Blending the qualitative and analytic research methods, this article critically examines the nature and limitations of inductive arguments within the field of logic. Inductive arguments, unlike their deductive counterparts, provide conclusions that extend beyond the premises, thus offering probabilistic rather than certain conclusions. This critique emphasises the weak inferential connections inherent in inductive reasoning, where premises give only partial or probable support to conclusions. The analysis highlights the ampliative value of inductive arguments, (...) illustrating how they broaden conceptual knowledge by introducing new information. However, the article also delves into reasons for their limited acceptance in logic, focusing on their probabilistic nature, contingency, experiential basis, and susceptibility to fallacies such as hasty generalisation, composition, false cause, and appeal to authority. Despite these criticisms, the article recognises that inductive arguments hold significant merit in scientific inquiry and everyday reasoning, contrasting with the firm, analytic nature of deductive arguments used in formal logic and philosophical analysis. The article concludes by advocating for a balanced approach in argumentation, underscoring the need to appreciate the contextual strengths and limitations of both inductive and deductive reasoning in the pursuit of effective logical analysis. (shrink)

This chapter attempts to establish an understanding of the concept of logic. Since there is no unanimous definition of logic, we shall consider some definitions which will facilitate our development of a working definition. This will be followed by a historical trajectory of logic as a discipline, and thereafter, an exposition of the scope and subject matter of logic, as well as the laws of thought shall be given. The aspects or divisions of logic shall also be examined here-in. This (...) chapter shall also include an overview of arguments in logic. This will be done by means of some definitions which will facilitate an understanding of the concept of argument. Since arguments are constituted by sentences which are called propositions, a distinction shall be made between sentences in the ordinary sense and in the propositional sense. This will be followed by a quick glance at the components of arguments. Inference, a term which denotes the relationship between the components of arguments, namely, premises and conclusions, shall also be considered. Also to be discussed in this chapter are: types and kinds of arguments, some instantiations of arguments, truth-value of propositions, validity and invalidity of arguments as well as sound and unsound arguments. A briefing on the usefulness of arguments to man shall also be given in this chapter. The end-point of this chapter shall be a discourse on the relevance or value of logic as it applies to our daily living. (shrink)

This chapter attempts to establish an understanding of the concept of logic. Since there is no unanimous definition of logic, we shall consider some definitions which will facilitate our development of a working definition. This will be followed by a historical trajectory of logic as a discipline, and thereafter, an exposition of the scope and subject matter of logic, as well as the laws of thought shall be given. The aspects or divisions of logic shall also be examined here-in. This (...) chapter shall also include an overview of arguments in logic. This will be done by means of some definitions which will facilitate an understanding of the concept of argument. Since arguments are constituted by sentences which are called propositions, a distinction shall be made between sentences in the ordinary sense and in the propositional sense. This will be followed by a quick glance at the components of arguments. Inference, a term which denotes the relationship between the components of arguments, namely, premises and conclusions, shall also be considered. Also to be discussed in this chapter are: types and kinds of arguments, some instantiations of arguments, truth-value of propositions, validity and invalidity of arguments as well as sound and unsound arguments. A briefing on the usefulness of arguments to man shall also be given in this chapter. The end-point of this chapter shall be a discourse on the relevance or value of logic as it applies to our daily living. (shrink)

The thesis of this paper is that we can justify induction deductively relative to one end, and deduction inductively relative to a different end. I will begin by presenting a contemporary variant of Hume ’s argument for the thesis that we cannot justify the principle of induction. Then I will criticize the responses the resulting problem of induction has received by Carnap and Goodman, as well as praise Reichenbach ’s approach. Some of these authors compare induction to deduction. Haack compares (...) deduction to induction, and I will critically discuss her argument for the thesis that we cannot justify the principles of deduction next. In concluding I will defend the thesis that we can justify induction deductively relative to one end, and deduction inductively relative to a different end, and that we can do so in a non-circular way. Along the way I will show how we can understand deductive and inductive logic as normative theories, and I will briefly sketch an argument to the effect that there are only hypothetical, but no categorical imperatives. (shrink)

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part of a (...) larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

