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)

Terms of logic like 'all' and 'some' can be understood in relation to one another purely in thought. Individual objects of which one becomes aware by sense perception are not objects of pure thought. Hence they cannot be uniquely named. Since Aristotelian and mathematical logic deals with pure thought alone there are no singular terms in either of them. The individual constants of mathematical logic are not objects of sense perception. In Nyaya and Buddhist logic singular (...) terms are indispensable, because inference does not deal with pure thought, but with knowledge which includes both thought and sense perception. (shrink)

The author contends that the analogies between "the logical roles of singular and universal statements" are important but do not "justify the conclusion that singular statements are reducible to propositions" of the forms a, E, I, And o. (staff).

In this paper is constructed an analogue of the square of opposition for propositions about relations between two non-empty sets. Unlike the classical square of opposition, the proposed scheme uses all logically possible syllogistic constants, formulated in V.I. Markin’s universal language for traditional positive syllogistic theories. This scheme can be called “Logical lantern”. The basic constants of this language are representing the five basic relations between two non-empty sets: equity, strict inclusion, reversed strict inclusion, intersection and exclusion (considered are only (...) the relations between sets, but not the relations of the sets to the universe). The other relations between sets are combinations (disjunctions) of the basic ones. There are 32 constants and 32 respective propositions, and the proposed diagram expresses and/or allows to deduce the logical relations among those propositions. The constructed scheme, making it possible to visually see the logical relations among propositions about relations between two non-empty sets, allows us to suggest considering logical relations not discussed previously: many-place Aristotelian-like logical relations among propositions: exhaustive _n_-place contrariety and exhaustive _n_-place subcontrariety (_n_ is a natural number, \(n>2\) ). (shrink)

Based on Aristotle's analysis of the form and matter found in human thought, this book examines the three steps the mind takes in arriving at the truth: defining, judging, and reasoning. The author further analyzes the type of material demanded for scientific or demonstrative knowledge: universal, necessary, and proper propositions and applies this examination to modern science.

A formal analysis is offered of Pseudo-Scotus's theory of the conversion of (i) propositions containing singular terms (including propositions with a singular term as predicate): and (ii) propositions with a quantified predicate. An attempt is made to steer a middle course between using the Aristotelian logic as a framework for the analysis, and using a Fregean framework.

Another textbook of traditional logic, subject to the characteristic limitations of that tradition. Propositional logic receives scant attention, and polyadic predicates are ignored. Propositions are in places confused with terms, as when the transitivity of implication is analyzed in terms of the Barbara syllogism. Although professedly Aristotelian, the treatment departs from Aristotle on a number of points: syllogisms are presented as inference rules rather than as logical theses; singular statements are assimilated to universal ones; and modal syllogisms are (...) not covered. Only about one-third of the book treats formal logic; the remainder includes discussion of basic semiotic, definition and division, induction, and fallacies, along with the rudiments of Thomistic epistemology and metaphysics.--J. B. B. (shrink)

Another textbook of traditional logic, subject to the characteristic limitations of that tradition. Propositional logic receives scant attention, and polyadic predicates are ignored. Propositions are in places confused with terms, as when the transitivity of implication is analyzed in terms of the Barbara syllogism. Although professedly Aristotelian, the treatment departs from Aristotle on a number of points: syllogisms are presented as inference rules rather than as logical theses; singular statements are assimilated to universal ones; and modal syllogisms are (...) not covered. Only about one-third of the book treats formal logic; the remainder includes discussion of basic semiotic, definition and division, induction, and fallacies, along with the rudiments of Thomistic epistemology and metaphysics.--J. B. B. (shrink)

© The Authors 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: [email protected] Propositions, Trenton Merricks defends a certain vision of the metaphysics of propositions: propositions exist necessarily and they primitively and essentially represent the world as being a certain way. The book is compact but rich: it is packed with arguments, moves at a fast pace, yet is written with admirable clarity.While I am sympathetic to many of Merrick’s conclusions, (...) I found some of his arguments towards these conclusions lacking. In Section 1, I discuss his claim that arguments constituted by propositions cannot be logically valid. In Section 2, I discuss his argument against singular propositions as having the entities they are about as constituents, and in particular his commitment to necessarily existing singular propositions... (shrink)

