R. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....intelligence to formalise these propositional attitudes as predicates of first order predicate logic. This allows the agents to express stronger introspective beliefs and engage in stronger meta reasoning than in the classical modal operator approach. Results by Montague [1963] and Thomason [1980] show, however, that the predicate approach is prone to inconsistency. More recent results by des Rivi eres Levesque [1988] and Morreau Kraus [1998] show that we can maintain the predicate approach if we make suitable restrictions to our set of epistemic axioms. Their results are proved ....

....and others beliefs, which is important in meta reasoning. Unfortunately, the predicate approach easily becomes inconsistent, since the added expressive power allows the agents to express self referential beliefs that in some cases turn out to be paradoxical. This was proved by Montague [1963] and Thomason [1980] . They prove that certain axiom schemes describing natural properties of knowledge and belief are inconsistent with formal arithmetic. Their results are reviewed in Section 3. Des Rivi eres Levesque [1988] and Morreau Kraus [1998] have shown there to be a way out of these inconsistency ....

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Richmond H. Thomason. A note on syntactical treatments of modality. Synthese, 44(3):391--395, 1980.

....any extension of it. We therefore only need to prove that none of (1) 4) is universally consistent. That (1) and (2) are not universally consistent is a direct consequence of a theorem of Montague in [7] That (3) is not universally consistent is a direct consequence of a theorem of Thomason in [13]. Both these theorems are reviewed in [3] and [8] Finally, that (4) is not universally consistent is proved in Section 5.3. # The fact that (1) is not universally consistent is probably not a serious problem, since (R2) is a very strong principle that beliefs of agents would not be likely to ....

....Conclusion As argued in e.g. 2,8,9,10] representing beliefs of agents should be done syntactically through predicates of first order logic to ensure su#cient expressivity. Unfortunately it turns out that representing beliefs syntactically easily leads to inconsistency of the representing system [7,8,13]. This calls for work in finding consistent ways to treat beliefs syntactically, that is, to find restricted ways of representing beliefs in first order logic such that consistency will necessarily be retained. Some of the most important previous results in this direction can be found in [3,8,9] ....

Thomason, R. H., A note on syntactical treatments of modality, Synthese 44 (1980), pp. 391--395.

....used; 27, 24, 25] and [1, 39] are some of the many references on the use of modal logics respectively on the rst and the second topic. One way to interpret these results is that rst order languages are all we need to give a consistent theory of propositional attitudes and provability (see [35, 42, 38] for a description of the problems which may arise with rst order treatments of modalities) Instead of extending the language, as modal logics do, another solution is to add more structure to the logic. These results show also that the resulting systems t better with our intuitions. This work ....

R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391{ 395, 1980.

....in a context. MT is distinct from the other contexts, and in particular from PSC. The reasons for this are the same explained for PSC plus four further facts. First, a meta theory distinguished from PSC can describe reasoning within PSC and avoid selfreferential problems (see for instance [27, 31]) Second, a distinguished meta theory can explicitly represent the interactions (through bridge rules) among PSC and s contexts. Third, PSC language is not rich enough to represent deductions. In PSC we can represent and reason about provability in s contexts, but not about derivability of a fact ....

.... that the axioms represent rules that have premises in a context (e.g. T ( P SC ) and conclusion in a different context 25 (e.g. T ( c ) with c 6= P SC ) Moreover, in both AX # and AX , the condition c 6= P SC prevents these axioms to cause inconsistencies based on self reference [27, 31]. Besides reflection principles between s contexts and PSC, it is also possible to have a set of bridge rules among s contexts (see figure 1) For instance, consider the set of bridge rules ffi ij defined in (4) They are represented in MT as: AX ffi : 8s8c i 8c j (T (s; c i ) c i c j oe T ....

R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

....knows is represented by a set of formulas and hence is not constrained under consequence. But such approaches are difficult to analyze, since they are not guided by knowledge based principles 1 . A commendable syntactic approach is presented by Konolige 1 In addition, Montague and Thomason [Mon63, Tho80] proved that certain axiom schemes characterizing ideal (unbounded) agents will lead to contradictions, if the syntactic approach is taken. 3 in his deduction model [Kon83] which gives a formal characterization of explicit beliefs and captures how agents syntactically derive new beliefs, ....

R Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

....in K and in MK. Theorem 4.1 If A is an L(Bl) wff then for any i 2 , MBK hA; ii, if and only if K A . Corollary 4.1 (Consistency) For every i 2 , h ; ii is not a theorem in MBK. Notice that in [RL86] Des Rivieres and Levesque show how Montague and Thomason s negative results [Mon74, Tho80] can be avoided with a careful translation of modal logics 9 into first order logics. Besides the differences in the motivations, the main technical difference between the results stated by theorem 4.1 and corollary 4.1 and those described in [RL86] is that they keep a unique language while we ....

