Entailment with Near Surety of Scaled Assertions of High Conditional Probability

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z ⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL.     

Paraconsistency in Categories: Case of Relevance Logic

Categorical-theoretic semantics for the relevance logic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.     

Nonmonotonic probabilistic reasoning under variable-strength inheritance with overriding

We present new probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment, called Zλ- and lexλ-entailment, which are parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of λ = 0 and λ = 1, the notions of Zλ- and lexλ-entailment coincide with probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment that have been recently introduced by the author. We show that the notions of Zλ- and lexλ-entailment have similar properties as their classical counterparts. In particular, they both satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, Zλ-entailment is weaker than lexλ-entailment, and both Zλ- and lexλ-entailment are proper generalizations of their classical counterparts.     

The Power of Belnap: Sequent Systems for <Emphasis Type="BoldItalic">SIXTEEN</Emphasis><Subscript><Emphasis Type="Bold">3</Emphasis></Subscript>

The trilattice SIXTEEN₃\textit{SIXTEEN}_3 is a natural generalization of the well-known bilattice FOUR₂\textit{FOUR}_2. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃\textit{SIXTEEN}_3 are presented.     

Should knowledge entail belief?

The appropriateness of S5 as a logic of knowledge has been attacked at some length in the philosophical literature. Here one particular attack based on the interplay between knowledge and belief is considered: Suppose that knowledge satisfies S5, belief satisfies KD45, and both the entailment property (knowledge implies belief) and positive certainty (if the agent believes something, she believes she knows it) hold. Then it can be shown that belief reduces to knowledge: it is impossible to have false beliefs. While the entialment property has typically been viewed as perhaps the least controversial of these assumptions, an argument is presented that it can plausibly be viewed as the culprit. More precisely, it is shown that this attack fails if we weaken the entailment property so that it applies only to objective (nonmodal) formulas, rather than to arbitrary formulas. Since the standard arguments in favor of the entailment property are typically given only for objective formulas, this observation suggests that care must be taken in applying intuitions that seem reasonable in the case of objective formulas to arbitrary formulas.Research sponsored in part by the Air Force office of Scientific Research (AFSC), under Contract F49620-91-C-0080. The United States Government is authorized to reproduce and distribute reprints for governmental purposes.     

A Logic For Inductive Probabilistic Reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.     

Relational Science: A Synthesis

A synthesis of the two primary theory structures in Robert Rosen’s relational complexity, (1) relational entailment mapping based on category theory as described by Rosen and Louie, and (2) relational holism based on modeling relations, as described by Kineman, provides an integral foundation for relational complexity theory as a natural science and analytical method. Previous incompatibilities between these theory structures are resolved by re-interpreting Aristotle’s four causes, identifying final and formal causes as relations with context. Category theory is applied to introduce contextual entailment algebra needed to complete the synthesis. The modeling relation is represented as a recursive four-cause hierarchy, which is a unit of both whole and part analysis (a ‘holon’) that relates realized and contextual domains of nature as complementary inverse entailments between structure and function. Context is a non-localized domain of distributed potentials (models) for existence, as contrasted with the realized domain of localized interactive and measurable events. Synthesis is achieved by giving modeling relations an algebraic form in category theory and by expanding relational analysis to include contextual entailments. The revised form of analysis is applied and demonstrated to examine Rosen’s M-R diagram, showing that structure–function relations imply adaptive interaction with the environment, and that contextual relations imply three forms of the M-R entailment corresponding with the generally known three forms of life; Archaea, Bacteria, and Eukaryota, which can be represented by their holon diagrams. The result of this synthesis is a consistent foundation for relational science that should have important implications in many disciplines.     

Process Reliabilism,Virtue Reliabilism,and the Value of Knowledge

The value problem for knowledge is the problem of explaining why knowledge is cognitively more valuable than mere true belief. If an account of the nature of knowledge is unable to solve the value problem for knowledge, this provides a pro tanto reason to reject that account. Recent literature argues that process reliabilism is unable to solve the value problem because it succumbs to an objection known as the swamping objection. Virtue reliabilism (i.e., agent reliabilism), on the other hand, is able to solve the value problem because it can avoid the swamping objection. I argue that virtue reliabilism escapes the swamping objection only by employing what I call an entailment strategy. Furthermore, since an entailment strategy is open to the process reliabilist (in two different forms), I argue that the process reliabilist is also able to escape the swamping objection and thereby solve the value problem for knowledge.     

