Zur logikabh?ngigkeit wissenschafts-theoretischer Paradoxien

Investigations in meta-theoretical topics such as the definability of disposition terms or the explication of qualitative and quantitative concepts of confirmation, as well as discussions of various systems of modal logic, e.g., deontic logic, often deal with a number of well known paradoxes. In general, classical logic is used in deriving the paradox of the ravens, Goodman's paradox, the paradoxes of derived obligation, etc. The questions whether these paradoxes depend essentially on the use of classical logic and whether they can be avoided by using intuitionistic or minimal logic are considered.     

Action negation and alternative reductions for dynamic deontic logics

Dynamic deontic logics reduce normative assertions about explicit complex actions to standard dynamic logic assertions about the relation between complex actions and violation conditions. We address two general, but related problems in this field. The first is to find a formalization of the notion of ‘action negation’ that (1) has an intuitive interpretation as an action forming combinator and (2) does not impose restrictions on the use of other relevant action combinators such as sequence and iteration, and (3) has a meaningful interpretation in the normative context. The second problem we address concerns the reduction from deontic assertions to dynamic logic assertions. Our first point is that we want this reduction to obey the free-choice semantics for norms. For ought-to-be deontic logics it is generally accepted that the free-choice semantics is counter-intuitive. But for dynamic deontic logics we actually consider it a viable, if not, the better alternative. Our second concern with the reduction is that we want it to be more liberal than the ones that were proposed before in the literature. For instance, Meyer's reduction does not leave room for action whose normative status is neither permitted nor forbidden. We test the logics we define in this paper against a set of minimal logic requirements.     

Dynamic Deontic Logic and its Paradoxes

In Meyer’s promising account [7] deontic logic is reduced to a dynamic logic. Meyer claims that with his account “we get rid of most (if not all) of the nasty paradoxes that have plagued traditional deontic logic.” But as was shown by van der Meyden in [4], Meyer’s logic also contains a paradoxical formula. In this paper we will show that another paradox can be proven, one which also effects Meyer’s “solution” to contrary to duty obligations and his logic in general. Presented by Hannes Leitgeb     

The Consistency of The Naïve Theory of Properties

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics.     

An Intuitionistic Reformulation of Mally’s Deontic Logic

In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally’s deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable as a deontic logic, but it does make sense as a lax logic in the modern sense of the word.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

Minimalism and Paradoxes

This paper argues against minimalism about truth. It does so by way of acomparison of the theory of truth with the theory of sets, and considerationof where paradoxes may arise in each. The paper proceeds by asking twoseemingly unrelated questions. First, what is the theory of truth about?Answering this question shows that minimalism bears important similaritiesto naive set theory. Second, why is there no strengthened version ofRussell's paradox, as there is a strengthened Liar paradox? Answering thisquestion shows that like naive set theory, minimalism is unable to makeadequate progress in resolving the paradoxes, and must be replaced by adrastically different sort of theory. Such a theory, it is shown, must befundamentally non-minimalist.     

Theism and Dialetheism

The divine attributes of omniscience and omnipotence have faced objections to their very consistency. Such objections rely on reasoning parallel to semantic paradoxes such as the Liar or to set-theoretic paradoxes like Russell's paradox. With the advent of paraconsistent logics, dialetheism—the view that some contradictions are true—became a major player in the search for a solution to such paradoxes. This paper explores whether dialetheism, armed with the tools of paraconsistent logics, has the resources to respond to the objections levelled against the divine attributes.     

Expressivity results for deontic logics of collective agency

We use a deontic logic of collective agency to study reducibility questions about collective agency and collective obligations. The logic that is at the basis of our study is a multi-modal logic in the tradition of stit (‘sees to it that’) logics of agency. Our full formal language has constants for collective and individual deontic admissibility, modalities for collective and individual agency, and modalities for collective and individual obligations. We classify its twenty-seven sublanguages in terms of their expressive power. This classification enables us to investigate reducibility relations between collective deontic admissibility, collective agency, and collective obligations, on the one hand, and individual deontic admissibility, individual agency, and individual obligations, on the other.

     

A logic of comparative obligation

Normal systems of modal logic, interpreted as deontic logics, are unsuitable for a logic of conflicting obligations. By using modal operators based on a more complex semantics, however, we can provide for conflicting obligations, as in [9], which is formally similar to a fragment of the logic of ability later given in [2], Having gone that far, we may find it desirable to be able to express and consider claims about the comparative strengths, or degrees of urgency, of the conflicting obligations under which we stand. This paper, building on the formalism of the logic of ability in [2], provides a complete and decidable system for such a language.     

The Simple Consistency of Naive Set Theory using Metavaluations

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches.     