Judaic Logic is an original inquiry into the forms of thought determining Jewish law and belief, from the impartial perspective of a logician. Judaic Logic attempts to honestly estimate the extent to which the logic employed within Judaism fits into the general norms, and whether it has any contributions to make to them. The author ranges far and wide in Jewish lore, finding clear evidence of both inductive and deductive reasoning in the Torah and other books of the (...) Bible, and analyzing the methodology of the Talmud and other Rabbinic literature by means of formal tools which make possible its objective evaluation with reference to scientific logic. The result is a highly innovative work – incisive and open, free of clichés or manipulation. Judaic Logic succeeds in translating vague and confusing interpretative principles and examples into formulas with the clarity and precision of Aristotelean syllogism. Among the positive outcomes, for logic in general, are a thorough listing, analysis and validation of the various forms of a-fortiori argument, as well as a clarification of dialectic logic. However, on the negative side, this demystification of Talmudic/Rabbinic modes of thought (hermeneutic and heuristic) reveals most of them to be, contrary to the boasts of orthodox commentators, far from deductive and certain. They are often, legitimately enough, inductive. But they are also often unnatural and arbitrary constructs, supported by unverifiable claims and fallacious techniques. Many other thought-processes, used but not noticed or discussed by the Rabbis, are identified in this treatise, and subjected to logical review. Various more or less explicit Rabbinic doctrines, which have logical significance, are also examined in it. In particular, this work includes a formal study of the ethical logic (deontology) found in Jewish law, to elicit both its universal aspects and its peculiarities. With regard to Biblical studies, one notable finding is an explicit formulation (which, however, the Rabbis failed to take note of and stress) of the principles of adduction in the Torah, written long before the acknowledgement of these principles in Western philosophy and their assimilation in a developed theory of knowledge. Another surprise is that, in contrast to Midrashic claims, the Tanakh (Jewish Bible) contains a lot more than ten instances of qal vachomer (a-fortiori) reasoning. In sum, Judaic Logic elucidates and evaluates the epistemological assumptions which have generated the Halakhah (Jewish religious jurisprudence) and allied doctrines. Traditional justifications, or rationalizations, concerning Judaic law and belief, are carefully dissected and weighed at the level of logical process and structure, without concern for content. This foundational approach, devoid of any critical or supportive bias, clears the way for a timely reassessment of orthodox Judaism (and incidentally, other religious systems, by means of analogies or contrasts). Judaic Logic ought, therefore, to be read by all Halakhists, as well as Bible and Talmud scholars and students; and also by everyone interested in the theory, practise and history of logic. (shrink)

While the empirical evidence pointing to a gender gap in professional, academic philosophy in the English-speaking world is widely accepted, explanations of this gap are less so. In this paper, we aim to make a modest contribution to the literature on the gender gap in academic philosophy by taking a quantitative, corpus-based empirical approach. Since some philosophers have suggested that it may be the argumentative, “logic-chopping,” and “paradox-mongering” nature of academic philosophy that explains the underrepresentation of women in the discipline, (...) our research questions are the following: Do men and women philosophers make different types of arguments in their published works? If so, which ones and with what frequency? Using data mining and text analysis methods, we study a large corpus of philosophical texts mined from the JSTOR database in order to answer these questions empirically. Using indicator words to classify arguments by type, we search through our corpus to find patterns of argumentation. Overall, the results of our empirical study suggest that women philosophers make deductive, inductive, and abductive arguments in their published works just as much as male philosophers do, with no statistically significant differences in the proportions of those arguments relative to each philosopher’s body of work. (shrink)

John D. Norton is responsible for a number of influential views in contemporary philosophy of science. This paper will discuss two of them. The material theory of induction claims that inductive arguments are ultimately justified by their material features, not their formal features. Thus, while a deductive argument can be valid irrespective of the content of the propositions that make up the argument, an inductive argument about, say, apples, will be justified (or not) depending on facts (...) about apples. The argument view of thought experiments claims that thought experiments are arguments, and that they function epistemically however arguments do. These two views have generated a great deal of discussion, although there hasn’t been much written about their combination. I argue that despite some interesting harmonies, there is a serious tension between them. I consider several options for easing this tension, before suggesting a set of changes to the argument view that I take to be consistent with Norton’s fundamental philosophical commitments, and which retain what seems intuitively correct about the argument view. These changes require that we move away from a unitary epistemology of thought experiments and towards a more pluralist position. (shrink)

Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. -/- Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. -/- Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. -/- The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise (...) is logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. -/- As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. -/- But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)

Drawing on the epistemology of logic literature on anti-exceptionalism about logic, we set out to investigate the following metaphilosophical questions empirically: Is philosophy special? Are its methods (dis)continuous with science? More specifically, we test the following metaphilosophical hypotheses empirically: philosophical deductivism, philosophical inductivism, and philosophical abductivism. Using indicator words to classify arguments by type (namely, deductive, inductive, and abductive arguments), we searched through a large corpus of philosophical texts mined from the JSTOR database (N = 435,703) (...) to find patterns of argumentation. The results of our quantitative, corpus-based study suggest that deductive arguments are significantly more common than abductive arguments and inductive arguments in philosophical texts overall, but they are gradually and steadily giving way to non-deductive (i.e. inductive and abductive) arguments in academic philosophy. (shrink)

Recent years have seen a surge of attention to the problem of logical pluralism; most of which has been a reaction to Beall and Restall’s account of logical pluralism as the existence of more than one equally correct semantic relation of logical consequence. The underlying thesis is that the indeterminacy of the notion of validity goes beyond what the inductive-deductive distinction can precisify. The notion of deductive validity itself is indeterminate as well and this indeterminacy has its (...) roots in the indeterminacy of the more fundamental notion of case. Cases are what make the premisses and the conclusion of an argument true; the most notable example being Tarskian models for classical logic. Deductive validity is the preservation of truth across all cases. This paper argues that unless this account of logical pluralism is supplemented with an argument in favor of the equal legitimacy of the purported cases it becomes merely a semi-controversial exposition of how different logics can be generated. (shrink)

The definitions of ‘deduction’ found in virtually every introductory logic textbook would encourage us to believe that the inductive/deductive distinction is a distinction among kinds of arguments and that the extension of ‘deduction’ is a determinate class of arguments. In this paper, we argue that that this approach is mistaken. Specifically, we defend the claim that typical definitions of ‘deduction’ operative in attempts to get at the induction/deduction distinction are either too narrow or insufficiently precise. We (...) conclude by presenting a deflationary understanding of the inductive/deductive distinction; in our view, its content is nothing over and above the answers to two fundamental sorts of questions central to critical thinking. (shrink)

