This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

The conflict of narrowness and precision in direct inference occurs if a body of evidence contains estimates for frequencies in a certain reference class and less precise estimates for frequencies in a narrower reference class. To develop a solution to this conflict, I draw on ideas developed by Paul Thorn and John Pollock. First, I argue that Kyburg and Teng’s solution to the conflict of narrowness and precision leads to unreasonable direct inference probabilities. I then show that Thorn’s recent solution (...) to the conflict leads to unreasonable direct inference probabilities. Based on my analysis of Thorn’s approach, I propose a natural distribution for a Bayesian analysis of the data directly obtained from studying members of the narrowest reference class. (shrink)

The goal of this paper is to provide a detailed reading of John Venn's Logic of Chance as a work of logic or, more specifically, as a specific portion of the general system of so-called ‘material’ logic developed in his Principles of Empirical or Inductive Logic and to discuss it against the background of his Boolean-inspired views on the connection between logic and mathematics. It is by means of this situating of Venn 1866 [The Logic of Chance. An Essay on (...) the Foundations and Province of the Theory of Probability. With Especial Reference to Its Application to Moral and Social Science, London: Macmillan] within the entirety of his oeuvre that it becomes both possible to revisit and necessary to re-articulate its place in the history of the frequency interpretation of probability. For it is clear that if Venn's approach to logic not only allowed him to establish its foundations on the basis of a process of idealization and define it as consisting of so-called hypothetical infinite series, bu.. (shrink)

This paper is concerned with De Morgan’s explanation of the validity of arguments that involve relational notions. It discusses De Morgan’s expansion of traditional logic aimed at accommodating those inferences, and makes the point that his endeavour is not successful in that the rules that made up his new logic are not sound. Nevertheless, the most important scholarly work on De Morgan’s logic, and contrary to that De Morgan’s mistake is not beyond repair. The rules that determine his new logic (...) are in fact monotonie replacement rules. And provided they are restricted in the correct way, these rules are demonstrably sound. (shrink)

In attempting to form rational personal probabilities by direct inference, it is usually assumed that one should prefer frequency information concerning more specific reference classes. While the preceding assumption is intuitively plausible, little energy has been expended in explaining why it should be accepted. In the present article, I address this omission by showing that, among the principled policies that may be used in setting one’s personal probabilities, the policy of making direct inferences with a preference for frequency information for (...) more specific reference classes yields personal probabilities whose accuracy is optimal, according to all proper scoring rules, in situations where all of the relevant frequency information is point-valued. Assuming that frequency information for narrower reference classes is preferred, when the relevant frequency statements are point-valued, a dilemma arises when choosing whether to make a direct inference based upon relatively precise-valued frequency information for a broad reference class, R, or upon relatively imprecise-valued frequency information for a more specific reference class, R*. I address such cases, by showing that it is often possible to make a precise-valued frequency judgment regarding R* based on precise-valued frequency information for R, using standard principles of direct inference. Having made such a frequency judgment, the dilemma of choosing between and is removed, and one may proceed by using the precise-valued frequency estimate for the more specific reference class as a premise for direct inference. (shrink)

The applicability of Bayesian conditionalization in setting one’s posterior probability for a proposition, α, is limited to cases where the value of a corresponding prior probability, PPRI(α|∧E), is available, where ∧E represents one’s complete body of evidence. In order to extend probability updating to cases where the prior probabilities needed for Bayesian conditionalization are unavailable, I introduce an inference schema, defeasible conditionalization, which allows one to update one’s personal probability in a proposition by conditioning on a proposition that represents a (...) proper subset of one’s complete body of evidence. While defeasible conditionalization has wider applicability than standard Bayesian conditionalization (since it may be used when the value of a relevant prior probability, PPRI(α|∧E), is unavailable), there are circumstances under which some instances of defeasible conditionalization are unreasonable. To address this difficulty, I outline the conditions under which instances of defeasible conditionalization are defeated. To conclude the article, I suggest that the prescriptions of direct inference and statistical induction can be encoded within the proposed system of probability updating, by the selection of intuitively reasonable prior probabilities. (shrink)