Moral dilemmas and nonmonotonic logic

Conclusion From a philosophical standpoint, the work presented here is based on van Fraassen [26]. The bulk of that paper is organized around a series of arguments against the assumption, built into standard deontic logic, that moral dilemmas are impossible; and van Fraassen only briefly sketches his alternative approach. His paper ends with the conclusion that the problem of possibly irresolvable moral conflict reveals serious flaws in the philosophical and semantic foundations of orthodox deontic logic, but also suggests a rich set of new problems and methods for such logic. My goal has been to suggest that some of these methods might be found in current research on nonmonotonic reasoning, and that some of the problems may have been confronted there as well.I have shown that nonmonotonic logics provide a natural framework for reasoning about moral dilemmas, perhaps even more useful than the ordinary modal framework, and that the issues surrounding the treatment of exceptional information within these logics run parallel to some of the problems posed by conditional oughts. However, there is also another way in which deontic logic might benefit from a connection to nonmonotonic reasoning. A familiar criticism among ethicists of work in deontic logic is that it is too abstract, and too far removed from the kind of problems confronted by real agents in moral deliberation. It must be said that similar criticisms of abstraction and irrelevance are often lodged against work in nonmonotonic reasoning by more practically minded researchers in artificial intelligence; but here, at least, the criticisms are taken seriously. Nonmonotonic logic aims at a qualitative account of commonsense reasoning, which can be used to relate planning and action to defeasible goals and beliefs; and at least some of the theories developed in this area have been tested in realistic situations. By linking the subject of deontic logic to this research, it may be possible also to relate the idealized study of moral reasoning typical of the field to a more robust treatment of practical deliberation.     

Deontic action logic, atomic boolean algebras and fault-tolerance

We introduce a deontic action logic and its axiomatization. This logic has some useful properties (soundness, completeness, compactness and decidability), extending the properties usually associated with such logics. Though the propositional version of the logic is quite expressive, we augment it with temporal operators, and we outline an axiomatic system for this more expressive framework. An important characteristic of this deontic action logic is that we use boolean combinators on actions, and, because of finiteness restrictions, the generated boolean algebra is atomic, which is a crucial point in proving the completeness of the axiomatic system. As our main goal is to use this logic for reasoning about fault-tolerant systems, we provide a complete example of a simple application, with an attempt at formalization of some concepts usually associated with fault-tolerance.     

Deontic logic and the possibility of moral conflict

Standard dyadic deontic logic (as well as standard deontic logic) has recently come under attack by moral philosophers who maintain that the axioms of standard dyadic deontic logic are biased against moral theories which generate moral conflicts. Since moral theories which generate conflicts are at least logically tenable, it is argued, standard dyadic deontic logic should be modified so that the set of logically possible moral theories includes those which generate such conflicts. I argue that (1) there are only certain types of moral conflicts which are interesting, and which have worried moral theorists, (2) the modification of standard dyadic deontic logic along the lines suggested by those who defend the possibility of moral conflicts makes possible only uninteresting types of moral conflicts, and (3) the general strategy of piecemeal modification standard dyadic deontic logic is misguided: the possibility of interesting moral conflicts cannot be achieved in that way.     

A qualitative examination of borderline personality disordered (BPD) patients: interpersonal dynamics and underlying paradoxes.

Theoretical conceptualizations of symptomatology in patients with borderline personality disorder (BPD) have noted an inability to integrate contradictory perceptions (splitting, or dichotomous thinking) as a hallmark of the disorder. This study investigated contradictions manifest in the thinking and behavior of BPD patients, using the concept of paradox. A paradox occurs when an apparent contradiction contains an underlying logic which makes the contradiction comprehensible. Using qualitative methods of analysis, this study explored paradoxes evident in 10 BPD patient narratives about relationship events. Specific paradoxes relating to interpersonal conflicts and self-destructiveness are presented, along with the underlying logic of each paradox as described by patients. Implications for treatment are discussed.     

On the non-availability of Dawson-modeling into certain relevance alethic modal logics

This paper shows that the Dawson technique of modelling deontic logics into alethic modal logics to gain insight into deontic formulas is not available for modelling a normal (in the spirit of Anderson) relevance deontic modal logic into either of the normal relevance alethic modal logics R _S4or R _M. The technique is to construct an extension of the well known entailment matrix set M ₀and show that the model of the deontic formula P (A v B). PA v PB is excluded.     

Combinations of tense and deontic modality: On the approach to temporal logic with historical necessity and conditional obligation

We consider three infinite hierarchies of what I call “two-dimensional temporal logics with explicit realization operators”, viz. (i) one without historical or deontic modalities, (ii) one with historical but without deontic modalities, and (iii) one with historical and with dyadic deontic modalities for conditional obligation and permission. Sound and complete axiomatizations are obtained for all three hierarchies relative to a simplified version of the finite co-ordinate-system semantics given for so-called T × W logic of historical necessity in [L. Åqvist, The logic of historical necessity as founded on two-dimensional modal tense logic, J. Philos. Logic 28 (1999) 329–369].     

No Justificatory Closure without Truth

It is well-known that versions of the lottery paradox and of the preface paradox show that the following three principles are jointly inconsistent: (Sufficiency) very probable propositions are justifiably believable; (Conjunction Closure) justified believability is closed under conjunction introduction; (No Contradictions) propositions known to be contradictory are not justifiably believable. This paper shows that there is a hybrid of the lottery and preface paradoxes that does not require Sufficiency to arise, but only Conjunction Closure and No Contradictions; and it argues that, given any plausible solution to this paradox, if one is not ready to deny Conjunction Closure (and analogous consistency principles), then one must endorse the thesis that justified believability is factive.