Many philosophers subscribe to the view that philosophy is a priori and in the business of discovering necessary truths from the armchair. This paper sets out to empirically test this picture. If this were the case, we would expect to see this reflected in philosophical practice. In particular, we would expect philosophers to advance mostly deductive, rather than inductive, arguments. The paper shows that the percentage of philosophy articles advancing deductive arguments is higher than those (...) advancing inductive arguments, which is what we would expect from the vantage point of the armchair philosophy picture. The results also show, however, that the percentages of articles advancing deductive arguments and those advancing inductive arguments are converging over time and that the difference between inductive and deductive ratios is declining over time. This trend suggests that deductive arguments are gradually losing their status as the dominant form of argumentation in philosophy. (shrink)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of probability through a (...) heterogeneous sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

This paper deals with the study of the nature of mind, its processes and its relations with the other filed known as logic, especially the contribution of most notable contemporary analytical philosophy Ludwig Wittgenstein. Wittgenstein showed a critical relation between the mind and logic. He assumed that every mental process is logical. Mental field is field of space and time and logical field is a field of reasoning (inductive and deductive). It is only with the advancement in logic, (...) we are today in the era of scientific progress and technology. Logic played an important role in the cognitive part or we can say in the ‗philosophy of mind‘ that this branch is developed only because of three crucial theories i.e. rationalism, empiricism, and criticism. In this paper, it is argued that innate ideas or truth are equated with deduction and acquired truths are related with induction. This article also enhance the role of language in the makeup of the world of mind, although mind and the thought are the terms that are used by the philosophers synonymously but in this paper they are taken and interpreted differently. It shows the development in the analytical tradition subjected to the areas of mind and logic and their critical relation. (shrink)

Many scholars believe that "On a New List of Categories" is a metaphysical or transcendental deduction. This essay will argue that Peirce derives the categories by induction and validates their order by precision . Afterwards, the article will draw support from Peirce's youthful and mature writings to explain how the new way of listing the categories can serve as a genealogy of meaning : how different types of terms, propositions and arguments emerge in the reasoning process as different types (...) of signs. In this way, the genealogy of meaning would then qualify as both a phenomenology of logic and a science of semiotics.. Such a science of semiotics will have three types of comparison corresponding to the sign-relation in inference: namely, uniparance, diaparance and comparance. Then, the three types of comparison will give rise to three types of relative in different types of proposition: namely, concurrent, disquiparant, and equal. Finally, the three types of relative will give rise to different types of signs corresponding to different types of terms: namely, icons, indices and symbols. With this classification, then, there is an explanation of how the reasoning process is a semiotic process with three forms of valid arguments: namely, hypothesis, induction and deduction . (shrink)

In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotle’s philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek (...) culture. Both claimed there was a close relationship between the culture and mathematicians; mathematics was understood through imaginative processes, experiencing the proofs in tangible ways, and establishing a consistent unified form of argumentation. These pieces of evidence provide the context in which Aristotle worked and their contributions lend support to the argument that mathematical premises as inductively available is a better way of understanding the origins of deductive practices, opposed to the Platonic tradition. (shrink)

Logic is the study of the quality of arguments. An argument consists of a set of premises and a conclusion. The quality of an argument depends on at least two factors: the truth of the premises, and the strength with which the premises confirm the conclusion. The truth of the premises is a contingent factor that depends on the state of the world. The strength with which the premises confirm the conclusion is supposed to be independent of the state (...) of the world. Logic is only concerned with this second, logical factor of the quality of arguments. (shrink)

A key question in the philosophy of logic is how we have epistemic justification for claims about logical entailment (assuming we have such justification at all). Justification holism asserts that claims of logical entailment can only be justified in the context of an entire logical theory, e.g., classical, intuitionistic, paraconsistent, paracomplete etc. According to holism, claims of logical entailment cannot be atomistically justified as isolated statements, independently of theory choice. At present there is a developing interest in—and endorsement of—justification holism (...) due to the revival of an abductivist approach to the epistemology of logic. This paper presents an argument against holism by establishing a foundational entailment-sentence of deduction which is justified independently of theory choice and outside the context of a whole logical theory. (shrink)

The traditional No-Miracles Argument (TNMA) asserts that the novel predictive success of science would be a miracle, and thus too implausible to believe, if successful theories were not at least approximately true. The TNMA has come under fire in multiple ways, challenging each of its premises and its general argumentative structure. While the TNMA relies on explaining novel predictive success via the truth of the theories, we put forth a deductive version of the No-Miracles argument (DNMA) that avoids inference (...) to the best explanation entirely. Instead, a relatively simple empirical framework and a probabilistic analysis can accomplish the ambitious goals of the TNMA while entirely sidestepping its problems. This close-but-distinct argument has many independent strengths and comparatively few weaknesses. Indeed, objections tailored specifically to the DNMA reveal surprising insights into how exactly NMAs are neither circular nor question-begging, as has been widely speculated. (shrink)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with (...) a modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