A new theoretical approach to Aristotelian Logic based on three axioms has been recently introduced. This formalization of the theory allowed for the unification of its uncommunicated traditional branches, thus restoring the theoretical unity of AL. In this brief paper, the applicability of the three AL axioms to Propositional Logic is explored. First, it is shown how the AL axioms can be applied to some simple PL arguments in a straightforward manner. Second, the development of a proof method (...) for PL inspired by the AL axioms is presented. This method mimics the underlying mechanics of the proof method from AL, and offers a complementary alternative to proof methods such as truth trees. (shrink)

The author claims to prove by example that, Contrary to what is generally maintained, A singular preposition of an observational kind is in some cases deducible from a natural law alone. On this basis he raises the question whether the universe might not be deterministic in a 'hyper-Laplacean' sense: that is, Whether the laws of nature might not be logically sufficient on their own to determine every actual state of the universe.

Intermediate Quantitifiers presents and analyzes the logical and linguistic features of intermediate quantifiers, in a fashion typical of traditional logic. Intermediate quantifiers express logical quantities which fall between Aristotle's two quantities of categorical propositions - the universal and the particular. This book is the first to use traditional methods to integrate the logic and semantics of intermediate quantifiers with the two traditional quantities. Few, many and most express the most commonly referred to intermediate quantities, yet in this book (...) Peterson argues that an infinite number of additional quantities can be understood through a deeper examination of the logical nature of all intermediate quantifiers. Drawing on the work of Robert Carnes and taking a critical look at James McCawley's grammatical analysis, Peterson concludes that a new approach to foundations of mathematics based on the syllogistic logic of quantifiers is possible.Offering invaluable insights for those studying across areas of logic, linguistics, and the philosophy or semantics of natural language, this book presents the completion of an explanation for extending traditional syllogistic logic to handle intermediate quantifiers. (shrink)

This book explores the reaction of tenth-century Arab philosopher Abu Nasr Alfarabi to the logical works of Aristotle. From numerous short treatises the author develops a systematic and comprehensive topical survey of Alfara bi's logical writings. The book is divided into two major parts: language as a tool of logic and logic as a tool with which to analyze language. The first five chapters deal with Alfarabi's analysis of the meanings of various terms as they are used in (...) logic and philosophy. Alfarabi refutes the Arab grammarians who claimed that Arab logicians were building a language within a language and shows that the philosophical meanings of terms are in fact their most original and essential meanings. The final chapter deals with Alfarabi's analysis of certain aspects of the Arabic language and demonstrates that Arabic, like any natural language, conforms to universal logical structures of which natural languages are only a concrete expression. (shrink)

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by (...) relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square. (shrink)

It is plausible that the universe exists: a thing such that absolutely everything is a part of it. It is also plausible that singular, structured propositions exist: propositions that literally have individuals as parts. Furthermore, it is plausible that for each thing, there is a singular, structured proposition that has it as a part. Finally, it is plausible that parthood is a partial ordering: reflexive, transitive, and anti-symmetric. These plausible claims cannot all be correct. We canvass some costs of denying (...) each claim and conclude that parthood is not a partial ordering. Provided that the relevant entities exist, parthood is not anti-symmetric and proper parthood is neither asymmetric nor transitive. (shrink)