In this paper the strategy for the eliminative reduction of the alethic modalities suggested by John Venn is outlined and it is shown to anticipate certain related contemporary empiricistic and nominalistic projects. Venn attempted to reduce the alethic modalities to probabilities, and thus suggested a promising solution to the nagging issue of the inclusion of modal statements in empiricistic philosophical systems. However, despite the promise that this suggestion held for laying the ‘ghost of modality’ to rest, this general approach, tempered (...) modal eliminativism, is shown to be inadequate for that task.<br><br>. (shrink)

In the nineteenth century, the separation of naturalist or psychological accounts of validity from normative validity came into question. In his 1877 Logical Studies (Logische Studien), Friedrich Albert Lange argues that the basis for necessary inference is demonstration, which takes place by spatially delimiting the extension of concepts using imagined or physical diagrams. These diagrams are signs or indications of concepts' extension, but do not represent their content. Only the inference as a whole captures the objective content of the proof. (...) Thus, Lange argues, the necessity of an inference is independent of psychological accounts of how we grasp the content of a proposition. (shrink)

According to the received view (Bocheński, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good reasons, it (...) overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Bocheński characterizes as ‘classical logic’, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed to provide a satisfactory (...) symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (shrink)

It is well known that Charles L. Dodgson (alias Lewis Carroll, 1832–1898) worked on a logic treatise that would popularise the subject of symbolic logic. The first part appeared in 1896 but the next parts never appeared. It has been claimed that Carroll worked in isolation and did not read the main works of his time. The object of this paper is to inquire what Carroll’s private library teaches us on his readings. The content of this library is known thanks (...) to the sale catalogues that were issued when the library was auctioned at Carroll’s death. This paper provides an overview of the logic books owned by Carroll. Then, it investigates the extent to which Carroll was acquainted with the main logic works of his time. Finally, the paper considers some methodological issues related to the use of ‘library arguments’ in intellectual history. (shrink)

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in (...) the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

Of late, probability subjectivism was resuscitated by the development of statistical decision theory. In the decision model, which is briefly described in the paper, the knowledge of a probability distribution over the states of nature plays a decisive role. What sources of probability knowledge are legitimate, or at all possible, is the main point at issue. Different definitions, evaluations, and foundations of probability are narrated, discussed, and weighed against each other. The typical research strategy of the statistician is set against (...) axiomatics of subjective or mathematical probability. Finally, the epistemological roots of the probability concept are located by the author in what he calls the “etiality principle”. (shrink)

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression the entire knowledge (...) about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800. (shrink)

From the mid-1600s to the beginning of the eighteenth century, there were two main circles of German scholars which focused extensively on diagrammatic reasoning and representation in logic. The first circle was formed around Erhard Weigel in Jena and consists primarily of Johann Christoph Sturm and Gottfried Wilhelm Leibniz; the second circle developed around Christian Weise in Zittau, with the support of his students, particularly Samuel Grosser and Johann Christian Lange. Each of these scholars developed an original form of using (...) geometric diagrams in logic. In this paper, I will trace the historical notes of John Venn and other modern logicians back to the original works published in the Weigel and Weise circles and describe the development of using geometric figures for logical reasoning and representation in that period of time. (shrink)

I. INTRODUCTION The notion of randomness has always been rather perplexing. Altho it is frequently used in natural and social science, both technically a informally, it seems to have been somewhat neglected by philosophers o science ever since the discussion of the foundations of the so-called fre- quency theory of probability, in which it was assigned a basic role, faded. Yet this discussion is of such significance that any attempt clarifying the notion of randomness will have to relate to it. (...) After a fe preliminary remarks on some of the problems and puzzles of randomne I shall, therefore, expound and discuss a concept of random distribution of a property in classes and sequences, defined in terms of relative f quencies and their limits. Because of certain shortcomings of this conce it appears advisable to turn to probabilities, in terms of which a qu different concept, viz., that of random conjunction of properties, can rea ily be defined as stochastic independence. This concept still has featu clashing with the ordinary sense of 'randomness' which become ma fest in cases where certain probabilities assume extreme values. However when we take measures defined in information theory as measuring the degree of randomness, to which purpose they lend themselves parti larly well, we find that these seemingly troublesome cases are rath harmless. A by-product of the discussion of measures of randomnes the concept of primitive randomness. The conclusion points out som further problems. (shrink)

