Future determination of entities in Talmudic public announcement logic

Ordinary dynamic action logics deal with states and actions upon states. The actions can be deterministic or non-deterministic, but it is always assumed that the possible results of the actions are clear cut.Talmudic logic deals with actions (usually legally meaningful actions which can change the legal status of an entity) which depend on the future and therefore may be not clear cut at the present and need future clarifications.The clarification is modelled by public announcement which comes at a later time after the action has taken place.The model is further complicated by the need to know what is the status of formulas at a time before the results of the action is clarified, as we do not know at which state we are in. Talmudic logic treats such states much like the quantum superposition of states and when clarification is available we get a collapse onto a pure state.The Talmudic lack of clarity of actions arises from applying an action to entities defined using the future, like the statement of a dying man on his death bed:

Let the man who will win the jackpot in the lottery next week be the sole heir in my will now

We need to wait a week for the situation to clarify.There is also the problem of legal backwards causality, as this man, if indeed he exists, unaware of his possible good fortune, may have himself meanwhile donated all his property to a charity. Does his donation include this unknown inheritance?This paper will offer a model and a logic which can represent faithfully the Talmudic reasoning in these matters.We shall also see that we get new types of public announcement logics and (quantum-like) action logics. Ordinary public announcement logic deletes possible worlds after an announcements. Talmudic public announcement logic deletes accessibility links after an announcement. Technically these two approaches are similar but not equivalent.     

Absolutely independent axiomatizations for countable sets in classical logic

The notion of absolute independence, considered in this paper has a clear algebraic meaning and is a strengthening of the usual notion of logical independence. We prove that any consistent and countable set in classical prepositional logic has an absolutely independent axiornatization.     

On qualitative axiomatizations for probability theory

In the literature, there are many axiomatizations of qualitative probability. They all suffer certain defects: either they are too nonspecific and allow nonunique quantitative interpretations or are overspecific and rule out cases with unique quantitative interpretations. In this paper, it is whown that the class of qualitative probability structures with nonunique quantitative interpretations is not first order axiomatizable and that the class of qualitative probability structures with a unique quantitative interpretation is not a finite, first order extension of the theory of qualitative probability. The idea behind the method of proof is quite general and can be used in other measurement situations.This research was partially supported by the national Science Foundation grant NSF BNS7702911 and by the joint NSF-NIE grant NSF SED 78-22271 to the University of California, Irvine.     

Complete axiomatizations for XPath fragments

We provide complete axiomatizations for several fragments of Core XPath, the navigational core of XPath 1.0 introduced by Gottlob, Koch and Pichler. A complete axiomatization for a given fragment is a set of equivalences from which every other valid equivalence is derivable; equivalences can be thought of as (undirected) rewrite rules. Specifically, we axiomatize single axis fragments of Core XPath as well as full Core XPath. Our completeness proofs use results and techniques from modal logic.     

Public announcement logic with distributed knowledge: expressivity,completeness and complexity

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.     

On translating logic

The paper comments on Dummett's Significance of Quine's Indeterminacy Thesis and discusses Quine's views on the translation of logical connectives. Some difficulties about the latter related to those raised by Morton (J. Phil. 70 (1973), 503–510) are considered. Quine seems here to be in a position considered by Dummett of not allowing a foreigner to be translated as conflicting with one's own firm theoretical commitment (in this case classical logic). But Dummett seems wrong in holding that entrenched theoretical statements must be stimulus analytic.A revised and expanded version of remarks at the Conference on Language, Intentionality, and Translation Theory, University of Connecticut, Storrs, March 2, 1973, commenting on Michael Dummett, The Significance of Quine's Indeterminacy Thesis. I am greatly indebted to the discussion at the Conference.     

On deontic logic

Some requirements concerning deontic logic are formulated and discussed. Stress is laid on the need to distinguish between theories and deductive systems. It is argued that deontic theories need not be closed under the rule of detachment. Two deontic calculi, called DSC₁, DSC², are presented and talked over.This paper was delivered before the Conference on Practical and Philosophical Motivations of Non-Classical Logics, Toru, August 16, 1979.     

On the logic of adverbs

In the paper I investigate aspects of adverbial modification as an operation applying an adverb or adverbial phrase to a predicate and thereby creating a new predicate. The logic of adverbial modification, on this view, belongs to the logic of predicate modifiers. The theory I present is intended to cover not only adverbial modification but also attributive modification, but problems concerning the latter will not be given any special attention.     

On the logic of theocracy

I identify the objectionable element in theocracy, not with reliance on God as such, nor with the idea that God might have something to do with morality, but with the anti-human propensity to issue orders without communicating good reasons for them. In medieval discussion prohibitions not based on good reasons attracted the labelmalum quia prohibitum, bad because forbidden and I take this to be the criterion of theocracy in its objectionable form. I examine, in part of the Vatican’s doctrine against contraception, a persistent tendency towards this approach, a tendency incompatible with the tradition of the Church and ultimately incompatible even with the thirteenth century discussion of such issues in the work of Thomas Aquinas.     

On logic of complex algorithms

An algebraic approach to programs called recursive coroutines — due to Janicki [3] — is based on the idea to consider certain complex algorithms as algebraics models of those programs. Complex algorithms are generalizations of pushdown algorithms being algebraic models of recursive procedures (see Mazurkiewicz [4]). LCA — logic of complex algorithms — was formulated in [11]. It formalizes algorithmic properties of a class of deterministic programs called here complex recursive ones or interacting stacks-programs, for which complex algorithms constitute mathematical models. LCA is in a sense an extension of algorithmic logic as initiated by Salwicki [14] and of extended algorithmic logic EAL as formulated and examined by the present author in [8], [9], [10]. In LCA — similarly as in EAL-ω ⁺ -valued logic is applied as a tool to construct control systems (stacks) occurring in corresponding algorithms. The aim of this paper is to give a complete axiomatization. of LCA and to prove a completeness theorem. Logic of complex algorithms was presented at FCT'79 (International Symposium on Fundamentals of Computation Theory, Berlin 1979)     

On fragments of Medvedev's logic

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that \(\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} \) theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (?a→b∨c) →(?a→b)∨(?a → c) separable?