This essay (a revised version of my undergraduate honors thesis at Stanford) constructs a theory of analogy as it applies to argumentation and reasoning, especially as used in fields such as philosophy and law. The word analogy has been used in different senses, which the essay defines. The theory developed herein applies to analogia rationis, or analogical reasoning. Building on the framework of situation theory, a type of logical relation called determination is defined. This determination relation solves a puzzle about (...) analogy in the context of logical argument, namely, whether an analogous situation contributes anything logically over and above what could be inferred from the application of prior knowledge to a present situation. Scholars of reasoning have often claimed that analogical arguments are never logically valid, and that they therefore lack cogency. However, when the right type of determination structure exists, it is possible to prove that projecting a conclusion inferred by analogy onto the situation about which one is reasoning is both valid and non-redundant. Various other properties and consequences of the determination relation are also proven. Some analogical arguments are based on principles such as similarity, which are not logically valid. The theory therefore provides us with a way to distinguish between legitimate and illegitimate arguments. It also provides an alternative to procedures based on the assessment of similarity for constructing analogies in artificial intelligence systems. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Havstad (2022) argues that the argument from inductive risk for the claim that non-epistemic values have a legitimate role to play in the internal stages of science is deductively valid. She also defends its premises and thus soundness. This is, as far as we are aware, the best reconstruction of the argument from inductive risk in the existing literature. However, there is a small flaw in this reconstruction of the argument from inductive risk which appears to render (...) the argument invalid. This flaw is superficial, and a small amendment to it rescues the claim of validity. (shrink)

The resolving of the main problem of quantum mechanics about how a quantum leap and a smooth motion can be uniformly described resolves also the problem of how a distribution of reliable data and a sequence of deductive conclusions can be uniformly described by means of a relevant wave function “Ψdata”.

Logic has a fundamental role in the philosophy of Franjo Marković (1845-1914). His theory of concepts and reasoning is analyzed, especially with respect to the essential role of the principle of sufficient reason and in connection with the concept of causality. The interplay of various types of evidence in Marković's inductive-deductive logic is analysed by means of contemporary justification logic tools.

SummaryIn this paper, two different theoretical problems of induction are delineated. The first problem is addressed; the second problem is deferred to the sequel to this paper. The first problem of induction is taken to be the seemingly unformalizable nature of traditional inductive arguments. It is shown that the problem does not arise out of some particularly dubious argument form , but rather from the presupposition that inductive “logic” is, like deductive logic, assertoric. Rather , (...) class='Hi'>inductive logic is dialectical in nature.RésuméDans cet article, deux problèmes théoriques de l'induction sont formulés. Le premier est traité, le second le sera dans une suite à cet article. Le premier problème de l'induction est ici la nature apparemment non formalisable des arguments inductifs traditionnels. L'auteur montre que le problème ne provient pas d'une forme d'argumentation particulièrement douteuse , mais au contraire de la présupposition que la «logique» inductive est, comme la logique déductive, assertorique. Il soutient au contraire qu'elle est par nature dialectique.ZusammenfassungIn der vorliegenden Arbeit werden zwei theoretische Probleme der Induktion formuliert, wovon das zweite in einem anderen Artikel behandelt werden soll. Das erste Problem betrifft die scheinbar nicht formalisierbare Natur der üblichen induktiven Argumente. Der Verfasser zeigt dass das Problem nicht von einer besonders zweifelhaften Argumentationsform herrührt , sondern von der Voraussetzung, dass die induktive «Logik» — wie die deduktive Logik — assertorisch sei. Es wird dagegen gezeigt, dass die induktive Logik dialektischer Natur ist. (shrink)

A Fortiori Logic: Innovations, History and Assessments is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all dispel the many fallacious (...) ideas circulating regarding the nature of a fortiori reasoning. -/- The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged. -/- The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah. -/- The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed. -/- This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; 4. (...) Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of (...) deduction can be found in S. Bobzien, Stoic Syllogistic, OSAP 1996. (shrink)

Philosophy is often divided into two traditions: analytic and continental philosophy. Characterizing the analytic-continental divide, however, is no easy task. Some philosophers explain the divide in terms of the place of argument in these traditions. This raises the following questions: Is analytic philosophy rife with arguments while continental philosophy is devoid of arguments? Or can different types of arguments be found in analytic and continental philosophy? This paper presents the results of an empirical study of a large (...) corpus of philosophical texts mined from the JSTOR database (n = 53,260) designed to find patterns of argumentation by type. Overall, the results suggest that there are no significant differences between the types of arguments advanced in analytic and continental philosophy journal articles. The findings, therefore, provide no empirical support to the hypothesis that the divide between analytic and continental philosophy has to do with the place of argument in these traditions. (shrink)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...) by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

In a world filled with information overload and complex problems, the ability to think logically is a superpower. "A Little More Logical" is your guide to mastering this essential skill. This engaging and accessible open educational resource is perfect for students, teachers, and lifelong learners who want to improve their critical thinking abilities and make better decisions in all aspects of life. -/- Through a series of fun and interactive chapters, "A Little More Logical" covers a wide range of topics, (...) from the basics of constructing sound arguments to the application of logic in fields like ethics, science, and computing. You'll learn how to spot and avoid common fallacies, analyze real-world arguments, and use logical reasoning to solve problems and make informed choices. -/- But this book isn't just about abstract concepts – it's about the human story behind logic. You'll discover the fascinating lives and ideas of diverse thinkers, from ancient philosophers to modern-day pioneers, who have shaped the development of logical reasoning throughout history. With thought-provoking examples, engaging exercises, and plenty of humor along the way, "A Little More Logical" makes the study of logic both rigorous and entertaining. -/- Whether you're a student looking to ace your philosophy class, a teacher searching for high-quality educational resources, or a curious reader who wants to sharpen your mind, "A Little More Logical" is the perfect companion on your journey to becoming a more effective thinker. Unlock the power of logical thinking today with this one-of-a-kind open educational resource! (shrink)