There are many ways of attaching two objects together: for example, they can be connected, linked, tied or bound together; and the connection, link, tie or bind can be made of chain, rope, or cement. Every one of these binding methods has been used as a metaphor for causation. What is the real significance of these metaphors? They express a commitment to a certain way of thinking about causation, summarized in the following thesis: ‘In any concrete situation, there is an (...) objective fact of the matter as to whether two events are in fact bound by the causal relation. It is the aim of philosophical inquiry to analyze this objective relation.’ Through a variety of examples, I hope to cast doubt on this seemingly innocuous thesis. The problem lies not with the word ‘objective’, but with the word ‘the’. The goal of a philosophical account of causation should not be to capture the causal relation, but rather to capture the many ways in which the events of the world can be bound together. 1 The metaphors 2 Unpacking the metaphors 3 Theories of causation 4 The two assassins 5 The birth control pills 6 The smoker-protector gene 7 The bicycle thief 8 Further examples 8.1 Indeterminism 8.2 Probability-lowering causes 8.3 Parts vs wholes 8.4 Symmetric overdetermination 8.5 Delayers 8.6 Causation by omission 8.7 Double prevention/disconnection 8.8 Preemptive prevention 8.9 Quantitative variables 9 Conclusion. (shrink)

A generation before Beardsley, legal scholar John Henry Wigmore invented a scheme for representing arguments in a tree diagram, aimed to help advocates analyze the proof of facts at trial. In this essay, I describe Wigmore's "Chart Method" and trace its origin and influence. Wigmore, I argue, contributes to contemporary theory in two ways. His rhetorical approach to diagramming provides a novel perspective on problems about the theory of reasoning, premise adequacy, and dialectical obligations. Further, he advances a novel solution (...) to the problem of assessing argument quality by representing the strength of argument in meeting objections. (shrink)

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

One very popular framework in contemporary epistemology is Bayesian. The central epistemic state is subjective confidence, or credence. Traditional epistemic states like belief and knowledge tend to be sidelined, or even dispensed with entirely. Credences are often introduced as familiar mental states, merely in need of a special label for the purposes of epistemology. But whether they are implicitly recognized by the folk or posits of a sophisticated scientific psychology, they do not appear to fit well with perception, as is (...) often noted. -/- This paper investigates the tension between probabilistic cognition and non-probabilistic perception. The tension is real, and the solution—to adapt a phrase from Quine and Goodman—is to renounce credences altogether. (shrink)

Contemporary logicians continue to address problems associated with the existential import of categorical propositions. One notable problem concerns invalid instances of subalternation in the case of a universal proposition with an empty subject term. To remedy problems, logicians restrict first-order predicate logics to exclude such terms. Examining the historical origins of contemporary discussions reveals that logicians continue to make various category mistakes. We now believe that no proposition per se has existential import as commonly understood and thus it is unnecessary (...) to restrict first-order predicate logics to non-empty classes. After introducing the problem, we trace some nineteenth century treatments of the issue to locate a source of misconstruing propositional import in misconceptions of ‘implies’ and ‘affirms’ and name the process/product fallacy, along with the translation of categorical sentences using quantifiers and accommodating an empty class. Next we treat some metalogical matters to orient our discussion by which we provide a more precise nomenclature about ‘sentence’ and ‘proposition’ to correct previous misconceptions; here we uncover a common category mistake in respect of a proposition’s efficacy. The semantic distinction between agent and force is helpful in this connection. We conclude by showing that logicians have reinserted existence as a predicate, a position previously excised by Kant, and that the Frege-Russell ambiguity thesis applies only to relationships within a categorical sentence between grammatical predicate and subject. (shrink)

Venn diagram system has been extended by introducing names of individuals and their absence. Absence gives a kind of negation of singular propositions. We have offered here a non-classical interpretation of this negation. Soundness and completeness of the present diagram system have been established with respect to this interpretation.

