial Intelligence, 34:179--212, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Depending on the choice of the languages where the premise, the conclusion and the statement itself is formulated we have di erent approaches. Examples where the object theory, the meta theory coincide and the re ection principle is stated in the same language are [12, 32, 6] in formal logic) and [27, 28] (in Arti cial Intelligence) In other approaches the object theory and the metatheory coincide but the re ection principle is stated externally to them, by means of a set of inference rules. In this case we have an informal representation of (1) which is not a formula in a formal system. Such ....

....axioms or a set of inference rules, is exploited to generate the object theory and the metatheory in the 2 relation expressed by the re ection principle itself. Examples of the rst approach can be found in the area of Formal Logics [12, 32, 6] Logic Programming [9] and Arti cial Intelligence [27, 28]. Examples of re ection principle with a generative role can be found in the area of Arti cial Intelligence [36, 17, 2, 26] The third and last observation about (1) concerns its strength. A re ection principle can be stated for a single formula. This is the case, for instance, of the consistency ....

D. Perlis. Languages with Self-Reference II: Knowledge, Belief, and Modality. Articial Intelligence, 34:179-212, 1988.

....one application of the applicable rules of inference. As previously stated, the agent is to have only 1 For more details on SLn theories, see [Drapkin and Perlis, 1986, Elgot Drapkin, 1988] 2 known, believed, or concluded. The distinctions between these (see [Gettier, 1963, Perlis, 1986, Perlis, 1988] are not addressed here. 3 For simplicity, in the remainder of the paper we drop the quotes around the second argument of predicates U and K j . a finite number of theorems (conclusions, beliefs, or simply wffs) at any given step. We write: i : ff i 1 : fi to mean that ff ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

....of object oriented software components. 2 Reflection Reflection in computer science in general has been first suggested in the context of artificial intelligence where the construction of self aware systems can be assimilated to the end goal of the field (see the two papers of Perlis [Per85, Per88]) In the context of programming languages, the seminal work of Brian Smith [Smi82, Smi84] has been the departure point of a large research effort both in the functional and the object oriented programming community. It is not our goal to introduce reflection here; the reader is refered to the ....

D. Perlis. Languages with self-reference II: Knowledge, Belief, and Modality. Artificial Intelligence, 34:179--212, 1988.

....is to provide semantics for HFP and Prolog practice in order to do our computational classical logic , rather than construct self reflective languages which have their own truth predicates. So we do not mind to have the Hold predicate not to apply to itself. See the Perlis papers [Perlis, 1985, Perlis, 1988] for a good coverage of the subject. Second, in the Horn clause fragment of HFP negation is not available and it is possible to have a truth predicate in the object level for the object language itself, without leading to contradiction. The worst we can get are loops, if we use negation as ....

D. Perlis. Languages with self reference II: knowledge, belief and modality. Artificial Intelligence, 34:179--212, 1988.

....2 We ask, furthermore, that if such fixed points are expressible in the logic 6 then the Delta i are computable from T [C] itself. The problem is to determine the restrictions that has to be imposed so as to avoid the obvious paradoxes of a self referential language. Both in (Perlis, 1985; Perlis, 1988) and in (Montague, 1963) the difficulties of dealing with a first order self referential language are thoroughly analyzed. In particular Montague points out the problems concerned with substitution; we discuss this point in the last section of this paper. It is possible to circumvent the ....

....at all worlds in A. A modal formula A is valid on a frame F if A is true at every worlds of F, under every valuation. 10 Most of the research in commonsense reasoning dealing with knowledge, belief and self reference takes into account normal modal logics (e.g. McDermott, 1982; Moore, 1987; Perlis, 1988; Schwarz, 1990; Truszczynski, 1991; Marek and Truszczynski, 1993; Stalnaker, 1993) A normal modal logic is a collection of well formed formulae of the modal language L that extends propositional logic with the axiom schema K: 2A2(A B) 2B and it is closed under the rules of: Necessitation ....

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Perlis, D. (1988). Languages with self-reference II: Knowledge, belief, and modality. Artificial Intelligence Journal, 34:179--212.

....or somewhere else. We just need the minimal amount of information. Analogously we do not need the context in the water . The only necessary and sufficient context is heat on . This self referential view of contexts is not new, both Smullyan and Perlis in their papers on self reference [26] and [21] refer to contexts. Now, 3) make sense if their definitional form is, indeed, a fixed point equation, and thus they are logical truths in some logical system. The only way to ensure this is to show that, for each context, there is at least a fixed point equation that satisfies it. That is, 3) ....

D. Perlis. Languages with self-reference II: Knowledge, belief, and modality. Artificial Intelligence Journal, 34:179--212, 1988.