Learning from Arguments advances accessible versions of key philosophical arguments, in a form that students can emulate in their own writing, and with the primary aim of cultivating an understanding of the dynamics of philosophical argumentation. -/- The book contains ten core chapters, covering the problem of evil, Pascal’s wager, personal identity, the irrationality of fearing death, free will and determinism, Cartesian skepticism, the problem of induction, the problem of political authority, the violinist argument, the future-like-ours argument, the (...) ethics of eating meat, utilitarianism (both act and rule), and the trolley problem. Additionally, there is an introductory chapter explaining what arguments are and surveying some common argumentative strategies, an appendix on logic explaining the mechanics and varieties of valid arguments, and an appendix providing detailed advice for writing philosophy papers. -/- Each of the ten core chapters offers a sustained argument for some controversial thesis, specifically written for an audience of beginners. The aim is to introduce newcomers to the dynamics of philosophical argumentation, using some of the arguments standardly covered in an introductory philosophy course, but without the additional hurdles one encounters when reading the primary sources of the arguments: challenging writing, specialized jargon, and references to unfamiliar books, philosophers, or schools of thought. (shrink)

Logical abductivism is the epistemic view about logic according to which logical theories are justified by abduction (or Inference to the Best Explanation), that is on how well they explain the relevant evidence, so that the correct logical theory turns out to be the one that explains it best. Arguably, this view should be equally applied to both deductive and non-deductive logics, abduction included. But while there seems to be nothing wrong in principle in using abduction to determine (...) the correct logical theories of deduction and induction, things might be more complicated regarding logical theories of abduction. We may wonder whether allowing for an abductive justification of a theory of abduction is an epistemically legitimate move, since here circularity casts its shadow and makes the situation darker. This is the issue to which this work is devoted. I will defend that, to be effective, an abductive justification for a theory of abduction calls for a justification of abduction in advance, which we do not yet have. (shrink)