According to the received view, Charles S. Peirce's theory of diagrammatic reasoning is derived from Kant's philosophy of mathematics. For Kant, only mathematics is constructive/synthetic, logic being instead discursive/analytic, while for Peirce, the entire domain of necessary reasoning, comprising mathematics and deductive logic, is diagrammatic, i.e. constructive in the Kantian sense. This shift was stimulated, as Peirce himself acknowledged, by the doctrines contained in Friedrich Albert Lange's Logische Studien (1877). The present paper reconstructs Peirce's reading of Lange's book, and illustrates (...) what, according to Peirce, was right and what was problematic in Lange's account of reasoning. It further seeks to explain how Peirce's theory of deductive reasoning was a combination of Kant's philosophy of mathematics and Lange's philosophy of logic. (shrink)

. In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by (...) the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it? (shrink)

Hypotheticals, conditionals, and their connecting relation, implication, dramatically changed their meanings during the nineteenth and early part of the twentieth century. Modern logicians ordinarily do not distinguish between the terms hypothetical and conditional. Yet in the late nineteenth century their meanings were quite different, their ties to the implication relation either were unclear, or the implication relation was used exclusively as a logical operator. I will trace the development of implication as an inference operator from these earlier notions into the (...) first third of the twentieth century using as the starting point the ideas primarily of R. Whately and W. Hamilton before discussing the work of the transitional logicians, A. De Morgan and G. Boole on these topics. Then we discuss the relevant views of four prominent but relatively unknown nineteenth century British logicians, W.E. Johnson, J.N. Keynes, E.E.C. Jones, and H. MacColl, as well as those of the more influential logicians, W.S. Jevons and J. Venn, closing with a section on ‘implication as inference’ where we explore some key ideas of B. Russell and sketch the work of D. Hilbert, P. Hertz, and G. Gentzen who together are responsible for the development of the modern ideas related to the subjects of this paper. (shrink)

There are two central questions concerning probability. First, what are its formal features? That is a mathematical question, to which there is a standard, widely (though not universally) agreed upon answer. This answer is reviewed in the next section. Second, what sorts of things are probabilities---what, that is, is the subject matter of probability theory? This is a philosophical question, and while the mathematical theory of probability certainly bears on it, the answer must come from elsewhere. To see why, observe (...) that there are many things in the world that have the mathematical structure of probabilities---the set of measurable regions on the surface of a table, for example---but that one would never mistake for being probabilities. So probability is distinguished by more than just its formal characteristics. The bulk of this essay will be taken up with the central question of what this “more” might be. (shrink)

It is little known that Schopenhauer (1788–1860) made thorough use of Euler diagrams in his works. One specific diagram depicts a high number of concepts in relation to Good and Evil. It is, hence, uncharacteristic as logicians of that time seldom used diagrams for more than three terms (the number demanded by syllogisms). The objective of this paper is to make sense of this diagram by explaining its function and inquiring whether it could be viewed as an early serious attempt (...) to construct complex diagrams. (shrink)

In this paper I address the question of whether the probabilities that appear in models of stochastic gene expression are objective or subjective. I argue that while our best models of the phenomena in question are stochastic models, this fact should not lead us to automatically assume that the processes are inherently stochastic. After distinguishing between models and reality, I give a brief introduction to the philosophical problem of the interpretation of probability statements. I argue that the objective vs. subjective (...) distinction is a false dichotomy and is an unhelpful distinction in this case. Instead, the probabilities in our models of gene expression exhibit standard features of both objectivity and subjectivity. (shrink)

The aim of this talk unpublished until now, and that constitutes the last contribution of the author to this topic, is to show how Hans Reichenbach’s work “Die Kausalstruktur der Welt und der Unterschied zwischen Vergangenheit und Zukunft” , published in 1925, and significantly expanded and revised in the posthumously published book in 1956, The Direction of Time, inspires substantially part of the work that is carried out in the domain of causality at the beginning of the twenty first century.

Symbolic logic faced great difficulties in its early stage of development in order to acquire recognition of its utility for the needs of science and society. The aim of this paper is to discuss an early attempt by the British logician Lewis Carroll (1832–1898) to promote symbolic logic as a social good. This examination is achieved in three phases: first, Carroll’s belief in the social utility of logic, broadly understood, is demonstrated by his numerous interventions to fight fallacious reasoning in (...) public debates. Then, Carroll’s attempts to promote symbolic logic, specifically, are revealed through his work on a treatise that would make the subject accessible to a wide and young audience. Finally, it is argued that Carroll’s ideal of logic as a common good influenced the logical methods he invented and allowed him to tackle more efficiently some problems that resisted to early symbolic logicians. (shrink)