Alternative semantics for quantified first degree relevant logic

A system FDQ of first degree entailment with quantification, extending classical quantification logic Q by an entailment connective, is axiomatised, and the choice of axioms defended and also, from another viewpoint, criticised. The system proves to be the equivalent to the first degree part of the quantified entailmental system EQ studied by Anderson and Belnap; accordingly the semantics furnished are alternative to those provided for the first degree of EQ by Belnap. A worlds semantics for FDQ is presented, and the soundness and completeness of FDQ proved, the main work of the paper going into the proof of completeness. The adequacy result is applied to yield, as well as the usual corollaries, weak relevance of FDQ and the fact that FDQ is the common first degree of a wide variety of (constant domain) quantified relevant logics. Finally much unfinished business at the first degree is discussed.     

Distributive Terms,Truth, and the Port Royal Logic

The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension.     

A Lewisian Semantics for S2

This paper sets out a semantics for C.I. Lewis's logic S2 based on the ontology of his 1923 paper ‘Facts, Systems, and the Unity of the World’. In that article, worlds are taken to be maximal consistent systems. A system, moreover, is a collection of facts that is closed under logical entailment and conjunction. In this paper, instead of defining systems in terms of logical entailment, I use certain ideas in Lewis's epistemology and philosophy of logic to define a class of models in which systems are taken to be primitive elements but bear certain relations to one another. I prove soundness and completeness for S2 over this class of models and argue that this semantics makes sense of at least a substantial fragment of Lewis's logical theory.     

Neighborhoods for Entailment

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood semantics meets the Routley–Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable.     

Projective identification and the therapist's use of self

This article addresses how therapists can recognize and therapeutically utilize client projective identification in therapy. Splitting and projective identification are conceptualized as occurring in sequence. Transference, countertransference, and projective identification are discussed as interrelated phenomena, which underlie the therapist's effective use of self as an instrument in therapy. A three phase intervention process, which emphasizes the importance of the timing of interventions, is introduced.     

Domination and Dialogue in Merleau-Ponty's Phenomenology of Perception

Merleau-Ponty's claim in Phenomenology of Perception (1962) that the anonymous body guarantees an intersubjective world is problematic because it omits the particularities of bodies. This omission produces an account of “dialogue” with another in which I solipsistically hear only myself and dominate others with my intentionality. This essay develops an alternative to projective intentionality called “hypothetical construction,” in which meaning is socially constructed through an appreciation of the differences of others.     

Toward Development of a Projective Set

An exploration of the possibility of developing a projective set in a client that will allow for lowering of his defenses and thus possibly eliciting “richer projective materials.” A “anxious tension” set was produced in one group of Ss by means of an intelligence type test and “playful tension” was produced in another group by means of a fantasy task. A projective technique (Part A of the Rosenzweig F-Battery) was then given to both groups. The results indicate that this projective technique was taken at a more implicit (or deeper) level following the “playful tension” than following the “anxious tension.” Implications for the pre-test environment of projective techniques are given.     

Consequentialism and the World in Time

Consequentialism is a general approach to understanding the nature of morality that seems to entail a certain view of the world in time. This entailment raises specific problems for the approach. The first seems to lead to the conclusion that every actual act is right – an unacceptable result for any moral theory. The second calls into question the idea that consequentialism is an approach to morality, for it leads to the conclusion that this approach produces a theory whose truth does not depend in any way on the nature of rational beings or value.     

Naive Modus Ponens and Failure of Transitivity

In the recent paper “Naive modus ponens”, Zardini presents some brief considerations against an approach to semantic paradoxes that rejects the transitivity of entailment. The problem with the approach is, according to Zardini, that the failure of a meta-inference closely resembling modus ponens clashes both with the logical idea of modus ponens as a valid inference and the semantic idea of the conditional as requiring that a true conditional cannot have true antecedent and false consequent. I respond on behalf of the non-transitive approach. I argue that the meta-inference in question is independent from the logical idea of modus ponens, and that the semantic idea of the conditional as formulated by Zardini is inadequate for his purposes because it is spelled out in a vocabulary not suitable for evaluating the adequacy of the conditional in semantics for non-transitive entailment. I proceed to generalize the semantic idea of the conditional and show that the most popular semantics for non-transitive entailment satisfies the new formulation.     

Gentzenization of Trilattice Logics

Sequent calculi for trilattice logics, including those that are determined by the truth entailment, the falsity entailment and their intersection, are given. This partly answers the problems in Shramko-Wansing (J Philos Logic 34:121–153, 2005).     

A Naturalist Ontology of Instantiation

The aim of this paper is to defend a naturalistic approach to instantiation and the Principle of Instantiation. I argue that the instantiation of an ordinary property F consists of two coordinated relationships at the levels of the manifest and scientific images, namely, constituency and entailment. Also, I offer an account of the Principle of Instantiation related to this conception of instantiation based on the notion of scientific prediction.