R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391-- 395, 1980. 14

....a model has another important consequence, the consistency of T . This provides an interesting contrast with a result of Montague [44] He showed that under certain fairly intuitive conditions a syntactic formalization of modal notions such as knowledge becomes contradictory. Thomason later showed [54] that under even less restrictive conditions a syntactic formalization yields a contradictory set of beliefs. In Montague s case the meta theory is inconsistent, and in Thomason s the object (agent) theory is inconsistent. 4 Yet our meta theory is always consistent. Moreover an agent theory need ....

R. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

....to making DRSs themselves into the objects of the attitudes in a formal theory of attitude reports. This can easily lead to a theory which suffers from all the familiar pitfalls of self reference, which had come into full view through the results of [ Montague, 1963 ] and against which Thomason [ Thomason, 1980 ] launched a crusade (much justified ) in the early seventies. It was argued by [ Asher and H.Kamp, 1989 ] a paper which neither uses nore mentions DRT, but which was motivated by the self reference problem that arises when attitudinal predicates are construed as predicates of DRSs that trying ....

Thomason, R. 1980. A note on syntactical treatments of modality. Synthese 44:391--395.

....used; 27, 24, 25] and [1, 39] are some of the many references on the use of modal logics respectively on the first and the second topic. One way to interpret these results is that first order languages are all we need to give a consistent theory of propositional attitudes and provability (see [35, 42, 38] for a description of the problems which may arise with first order treatments of modalities) Instead of extending the language, as modal logics do, another solution is to add more structure to the logic. These results show also that the resulting systems fit better with our intuitions. This ....

R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391-- 395, 1980.

....d : e , are used to indicate that their contents are a meta language term denoting the corresponding object language formula. Unfortunately, meta language formalisms have their own package of problems, not the least of which is that they tend to fall prey to inconsistency (Montague, 1963; 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; ....

Thomason, R. (1980). A note on syntactical treatments of modality. Synthese, 44:391--395.

....operators which represent meta level notions are translated into meta level predicates. Montague [Mon63] has shown that any theory obtained through this translation is inconsistent if it contains, at least, the system T . Further results along the same line have been obtained by Thomason [Tho80]. For a while these results discouraged any attempt of treating modalities as syntactical objects, but recent work on this topic overcomes the above mentioned problems. Des Rivieres and Levesque [dRL88] point out that the causes of inconsistency in the theories resulting from the translation of a ....

R. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

....The quote marks, d . e , are used to indicate that their contents are a meta language term denoting the corresponding object language formula. Unfortunately, meta language formalisms have their own package of problems, not 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 ....

R. Thomason. A note on syntactical treatments of modality. Synthese, 44, 1980.

....that any moderately sophisticated first order theory (and in particular, basic arithmetic) containing the above axiom and inference rule is inconsistent. This result appears to be devastating, and for a long time dampened down research into first order meta languages. A similar result by Thomason ([169]) is similarly discouraging. However, des Rivieres and Levesque have recently shown that, while 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 ....

R. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

....52900 Ramat Gan, Israel sarit umiacs.umd.edu c Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742 Abstract Syntactical treatments of propositional attitudes are attractive to artificial intelligence researchers. But results of Montague [12] and Thomason [16] seem to show that syntactical treatments are not viable. They show that if representation languages are sufficiently expressive, then axiom schemes characterizing knowledge and belief give rise to paradox. De Rivi eres and Levesque [6] characterized a class of sentences within which these schemes ....

....see, Richard Montague [12] shows that if the language in which known facts are represented is sufficiently expressive, then familiar axiom schemes characterizing knowledge are inconsistent. He argues that knowledge can be given a syntactical treatment only on pain of triviality. Richmond Thomason [16] argues under similar assumptions that a syntactical treatment also trivializes belief. Montague and Thomason obtain their limiting results by instantiating their axiom schemes with self referential formulas. In response, Jim de Rivi eres and Hector Levesque [6] show that syntactical treatments ....

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R Thomason. A note on syntactical treatments of modality. Synthese, 44:391-- 395, 1980.

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Thomason, R., "A note on syntactical treatments of modality."Synthese 44 (1980), pp. 391-395. 12

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R. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

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R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391-- 395, 1980.

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R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391--395, 1980.

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R.H. Thomason. A note on syntactical treatments of modality. Synthese, 44:391---395, 1980.

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Richmond H. Thomason. A note on syntactical treatments of modality. Synthese, 44(3):391--395, 1980.

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Richmond H. Thomason, A note on syntactical treatments of modality, Synthese, vol. 44 (1980), pp. 391-396.

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Thomason, R. 1980. A note on syntactical treatments of modality. Synthese 44:391--395.

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