THE INDICATIVE CONDITIONAL. A PROBABILISTIC CRITERION OF SOUNDNESS FOR DEDUCTIVE INFERENCES Our objective in this section is to establish a prima facie case ...

The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the (...) premises in the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

It is argued that the "inner" negation $\mathord{\sim}$ familiar from 3-valued logic can be interpreted as a form of "conditional" negation: $\mathord{\sim}$ is read '$A$ is false if it has a truth value'. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g., "If Jim went to the party he had (...) a good time"). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from $A \rightarrow \nega B$ to $\nega(A\rightarrow B)$ to. (shrink)

The logic of indicative conditionals remains the topic of deep and intractable philosophical disagreement. I show that two influential epistemic norms—the Lockean theory of belief and the Ramsey test for conditional belief—are jointly sufficient to ground a powerful new argument for a particular conception of the logic of indicative conditionals. Specifically, the argument demonstrates, contrary to the received historical narrative, that there is a real sense in which Stalnaker’s semantics for the indicative did succeed in capturing (...) the logic of the Ramseyan indicative conditional. (shrink)

entry for the Entry for the Stanford Encyclopedia of Philosophy, 2007.

Conditional statements are ubiquitous, from promises and threats to reasoning and decision making. By now, logicians have studied them from many different angles, both semantic and proof-theoretic. This paper suggests two more perspectives on the meaning of conditionals, one dynamic and one geometric, that may throw yet more light on a familiar and yet in some ways surprisingly elusive and many-faceted notion.1.

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

In some recent works, Crupi and Iacona have outlined an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.

This article begins by outlining some of the history—beginning with brief remarks of Quine's—of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. (...) The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it. (shrink)

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. (...) counterfactuals) to the exotic (e.g. conditional obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- (...) ditional counterparts) cannot generate a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establishthe minimum system $C2L_{m}$ of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of $C2L_{m}$ , introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of $C2L_{m}$ by the canonical model. Finally, according to the technical results of the minimum (...) system $C2L_{m}$ , the authors discuss some of the important problems concerning primary-conditionals and secondary- onditionals. (shrink)

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an (...) algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates. (shrink)

This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.

In order to resolve the controversial discussion regarding content effects in deductive reasoning, we propose distinguishing between two inferential sources—an argument's form , and additional relations people associate with the argument's content —and analysing their interplay. Both sources are equally necessary in order to understand the role content plays in deductive reasoning. People make valid deductions from the content relations ( content competence ), but in thematic reasoning tasks, these deductions lead to the intriguing phenomenon known as content effects . (...) Focusing on the interplay of both sources of inferences, the dual source distinction enables a novel class of predictions to be made, namely the correct mastery of the logical connectors ( form competence ) in tasks that require the individual to think about an argument's form in relation to its content. To illustrate the dual source approach and its implications, the selection task paradigm of conditional reasoning with a point of view is used in combination with two content domains: conditional promises with cheating and non-cheating perspectives and technical systems with causal perspectives. Experimental findings corroborate all three phenomena: content competence, content effects, and form competence. The dual source distinction is discussed with regard to current theories of reasoning. (shrink)

In this work Ockham proposes a theory of simple predication, which he uses in explicating the truth conditions of progressively more complicated kinds of propositions. His discussion includes what he takes to be the correct semantic treatment of quantified propositions, past tense and future tense propositions, and modal propositions, all of which are receiving much attention from contemporary philosophers. He also illustrates the use of exponential analysis to deal with propositions that prove troublesome in both semantic theory and other disciplines, (...) such as metaphysics, physics, and theology. This type of analysis plays an essential role in his substantive philosophical and theological works, and in many cases then can hardly be understood without a prior acquaintance with this section of the Summa. An introduction by Alfred J. Freddoso clarifies and summarizes Ockham's theory, discusses certain philosophical problems which it engenders, and proposes new interpretations for parts of it. The essay will be helpful both to those interested in Ockham as a historical figure and to those interested in his substantive systematic contributions to logical theory. (shrink)

In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it (...) is not truth functionality as such that is the problem, but the philosophical foundation of modern logic. (shrink)