Filosofia limbajului are legătură cu studiul modului în care limbajul nostru se implică și interacționează cu gândirea noastră. Studierea logicii și relația dintre logică și vorbirea obișnuită poate ajuta o persoană să își structureze mai bine propriile argumente și să critice argumentele celorlalți. Înțelesul este modul în care pot fi considerate în mod corespunzător cuvinte, simboluri, idei și convingeri, definiția sa depinzând de teoria abordată, precum teoria corespondenței, teoria coerenței, teoria constructivistă, teoria consensului sau teoria pragmatică. Există mai multe explicații (...) distince despre ceea ce înseamnă un „sens” lingvistic, în funcție de diverse teorii (ideologice, adevăr-condiționale, de utilizare a limbajului, constructiviste, de referință, verificaționiste, pragmatice, etc.) Investigațiile privind modul în care limbajul interacționează cu lumea sunt numite teorii de referință. Sensul unei propoziții este gândul pe care îl exprimă. Un astfel de gând este abstract, universal și obiectiv. Sensurile determină referința și sunt, de asemenea, modurile de prezentare a obiectelor la care se referă expresiile. Referințele sunt obiectele din lume despre care vorbesc cuvintele. Filosofia limbajului explorează relația dintre limbă și realitate, în special filosofia problemelor de studiu lingvistic care nu pot fi abordate de alte domenii. Logica filozofică se ocupă de descrieri formale ale limbajului obișnuit, nespecializat („natural”). -/- CUPRINS: -/- 1. Filosofia limbajului - 1.1 Istorie - - Filosofia antică - - Filosofia medievală - - Filosofia modernă - - Filosofia contemporană - 1.2 Subiecte și sub-domenii majore - - Compoziție și părți - - Natura sensului - - Referinţă - - Mintea și limba - - - Înnăscut și învățat - - - Limba și gândul - - Interacțiunea socială și limba - 1.3 Limbajul și filozofia continentală - 1.4 Probleme în filosofia limbajului - - Imprecizia - - Problema universalului și a compoziției - - Natura limbajului - - Abordări formale versus informale - - Traducere și interpretare - 1.5 Nominalism - - Istorie - - - Filozofia greacă veche - - - Filozofia medievală - - - Filosofia modernă și contemporană - - Problema universalelor - - Tipuri - - - Filosofie analitică și matematică - - Critica originilor istorice ale termenului 2. Înțeles - 2.1 Adevăr și înțeles - - Teorii majore ale înțelesului - - - Teoria corespondenței - - - Teoria coerenței - - - Teoria constructivistă - - - Teoria consensului - - - Teoria pragmatică - - Teorii și comentarii asociate - - - Logica și limbajul - - - Gottlob Frege - - - Bertrand Russell - - - Alte teorii ale adevărului - - - Saul Kripke - - - Criticile teoriilor de adevăr ale înțelesului - 2.2 Bertrand Russell, Despre denotare - - "Fraza care denotă" - - - Concepția lui Russell despre o frază care denotă - - - Referința la ceva care nu există - - - Epistemologie - - Teoria descrierilor - - - Descrierea matematică - - - Ilustrare - - - Meinong - - Rezolvarea problemei existențialelor negative - - - Declarații despre concepte în care obiectul nu există - - - Ambiguitate - - - Nume fictive - - Critici - 2.3 Sens și referință - - Precursori - - - Antistene - - - John Stuart Mill - 2.3.1 Sens - - Sens și descriere - - Traducerea Bedeutung - - Natura sensului - 2.3.2 Referinţă - 2.4 Nume proprii - - Problema - 2.4.1 Teorii - - Teoria lui Mill - - Teoria bazată pe sensul numelor - - Teoria descriptivă - - Teoria cauzală a numelor - - Teorii de referință directă - - Filosofia continentală - 2.5 Gottlob Frege, Despre sens și referință - 2.6 Teorii cauzale ale referinței - - Motivaţie - - Variații - 2.6.1 Teoria cauzală a referinței a lui Saul Kripke - 2.6.2 Teoria cauzală a referinței a lui Gareth Evans - 2.6.3 Teoria cauzală a referinței a lui Michael Devitt - 2.6.4 Blockchain și arborele cauzal al referinței - 2.6.5 Perspective - 2.7 Saul Kripke, Numire și necesitate - 2.7.1 Prelegerea I - 2.7.2 Prelegerea II - 2.7.3 Prelegerea III - 2.7.4 Concluzii - 2.8 Kit Fine, Relaționismul semantic - - A. Antinomia variabilei - - B. Abordarea tarskiană - - C. Respingerea rolului semantic - - D. Abordarea instanțială - - E. Abordarea algebrică - - F. Abordarea relațională - - G. Semantica relațională pentru logica de primul ordin 3. Logica - Concepte - - Forma logică - - Semantică - - Inferență - - Sisteme logice - - Logică și raționalitate - - Concepte rivale - Tipuri - - Logica silogistică - - Logica propozițională - - Logica predicatelor - - Logica modală - - Raționament informal și dialectică - - Logica matematică - - Logica filozofică - - Logica computațională - - Logica non-clasică - Controverse - - "Este logica empirică?" - - Implicare: Strictă sau materială - - Tolerarea imposibilului - - Respingerea adevărului logic - 3.1 Filosofia logicii - - Adevăr - - - Purtătorii de adevăr - - - Adevăruri analitice, adevăr logic, valabilitate, consecință logică și implicare - 3.2 Logica propozițională - - Explicaţie - - Istorie - - Terminologie - - Noțiuni de bază - - - Închiderea sub operații - - - Argument - 3.3 Logica predicatelor - - Introducere - - Sintaxa - - - Alfabetul - - - - Simboluri logice - - - - Simboluri non-logice - - - Regulile formării - - - - Termeni - - - - Formule - - - - Convenții notaționale - - - Variabile libere și legate - - Semantica - - - Structuri de prim ordin - - - Evaluarea valorilor de adevăr - - - Valabilitate, satisfabilitate și consecință logică - - - Algebrizare - - - Teorii, modele și clase elementare de prim ordin - - - Domenii goale - 3.4 Logica modală - - Dezvoltarea logicii modale - - Semantica - - - Teoria modelului - - - Sisteme axiomatice - - - Teoria dovezilor structurale - - - Metode de decizie - - Tipuri de logici modale - - - Logica aletică - - - Logica epistemică - - - Logica temporală - - - Logica deontică - - - Logica doxastică - - - Alte logici modale - - Ontologia posibilității - - Controverse - 3.5 Declarații - - Declarația ca o entitate abstractă - 3.6 Dileme - - Utilizarea dilemei în logică - 3.7 Argumente - - Formal și informal - - Tipuri standard - - - Argumente deductive - - - Argumente inductive - 3.8 Actualism - - Exemplu - - Puncte de vedere filosofice - - Analiza indexicală a actualității - 3.9 Lumi posibile - - Posibilitate, necesitate și contingență - - Semantica formală a logicii modale - - De la logica modală la instrumentul filosofic - - Teoria lumii posibile în studiile literare - 3.9.1 Lumi reale, ne-reale dar posibile, și imposibile - - Lumi posibile logic - - Constituienții lumilor posibile - 3.10 Saul Kripke - - Wittgenstein - - Adevăr - 3.10.1 Logica modală - - Modele canonice - - Modele Carlson - 3.10.2 Logica intuiționistă - - Logica intuiționistă de ordinul întâi - 3.10.3 Nume și Necesitate - - "Un puzzle despre credință" Referințe Despre autor - Nicolae Sfetcu - - De același autor - - Contact Editura - MultiMedia Publishing . (shrink)

This essay presents deductive arguments to an introductory-level audience via a discussion of Aristotle's three types of rhetoric, the goals of and differences between deductive and non-deductive arguments, and the major features of deductive arguments (e.g., validity and soundness).