....Godel provability. Another property of a self representable language is that self referential sentences can be constructed [9] This is not harmful in itself, but a theory with such a language can express paradoxes. The concept of self reference has been studied quite extensively, e.g. by Perlis [70, 71]. There is an overview of self reference and incompleteness in logic by Smorynski [80] 2.3 Amalgamation: Theories that Represent Themselves Given a self representable language LA , we may go one step further and develop a theory TA that represents itself, i.e. having the property that Demo TA ....

Perlis, D., Languages with Self-Reference II: Knowledge, Belief and Modality, Artificial Intelligence, 34:179--212 (1988).

....to Figure 1. Wise man #1 knows the following: a. Wise man #2 uses the rule of modus ponens. b. Wise man #2 knows at step 1 that if my card is black, then his is white. 3 For more details on SLn step logics, see [ED88] 4 known, believed, or concluded. The distinctions between these (see [Get63, Per86, Per88]) are not addressed here. 5 For simplicity, in the remainder of the paper we drop the quotes around the second argument of predicates U and K j . OBSW 2 is defined as follows. OBSW 2 (i) 8 : 8 : 8i) 8x) 8y) K ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

....very few such experiences have been reported yet. 4 Formal notions for Reflection We discuss here of formal notions used to characterize reflection in programming languages of the three different paradigm, preceded by a historical discussion of reflection in logic freely inspired by [Per85, Per88] Wey80] Bow82] Fef62] and [Cos90] 4.1 Historical discussion Gottlieb Frege developed the first formal quantificational logic over a period of more than two decades culminating in 1903. The idea was to have a universal language for logic. For Frege, an object c and the properties P it may ....

....with some meta level languages. In Perlis [Per85] a theory of quotation and unquotation is presented to refer to certain statements as true and false. The author concentrates on the importance of the roles of truth and self reference in commonsense reasoning. In his following article [Per88] he has showed that modal logics are on no firmer ground than first order ones when equally endowed with substitutive self reference. Thus his work has tried to understand the relationship between self reference and modal logic. 4.2.1 Declarative semantics in the logical style The next attempts ....

D. Perlis. Languages with self-reference II: Knowledge, Belief, and Modality. Artificial Intelligence, 34:179--212, 1988.

.... Thomason, 1980) However, there have been some fairly successful meta language formalisms, including those by Konolige (Konolige, 1982) Haas (Haas, 1986) Morgenstern (Morgenstern, 1987) and Davies (Davies, 1993) Some results on retrieving consistency appeared in the late 1980s (Perlis, 1985; Perlis, 1988; des Rivieres and Levesque, 1986; Turner, 1990) 2.5 Pro attitudes: Goals and Desires An obvious approach to developing a logic of goals or desires is to adapt possible worlds semantics see, e.g. Cohen and Levesque, 1990a; Wooldridge, 1994) In this view, each goal accessible world ....

Perlis, D. (1988). Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212.

....is easily possible to do away with this unrealistic property. On the other hand, meta systems allow paradoxical sentences such as the liar sentence ( This sentence is false ) to be represented. This seems to be a crucial disadvantage. However, powerful modal logics are faced with the same problem [15]. In this paper, we shall see in more detail, why paradoxes occur in metasystems and what different ways out of the problems exist. Essentially I shall advocate a system, in which paradoxes are taken seriously and formulae like the liar sentence are admitted. They are neither true nor false, but ....

....not cause any formal problem anymore in this approach, although the solution is not intuitive. General problems about mutually related statements about each others beliefs (A says Everything B says is wrong. and B says Everything A says is right. persist, however. By the way, as Perlis showed [15], this problem is not specific to metasystems, but occurs in modal logics as well (unless there are serious restrictions on the expressive power of the modal logic) Third approach Kripke [13] and others attacked the problem by changing the semantic construction of the truth values of formulae. ....

Donald Perlis. Languages with self-reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

.... the least of which is that they tend to fall prey to inconsistency [95, 132] However, there have been some fairly successful meta language formalisms, including those by Konolige [79] Haas [61] Morgenstern [97] and Davies [32] Some results on retrieving consistency appeared in the late 1980s [101, 102, 36, 133]. 2.3 Towards a Theory of Agency All of the formalisms considered so far have focussed on just one aspect of intelligent agency: either knowledge or belief. However, it is to be expected that any realistic agent theory will be represented in a much richer logical framework. First, neither agents ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34, 1988.

....to treat meta language predicates as syntactic modalities run into inconsistency. These difficulties arise, for the large part, from the issue of self reference 14 . 14 A good discussion of these issues may be found in [131] the issue of self reference is examined in detail in [129] and [130]. Suppose one wished to use a meta language approach to represent and reason about knowledge and belief. Following standard practice, knowledge may be defined as true belief. One might begin by setting objectlanguage = meta language, so that the object and meta language are the same; in such ....