This paper is taken from a long section of the _Tattva-cintā-maṇi_ by Gaṅgeśa that is devoted to proving the existence of—to use an inadequate word—“God” in a somewhat minimalist sense. The _īśvara_, the “Lord,” is for Gaṅgeśa, following Nyāya predecessors, a divine agent, a self, responsible for much, not all, of the order in the world. Unseen Force, _adṛṣṭa_, which is in effect _karman_ made by human action, is also a powerful agent as well as things’ intrinsic natures. Moreover, ordinary (...) selves, atoms, ether, and universals are uncreated. But the _īśvara_ brings about just desert in reincarnation in actualizing Unseen Force, and is responsible for a broad swathe of what some see as accidental arrangements as well as the forming of the macro elements from eternal, naturally _dis_joined atoms. Thus the cosmos in its general existence and structure is viewed in Nyāya as the work of the Lord. Gaṅgeśa’s argument runs: Earth and the like [_a_ (_pakṣa_) = earth and the like (_kṣity-ādi_)] have a conscious agent as a cause [S (_sādhya_) = having an agential cause (_sakartṛkatva_) (S_a_)], _since_ they are effects [H (_sādhana_) = being an effect [(_kāryatva_) (H_a_)], like a pot [_b_ (_dṛṣṭānta_) = a pot (H_b_,S_b_)]. And so the _vyāpti_ rule is: [H → S (_vyāpti_)] Whatever is an effect has an agential cause. For earth and the like, it is reasoned that only an omniscient _īśvara_ could be that cause. The argument was a target of Buddhists who pointed to counterexamples such as growing grass. Growing grass exhibits the prover property, being-an-effect, but not the property to be proved, having-an-agential-cause. The long section is dominated by Gaṅgeśa’s rebutting this and other potential defeaters, in particular, the _upādhi_, having-a-living-body (God does not have a living body but all the agential causes with which we are familiar do), along with a counterinference (_sat-pratipakṣa_), I_a_ & (x)(Ix → ¬Sx), where I = not-produced-by-an-agent-with-a-body. (shrink)

Discontented people might talk of corruption in the Commons, closeness in the Commons and the necessity of reforming the Commons, said Mr. Spenlow solemnly, in conclusion; but when the price of wheat per bushel had been the highest, the Commons had been the busiest; and a man might lay his hand upon his heart, and say this to the whole world, – ‘Touch the Commons, and down comes the country!’.

This paper shows that the non-Boolean logic of quantum measurements is more naturally represented by a relatively new 4-operation system of Boolean fractions—conditional events—than by the standard representation using Hilbert Space. After the requirements of quantum mechanics and the properties of conditional event algebra are introduced, the quantum concepts of orthogonality, completeness, simultaneous verifiability, logical operations, and deductions are expressed in terms of conditional events thereby demonstrating the adequacy and efficacy of this formulation. Since conditional event algebra is nearly (...) Boolean and consists merely of ordered pairs of standard events or propositions, quantum events and the so-called “superpositions” of states need not be mysterious, and are here fully explicated. Conditional event algebra nicely explains these non-standard “superpositions” of quantum states as conjunctions or disjunctions of conditional events, Boolean fractions, but does not address the so-called “entanglement phenomena” of quantum mechanics, which remain physically mysterious. Nevertheless, separating the latter phenomena from superposition issues adds clarity to the interpretation of quantum entanglement, the phenomenon of influence propagated at faster than light speeds. With such treacherous possibilities present in all quantum situations, an observer has every reason to be completely explicit about the environmental–instrumental configuration, the conditions present when attempting quantum measurements. Conditional event algebra allows such explication without the physical and algebraic remoteness of Hilbert space. (shrink)

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, →, in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, 'C → B' holds just in case P[B | (...) C] ≥ r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti and Reichenbach. On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the (...) predicament: one pairs de Finetti’s conditional with validity as the preservation of non-false values, but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table proposed by Cooper and Cantwell. In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties. (shrink)

The aim of the essay is to devise a logic of conditionality which escapes the paradoxes which arise when the general conditional is identified with the universalization of the material conditional. The assumption I adopt is that the logic of one contingent form differs from that of another to the extent that the two forms have different confirmations and disconfirmations. The logic of conditionals is not, But that of their confirmations and disconfirmations is, At bottom truth-Functional; (...) and the logical relations between conditionals, And between conditionals and non-Conditionals, May be traced through the truth-Functional connections between their respective confirmations and disconfirmations. The approach necessitates the re-Definition of the notions of confirmation and entailment. A peculiarity of the system which emerges is that from it can be derived, Without existential assumptions, The twenty-Four valid forms of the syllogism, The relationships of the square of opposition and the other commonly accepted inferences of traditional logic. (shrink)

When one thinks—knows, believes, imagines—that something is the case, one’s thought has a topic: it is about something, towards which one’s mind is directed. What is the logic of thought, so understood? This book begins to explore the idea that, to answer the question, we should take topics seriously. It proposes a hyperintensional account of the propositional contents of thought, arguing that these are individuated not only by the set of possible worlds at which they are true, but also (...) by their topic: what they are about. The book then builds epistemic, doxastic, probabilistic, and conditional logics based on this view. It applies them to issues ranging from dogmatism, scepticism, and epistemic fallibilism, to imagination and suppositional reasoning, belief revision, framing effects, and the acceptability of indicative conditionals. (shrink)