Why we need a relevant theory of conditionals (original) (raw)
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Foundations of conditional logic
Journal of Philosophical Logic, 1984
Conditional statements occupy a central place in reasoning, and hence their proper analysis is a principal task of logic. Now, ever since material implication was proposed, and found wanting as an explication, new analyses of conditionality have been put forward by logicians and philosophers. The resulting variety of formal explications itself raises several background questions, and it is to this foundational theme that the present paper is devoted. General questions concerning existing semantic accounts of the meaning of conditionals are exemplified by the following. About the language of conditional statements, should conditionality be treated as an operation upon propositions, or rather as a relation between these? As for the semantic apparatus, how can one judge the need for, or the relative merits of the various types of model and truth definition proposed in the literature? Finally, with respect to the 'logical evidence', what is the status of the intuitions of validity, often invoked as a touch-stone for the conditional logic resulting from some particular analysis? These are issues which may give rise to lively, but also inconclusive philosophical debate. For instance, operational and relational views of conditionals both have their adherents, and some people even entertain both, to the point of confusing object-language and metalanguage of their formalization. (This is the familiar criticism of C. I. Lewis' account of entailment; mentioned, e.g., in Scott (1971).) To mention another example, the validity of a principle such as Conditional Excluded Middle ('if X, then Y, or, if X, then not Y') has strong intuitive support, but also provokes grave doubts. .. sometimes within the same observer. What we need, then, is a general unifying perspective, enabling us to arrive at more definite issues and results. Definite, not in the sense of a universal settling of old scores, but of establishing the true logical relations between various options. For instance, argument about the validity of some
Towards a New Logic of Indicative Conditionals
In this paper I will propose a refinement of the semantics of hypervaluations (Mura 2009), one in which a hypervaluation is built up on the basis of a set of valuations, instead of a single val-uation. I shall define validity with respect to all the subsets of valua-tions. Focusing our attention on the set of valid sentences, it may easily shown that the rule substitution is restored and we may use valid schemas to represent classes of valid sentences sharing the same logical form. However, the resulting semantical theory TH turns out to be throughout a modal three-valued theory (modal sym-bols being definable in terms of the non modal connectives) and a fragment of it may be considered as a three-valued version of S5 system. Moreover, TH may be embedded in S5, in the sense that for every formula ϕ of TH there is a corresponding formula ϕ' of S5 such that ϕ' is S5-valid iff ϕ is TH-valid. The fundamental property of this system is that it allows the definition of a purely semantical relation of logical consequence which is coextensive to Adams’ p-entailment with respect to simple conditional sentences, without be-ing defined in probabilistic terms. However, probability may be well be defined on the lattice of hypervaluated tri-events, and it may be proved that Adam’s p-entailment, once extended to all tri-events, coincides with our notion of logical consequence as defined in purely semantical terms.
The Inextricable Link Between Conditionals and Logical Consequence
There is a profound, but frequently ignored relationship between logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same kind of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) the traditional examples of valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound.