....Montague s results are technically correct, a careful reworking, with slightly different assumptions, leads back to consistency [40] 15 . Also, Perlis has shown how a similar technique to that proposed by him for recovering from Tarski s result can be used to recover from Montague s results [130]. Despite these results, the author is not aware of any recovery from Thomason s negative results. If nothing else, the above discussion illustrates that the whole issue of self referential languages is a difficult one. It is hardly surprising, therefore, that some researchers have rejected self ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

....metalanguage should be separated in an hierarchical fashion, so that selfreference can not occur. Moreover, Montague [25] and Thomason [35] showed that object language and metalanguage can not be combined consistently in the more restricted manner of modal logic. However, more recent studies (e.g. [26, 27, 7, 31]) indicate ways in which object language and metalanguage can be combined, provided that appropriate restrictions are imposed. At least two other objections have been raised against systems that combine object language and metalanguage. One is that the naming conventions necessary to distinguish ....

Perlis, D. Language with Self-Reference II: Knowledge, Belief and Modality, Artificial Intelligence, 34, 179--212.

....not well defined. That is, R 6 R does not obey Normal Order (for set membership rather than truth) This does not mean that no such set as R exists; on the contrary, in Gilmore s set theory it is provable that R exists. Kripke provided a similar treatment for truth. Later Feferman [2] and Perlis [10, 11] also working independently, unified these two treatments. The following schemata capture much of this approach: T(a) a where a has no effect on a unless there is an embedded and negated T inside, and then: Finally we also require o o That is, the assertion of the truth of the negated ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modaltry. Artificial In- telligence, 34:179-212, 1988.

.... similar to autocircumscription, but with an added feature, namely positive introspective (as well as negative introspective which is already present in autocircumscription) This certainly is an attractive addition, as long as inconsistency is not thereby introduced as pointed out in [4]: logics with both positive and negative introspection, and a modest amount of substitutive textual manipulations such as provided, for instance, by arithmetic, are always inconsistent. However, in the limited form employed by Lifschitz, consistency is preserved. Specifically, he does not have a ....

....predicate constants specified in advance. That is, for each such specified constant P , there is a companion predicate constant LP (meaning P is believed) but in general there is no constant LLP , etc. Whether unlimited nesting of predicate companions would lead to the inconsistencies shown in [4] is an open question. Note that autocircumscription does allow unlimited nesting of belief statements, but avoids contradiction by employing only negative, not positive, cases of introspection. Negative cases of introspection are, in general, the ones that give the green light to a pending ....

D. Perlis. Languages with self reference II: knowledge, belief, and modality. Artificial Intelligence, vol. 34 (1988), pp. 301--322.

....use of the turnstile here. 7 We see that the predicate letter K has two roles: in SL n and in SLn . The context will make the role clear. 8 In [5, 4] we used K(i; ff) for K(i; ff) 9 known, believed, or concluded. As already stated, we are not distinguishing between these terms. See [9, 18, 19] for a discussion of these. For time, we envision a clock which is ticking as the agent is reasoning. At each step in its reasoning, the agent looks at this clock to obtain the time. 10 The wff Now(i) is an i theorem. Now(i) corresponds intuitively to the statement The time is now i. ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

....7 . Finally, we use the notation SL 7 for any of a family of step logics whose OBS and INF involve the predicates Now and K and contain a retraction mechanism. Choosing OBS and INF therefore fixes the theory within the family. 20 known, believed, or concluded. The distinctions between these (see [23, 4, 26]) will not be addressed here. to mean that ff is an i theorem, and fi is an i 1 theorem. There is no implicit assumption that ff (or any other wff other than fi) is present (or not present) at step i 1. The ellipsis simply indicates that there might be other wffs present. Wffs are not ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

....medium We claim there is. 8 This is of course a necessary condition for any real world implementation, if we regard conclusions as physically stored entities; and it also is obeyed by our own formalism. 9 This sort of analysis is hinted at for formal theoretical reasons in [Perlis, 1986] and [Perlis, 1988] . One way to think of this is to imagine new data in conflict with already existing data, providing a momentary contradiction. This can be dealt with, roughly speaking, by looking for direct clashes between any two elements in the current set of beliefs (conclusions) at every step in the ....

D. Perlis. Languages with self reference II: Knowledge, belief, and modality. Artificial Intelligence, 34:179--212, 1988.

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D. Perlis. Languages with self-reference ii: Knowledge, belief and modality. Arti...cial Intelligence, 34:179--212, 1988.

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D. Perlis. Languages with self-reference ii:knowledge, belief and modality. Artificial Intelligence, (34):179--212, 1988.

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D. Perlis. \Languages with self reference II: knowledge, belief and modality". Art. Int., vol.34, pp.179-212, 1988.

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D. Perlis. Languages with self-reference II: Knowledge, belief, and modality. Artificial Intelligence Journal, 34:179--212, 1988.

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D. Perlis. Languages with self-reference ii:knowledge, belief and modality. Artificial Intelligence, 1988.

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D. Perlis. Languages with self-reference II: knowledge, belief, and modality. Artificial Intelligence, 34, 179-- 212, 1988.

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