The main aim of this paper is to examine an almost universal assumption concerning the structure of Paley’s argument for design. Almost all commentators suppose that Paley’s argument is an inductive argument---either an argument by analogy or an argument by inference to the best explanation. I contend, on the contrary, that Paley’s argument is actually a straightforwardly deductive argument. Moreover, I argue that, when Paley’s argument is properly understood, it can readily be seen that it is no good. (...) Finally---although I do not stress this very much---I note that the points that I make about Paley’s argument can carryover to modern design arguments that are based upon the argument that Paley actually gives. (shrink)

This paper raises obvious questions undermining any residual confidence in Mates work and revealing our embarrassing ignorance of true nature of Stoic deduction. It was inspired by the challenging exploratory work of JOSIAH GOULD.

Aristotle determines eristic argument as argument which either operates upon the basis of acceptable premisses and merely give the impression of being deductive, or argument which truly is deductive but operates upon the basis of premisses which seem to be acceptable, but are not. I attempt to understand what Aristotle has in mind when he says that someone is deceived into accepting premisses which seem to be acceptable but which are really not, and how this disqualifies such (...) class='Hi'>arguments from being dialectical. In the first section of the paper I interpret Aristotle’s notion of ἔνδοξα in terms of a relational concept of acceptability. Real ἔνδοα are propositions which are accepted by a qualified group or individual. False ἔνδοα may also be accepted by someone or some group, and may even be true, but they are used to serve the purposes of eristical argumentation, which departs from certain standards of dialectical argumentation articulated in the notion of ἔνδοξα as a norm for premiss-acceptance. In particular, eristic arguments may even be valid in the sense of a συλλογισμός while still failing to be proper dialectical arguments. In the second part of the paper I consider how this can be, in examining certain types of fallacies in the Sophistical Refutations and the relationship between fallacious argumentation and false ἔνδοξα. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

An ancient argument attributed to the philosopher Carneades is presented that raises critical questions about the concept of an all-virtuous Divine being. The argument is based on the premises that virtue involves overcoming pains and dangers, and that only a being that can suffer or be destroyed is one for whom there are pains and dangers. The conclusion is that an all-virtuous Divine (perfect) being cannot exist. After presenting this argument, reconstructed from sources in Sextus Empiricus and Cicero, this paper (...) goes on to model it as a deductively valid sequence of reasoning. The paper also discusses whet her the premises are true. Questions about the possibility and value of proving and disproving the existence of God by logical reasoning are raised, as well as ethical questions about how the cardinal ethical virtues should be defined. (shrink)

According to John D. Norton's Material Theory of Induction, all inductive inferences are justified by local facts, rather than their formal features or some grand principles of nature's uniformity. Recently, Richard Dawid (Found Phys 45(9):1101–1109, 2015) has offered a challenge to this theory: in an adaptation of Norton's own celebrated "Dome" thought experiment, it seems that there are certain inductions that are intuitively reasonable, but which do not have any local facts that could serve to justify them in accordance (...) with Norton's requirements. Dawid's suggestion is that “raw induction” might have a limited but important role for such inferences. I argue that the Material Theory can accommodate such inductions, because there are local facts concerning the combinatoric features of the induction’s target populations that can licence the inferences in an analogous way to existing examples of material induction. Since my arguments are largely independent of the details of the Dome, Norton's theory emerges as surprisingly robust against criticisms of excessive narrowness. (shrink)

JOHN CORCORAN AND WILIAM FRANK. Surprises in logic. Bulletin of Symbolic Logic. 19 253. Some people, not just beginning students, are at first surprised to learn that the proposition “If zero is odd, then zero is not odd” is not self-contradictory. Some people are surprised to find out that there are logically equivalent false universal propositions that have no counterexamples in common, i. e., that no counterexample for one is a counterexample for the other. Some people would be surprised to (...) find out that in normal first-order logic existential import is quite common: some universals “Everything that is S is P” —actually quite a few—imply their corresponding existentials “Something that is S is P”. Anyway, perhaps contrary to its title, this paper is not a cataloging of surprises in logic but rather about the mistakes that did or might have or might still lead people to think that there are no surprises in logic. The paper cataloging of surprises in logic is on our “to-do” list. -/- ► JOHN CORCORAN AND WILIAM FRANK, Surprises in logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] There are many surprises in logic. Peirce gave us a few. Russell gave Frege one. Löwenheim gave Zermelo one. Gödel gave some to Hilbert. Tarski gave us several. When we get a surprise, we are often delighted, puzzled, or skeptical. Sometimes we feel or say “Nice!”, “Wow, I didn’t know that!”, “Is that so?”, or the like. Every surprise belongs to someone. There are no disembodied surprises. Saying there are surprises in logic means that logicians experience surprises doing logic—not that among logical propositions some are intrinsically or objectively “surprising”. The expression “That isn’t surprising” often denigrates logical results. Logicians often aim for surprises. In fact, [1] argues that logic’s potential for surprises helps motivate its study and, indeed, helps justify logic’s existence as a discipline. Besides big surprises that change logicians’ perspectives, the logician’s daily life brings little surprises, e.g. that Gödel’s induction axiom alone implies Robinson’s axiom. Sometimes wild guesses succeed. Sometimes promising ideas fail. Perhaps one of the least surprising things about logic is that it is full of surprises. Against the above is Wittgenstein’s surprising conclusion : “Hence there can never be surprises in logic”. This paper unearths basic mistakes in [2] that might help to explain how Wittgenstein arrived at his false conclusion and why he never caught it. The mistakes include: unawareness that surprise is personal, confusing logicians having certainty with propositions having logical necessity, confusing definitions with criteria, and thinking that facts demonstrate truths. People demonstrate truths using their deductive know-how and their knowledge of facts: facts per se are epistemically inert. [1] JOHN CORCORAN, Hidden consequence and hidden independence. This Bulletin, vol.16, p. 443. [2] LUDWIG WITTGENSTEIN, Tractatus Logico-Philosophicus, Kegan Paul, London, 1921. -/-. (shrink)

