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 the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn.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)

This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses. This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or (...) graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature. (shrink)

The purpose of this note is to dispute Michael Ayers' claim that "there is no special problem of subjunctive conditionals".

Different inferences in probabilistic logics of conditionals 'preserve' the probabilities of their premisses to different degrees. Some preserve certainty, some high probability, some positive probability, and some minimum probability. In the first case conclusions must have probability I when premisses have probability 1, though they might have probability 0 when their premisses have any lower probability. In the second case, roughly speaking, if premisses are highly probable though not certain then conclusions must also be highly probable. In the third case (...) conclusions must have positive probability when premisses do, and in the last case conclusions must be at least as probable as their least probable premisses. Precise definitions and well known examples are given for each of these properties, characteristic principles are shown to be valid and complete for deriving conclusions of each of these kinds, and simple trivalent truthtable 'tests' are described for determining which properties are possessed by any given inference. Brief comments are made on the application of these results to certain modal inferences such as "Jones may own a car, and if he does he will have a driver's license. Therefore, he may have a driver's license.". (shrink)

This book is less about disjunction than about the English word ‘or’, and it is less for than against formal logicians—more exactly, against those who maintain that formal logic can be applied in certain ways to the evaluation of reasoning formulated in ordinary English. Nevertheless, there are many things to interest such of those persons who are willing to overlook the frequent animadversions directed against their kind in the book, and this review will concentrate on them.

Modifications of current theories of ordinal, interval and extensive measurement are presented, which aim to accomodate the empirical fact that perfectly exact measurement is not possible (which is inconsistent with current theories). The modification consists in dropping the assumption that equality (in measure) is observable, but continuing to assume that inequality (greater or lesser) can be observed. The modifications are formulated mathematically, and the central problems of formal measurement theory--the existence and uniqueness of numerical measures consistent with data--are re-examined. Some (...) results also are given on a problem which does not arise in current theories: namely that of determining limits of accuracy attainable on the basis of observations. (shrink)

How do concepts of topology such as that of a boundary apply to the empirical world? Take the example of a chess board, represented here with black squares in black and red squares in white. We see by looking at the board that the squares of any one color have common boundaries only with squares of the opposite color, but each square has corners in common with other squares of the same color, which are points at which their common boundaries (...) intersect. For example, the white square labelled ‘A’ has common boundaries with the black squares that surround it, and common corners with the white squares like square B that are diagonally adjacent to it. (shrink)

. Syllogisms like Barbara, “If all S is M and all M is P, then all S is P”, are here analyzed not in terms of the truth of their categorical constituents, “all S is M”, etc., but rather in terms of the corresponding proportions, e.g., of Ss that are Ms. This allows us to consider the inferences’ approximate validity, and whether the fact that most Ss are Ms and most Ms are Ps guarantees that most Ss are Ps. It (...) turns out that no standard syllogism is universally valid in this sense, but special ‘default rules’ govern approximate reasoning of this kind. Special attention is paid to inferences involving existential propositions of the “Some S is M” form, where it is does not make sense to say “Almost some S is M”, but where it is important that in everyday speech, “Some” does not mean “At least one”, but rather “A not insignificant number”. (shrink)

Aspects of a formal theory of approximate generalizations, according to which they have degrees of truth measurable by the proportions of their instances for which they are true, are discussed. The idealizability of laws in theories of fundamental measurement is considered: given that the laws of these theories are only approximately true "in the real world", does it follow that slight changes in the extensions of their predicates would make them exactly true?

That inferences of the form "If M then S and possibly M, therefore possibly S" are invalid in possible worlds modal logics can be viewed as another fallacy of material implication. However, this paper argues that properly analyzing this and related inferences requires treating the possibility involved as a practical modality. Specifically, ordinary language propositions of the form "It is possible that M" must be understood to mean that there is a non-negligible probability of M being the case. But this (...) entails reexamining the very idea of logical validity. (shrink)

I suppose that 'ravens are black' is an inexact generalization having a degree of truth measured by the proportion of ravens that are black, and a probability measured by its expected degree of truth in different 'possible worlds.' Given this, 'ravens are black' differs in truth, probability, and confirmation from 'non-black things are not ravens', and this suggests a new approach to Hempel's Paradox as well as to other aspects of confirmation. Basic concepts of a formal theory developing this approach (...) are described, and some of its fundamental laws are given, together with sketches of applications. (shrink)

An inexact generalization like ‘ravens are black’ will be symbolized as a prepositional function with free variables thus: ‘Rx ⇒ Bx.’ The antecedent ‘Rx’ and consequent ‘Bx’ will themselves be called absolute formulas, while the result of writing the non-boolean connective ‘⇒’ between them is conditional. Absolute formulas are arbitrary first-order formulas and include the exact generalization ‘(x)(Rx → Bx)’ and sentences with individual constants like ‘Rc & Bc.’ On the other hand the non-boolean conditional ‘⇒’ can only occur as (...) the main connective in a formula. We shall also need to consider formulas with more than one free variable such as ‘xHy ⇒ xTy,’ which might express ‘if x is the husband of y then x is taller than y.’ Though it is inessential, it will simplify things to work in ‘n-languages’ with a finite number of individual constants c1,…, cn, which are interpreted as denoting the elements of the domains of the ‘n-models’ to be described below. (shrink)

An informal theory is set forth of relations between abstract entities, includingcolors, physical quantities, times, andplaces in space, and the concrete things thathave them, or areat orin them, based on the assumption that there are close analogies between these relations and relations between abstractsets and the concrete things that aremembers of them. It is suggested that even standard scientific usage of these abstractions presupposes principles that are analogous to postulates of abstraction, identity, and other fundamental principles of set theory. Also (...) discussed is the significance of important disanalogies between sets and physical abstractions, including especiallymodal andtemporal aspects of physical abstractions, which is related to the problem of the characterizingconstancy, of colors, physical attributes, and locations in space. (shrink)

Grice's arguments that ordinary language indicative conditionals are logically equivalent to material conditionals are criticized. It is agreed that 'indirectness conditions' going beyond the material conditional can "sometimes" be detached' from ordinary language conditionals, but it is argued that this is not always possible. An example in which a speaker who knows that some mushrooms are non-poisonous tells a hearer "if you eat those mushrooms you will be poisoned", causing the hearer not to eat the mushrooms, is discussed, and it (...) is argued that this utterance should be regarded as factually unsatisfactory', and therefore, by Grice's own standards, false. (shrink)

Applying first-order logic to derive the consequences of laws that are only approximately true of empirical phenomena involves idealization of a kind that is akin to applying arithmetic to calculate the area of a rectangular surface from approximate measures of the lengths of its sides. Errors in the data, in the exactness of the lengths in one case and in the exactness of the laws in the other, are in some measure transmitted to the consequences deduced from them, and the (...) aim of a theory of idealization is to describe this process. The present paper makes a start on this in the case of applied first-order logic, and relates it to Plato's picture of a world or model of 'appearances' in which laws are only approximately true, but which to some extent resembles an ideal world or model in which they are exactly true. (shrink)