Axioms for a logic of consequential counterfactuals
2018
In §1 it is shown that Angell, Reichenbach, Chisholm and Goodman agreed in giving legitimacy to the (weak) Boethius' Thesis A → B -> -(A → -B) (BT), but did not perceive the difference between the properties of analytic and context-dependent (synthetic) conditionals, mainly exemplified by counterfactual conditionals. The basic idea of the paper is to define, into a logic which has BT as a basic axiom, synthetic conditionals in terms of the analytic ones and of the so-called "circumstantial" operator *, whose properties are derived from the properties of selection functions in the way shown in §2. Systems of analytic consequential implication, differing from the ones of connexive implication, are translatable into systems of normal modal logic via the definitions A → B =df □(A -> B) & X (A,B) (where X(A,B) stands for □A≡□B & <>A ≡ <> B) and □ A=df T → A. This fact allows defining the synthetic operators in terms of → and *, or alternatively in terms of □ and *. Plausible definitions of the synthetic operator such as A >' B =df *A → B and A > B =df *A => B (=> being a weak variant of →) are examined and rejected. The definitive proposal is to introduce, in a system named CI.0*Eq, a new operator >> by the definition A >> B = df □(*A -> B) & X (A,B). It is proved that the operator defined in this way satisfies all the properties required for the synthetic consequential operators, in the first place BT in the variant A >> B -> -(A >> -B). § 3 is devoted to identifying the >>-fragment of CI.0*Eq in the following way. A system CI.0>>, written in → and >> , is proved to be complete with respect to models <W, f, R,V>, where f is a selection function. It is then proved that there is an embedding function t from CI.0>> to CI.0*Eq, and beyond this another one, Tr, from CI.0*Eq to KT w , where KT w is a tableaux-decidable extension of the modal system KT written in a language which contains a new operator w which allows translating *A into w(A) & A. So there is a composed embedding function Trt of CI.0>> into KT w. On this basis, via a conversion of CI.0>>-models into KT w-models, it is proved (i) that A is a CI.0-thesis iff Tr t(A) is a KT w-thesis and (ii) tA is a CI.0*Eq-thesis iff A is a CI.0>>-thesis, which establishes the desired result. Thanks to Trt CI.0>> turns out then to be a tableaux-decidable system. The final section contains a discussion of Simplification of Disjunctive Antecedents and of other laws laws which mark important differences between classical and consequential conditional logic.
A. Introduction. Two key concepts and three perspectives in deontic logic. Situation, agency and agent oriented deontic theories. B. Concept of norm and its structure revisited. Norms as atomic and molecular entities. C. Two approaches to deontic logic: deterministic, focusing on alethic \ deontic regularities, and indeterministic, viewing agentive choice and alternative lines of behavior. D. Conclusion. Three faces of deontic logic pursue different perspective of normative codes' analysis, reasoning about norms and agentive behavior with preferences \ priorities accordingly.
Some Observations on Carlos Alchourrón’s Theory of Defeasible Conditionals
In this paper we review some aspects of the theory of defeasible conditionals that the late Carlos Alchourrón developed in the last years of his life. These include both philosophical intuitions and formal features of his theory. In particular, we discuss the concept of a contributory condition used by Alchourrón, his formalization of the notion of Prima facie duty and the connection between his theory of defeasible conditionals and the AGM logic of theory change.
1981
In a recent paper, Sven Danielsson argued that the 'original paradoxes' of deontic logic, in particular Ross's paradox and Prior's paradox of derived obligation, can be solved by restricting the modal inheritance rule. I argue that this does not solve the paradoxes.
A Conditional Logical Framework
Lecture Notes in Computer Science, 2008
The Conditional Logical Framework LF K is a variant of the Harper-Honsell-Plotkin's Edinburgh Logical Framemork LF. It features a generalized form of λ-abstraction where β-reductions fire under the condition that the argument satisfies a logical predicate. The key idea is that the type system memorizes under what conditions and where reductions have yet to fire. Different notions of β-reductions corresponding to different predicates can be combined in LF K. The framework LF K subsumes, by simple instantiation, LF (in fact, it is also a subsystem of LF!), as well as a large class of new generalized conditional λ-calculi. These are appropriate to deal smoothly with the side-conditions of both Hilbert and Natural Deduction presentations of Modal Logics. We investigate and characterize the metatheoretical properties of the calculus underpinning LF K , such as subject reduction, confluence, strong normalization.
Proceedings of the 18th International Joint Conference on Artificial Intelligence, 2003
We introduce a logical formalism of irreflexivc causal production relations that possesses both a standard monotonic semantics, and a natural nonmonotonic semantics. The formalism is shown to provide a complete characterization for the causal reasoning behind causal theories from [McCain and Turner, 1997]. It is shown also that any causal relation is reducible to its Horn sub-relation with respect to the nonmonotonic semantics. We describe also a general correspondence between causal relations and abductive systems, which shows, in effect, that causal relations allow to express abductive reasoning. The results of the study seem to suggest causal production relations as a viable general framework for nonmonotonic reasoning.