The paper provides an argument for the thesis that an agent’s degrees of disbelief should obey the ranking calculus. This Consistency Argument is based on the Consistency Theorem. The latter says that an agent’s belief set is and will always be consistent and deductively closed iff her degrees of entrenchment satisfy the ranking axioms and are updated according to the ranktheoretic update rules.

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the premises: (...) there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

In this paper I present and evaluate van Inwagen’s famous Consequence Argument, as presented in An Essay on Free Will. The grounds for the incompatibility of freewill and determinism, as argued by van Inwagen, is dependent on our actions being logical consequences of events outside of our control. Particularly, his arguments depend upon, in one guise or another, the transference of the modal property of not being possibly rendered false through the logical consequence relation, i.e. the β-principle. I argue (...) that, due to the symmetric nature of determinism, van Inwagen is exposed to what I call “reversibility arguments” in the literature. Such arguments reverse the β-principle and start from our apparent control over our own actions to our control over the initial conditions. Since van Inwagen does not endorse a particular theory of laws or logical consequence, he is open to such counterarguments. The plausibility of such reversibility arguments depends on what would be called a Wittgensteinian conception of logical consequence. In the _Remarks on the Foundation of Mathematics_, one of Wittgenstein’s main concerns is the normativity of logical inference i.e. proof. Such concerns with normativity and rule-following are generally a feature of his later philosophy. In the _Remarks_ Wittgenstein resists a conception of logical deduction which places the source of normativity outside of human practice. (shrink)

Nelson Goodman observed that we use only certain ‘good’ (viz. projectible) predicates during reasoning, with no obvious demarcation criterion in sight to distinguish them from the bad and gruesome ones. This apparent arbitrariness undermines the justifiability of our reasoning practices. Inspired by Quine’s 1969 paper on Natural Kinds, Peter Gärdenfors proposes a cognitive criterion based on his theory of Conceptual Spaces (CS). He argues the good predicates are those referring to natural concepts, and that we can capture naturalness in terms (...) of similarity. In contrast to Quine, he does not primarily rely on logic, but geometry. He frames his account as a descriptive project, however, and it is not obvious how it addresses the normative dimension of Goodman’s Riddle. This paper develops a charitable reconstruction of his argument, based on the idea that the instrumental success of our projectible concepts is grounded in their cognitive-pragmatic naturalness. It then explores three lines of reasoning against the argument: Evolutionarily motivated skepticism, the miracles argument, and the relation between instrumental and pragmatic success. I conclude that in its current form, the CS account fails to provide any justification of reasoning beyond appealing to its instrumental success, and that a metaphysically robust notion of naturalness helps to achieve the desired goal. (shrink)

Aristotle determines eristic argument as argument which either operates upon the basis of acceptable premisses (endoxa) and merely give the impression of being deductive, or argument which truly is deductive but operates upon the basis of premisses which seem to be acceptable, but are not (or, again, argument which uses both of these mechanisms). I attempt to understand what Aristotle has in mind when he says that someone is deceived into accepting premisses which seem to be acceptable but (...) which are really not, and how this disqualifies such arguments from being dialectical. In the first section of the paper I interpret Aristotle’s notion of endoxa in terms of a relational concept of acceptability. Real Índoxa are propositions which are accepted by a qualified group or individual. False endoxa may also be accepted by someone or some group, and may even be true, but they are used to serve the purposes of eristical argumentation, which departs from certain standards of dialectical argumentation articulated in the notion of endoxa as a norm for premiss-acceptance. In particular, eristic arguments may even be valid in the sense of a syllogismos while still failing to be proper dialectical arguments. In the second part of the paper I consider how this can be, in examining certain types of fallacies in the Sophistical Refutations and the relationship between fallacious argumentation and false endoxa. (shrink)

Aristotle's syllogistic theory, as developed in his Prior Analytics, is often regarded as the birth of logic in Western philosophy. Over the past century, scholars have tried to identify important precursors to this theory. I argue that Platonic division, a method which aims to give accounts of essences of natural kinds by progressively narrowing down from a genus, influenced Aristotle's logical theory in a number of crucial respects. To see exactly how, I analyze the method of division as it was (...) originally conceived by Plato and received by Aristotle. I argue that, while Plato allowed that some divisions fail to rigorously investigate the essence, he began a program continued by Aristotle (and others in antiquity and the middle ages) of seeking norms for division that would apply in any domain whatsoever. This idea of a rigorous, general method was taken up and developed by Aristotle in his syllogistic. Aristotle also used Plato's conception of predication as parthood in his semantics for syllogistic propositions. As part of my argument, I prove that a semantics based on Platonic divisional structures is sound and complete for the deduction system used in the literature to model Aristotle's syllogistic. (shrink)