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

Although most natural law ethical theories recognize moral absolutes, there is not much agreement even among natural law theorists about how to identify them. The author argues that in order to understand and determine the morality (or immorality) of a human action, it must be considered in relation to the organized system of human practices within which it is performed. Such an approach, he argues, is to be found in the natural law theory of Thomas Aquinas, especially once it is (...) recognized that the logical structure of Aquinas's ethical theory is basically that of an Aristotelian science. In order to depict this structure and to explain how it bears upon the analysis of action, the author investigates a number of issues that have attracted the attention of Thomistic and Aristotelian scholarship. He examines the nature of practical reason, its relationship with theoretical reason, the derivation of lower from higher ethical principles, the incommensurability of human goods, the relationship between will and intellect, and the principle of double effect. The book will be useful to students and scholars interested in ethics, especially from an Aristotelian and/or Thomistic perspective. One appendix reproduces the Leonine text of the De malo (question 6), with facing English translation. Another appendix provides facing Latin text and English translation of the Summa Theologiae I-II (question 94, article 2). ABOUT THE AUTHOR: Kevin L. Flannery, S.J., author of many works on the history of logic--particularly Aristotelian logic--is dean of the faculty of philosophy and professor of the history of ancient philosophy at the Pontifical Gregorian University in Rome. PRAISE FOR THE BOOK: ""I know of no more careful and detailed study of Thomas's principle that the realms of theoretical and practical rationality are parallel. This principle is indeed central to an understanding of Thomas Aquinas's ethical thought, and Father Flannery examines its ramifications most perspicaciously. His book constitutes a valuable and important addition to the Thomistic literature. Father Flannery's writing is lucid, if not always extremely engaging, and his arguments are well supported by references to, and quotations from, the Aristotelian and Thomistic texts upon which they are based.""--The Medieval Review ""It would be hard to overstate the importance of this book at the present juncture of Thomistic studies.""--Ralph McInerny, University of Notre Dame ""... of interest and of great value mostly to specialists in Aquinas's ethics or moral theory more generally. Flannery presents complex matters clearly, and his explanations of the logical presuppositions of Aquinas's moral thought are always illuminating.""--Jean Porter, Theological Studies ""This is a significant work that belongs on the bookshelf of any serious student of Aquinas.... An important contribution to Thomistic studies.""--Irish Theological Quarterly ""This scholastic text will particularly appeal to thinkers interested in Aristotelian mathematics and logic and their practical integration into Thomistic natural law ethics. However, all natural law scholars must address the questions of context and method raised by Flannery's insights and his careful, precise textual analyses."" -- Beverly Whelton, Review of Metaphysics. (shrink)

Originally published in 1938. This compact treatise is a complete treatment of Aristotle’s logic as containing negative terms. It begins with defining Aristotelian logic as a subject-predicate logic confining itself to the four forms of categorical proposition known as the A, E, I and O forms. It assigns conventional meanings to these categorical forms such that subalternation holds. It continues to discuss the development of the logic since the time of its founder and address traditional (...) class='Hi'>logic as it existed in the twentieth century. The primary consideration of the book is the inclusion of negative terms - obversion, contraposition etc. – within traditional logic by addressing three questions, of systematization, the rules, and the interpretation. (shrink)

According to many actualists, propositions, singular propositions in particular, are structurally complex, that is, roughly, (i) they have, in some sense, an internal structure that corresponds rather directly to the syntactic structure of the sentences that express them, and (ii) the metaphysical components, or constituents, of that structure are the semantic values — the meanings — of the corresponding syntactic components of those sentences. Given that reference is "direct", i.e., that the meaning of a name is its denotation, an apparent (...) consequence of this view is that any proposition expressed by a sentence containing a name that denotes a contingent being S is itself contingent — notably, the proposition [S does not exist]. Assuming that an entity must exist to have a property, necessarily, [S does not exist] must exist in order to be true. It seems to follow that, necessarily, [S does not exist] is not true and, hence, that S is not contingent after all. Past approaches to the problem — notably, those of Prior and Adams — lead to highly undesirable consequences for quantified modal logic. In this paper, several solutions to this puzzle are developed that preserve actualism, the structured view of propositions, the direct theory of reference, and the intuition that [S does not exist] is indeed possible without the adverse consequences for QML of previous solutions. (shrink)

Originally published in 1938. This compact treatise is a complete treatment of Aristotle’s logic as containing negative terms. It begins with defining Aristotelian logic as a subject-predicate logic confining itself to the four forms of categorical proposition known as the _A, E, I _and_ O_ forms. It assigns conventional meanings to these categorical forms such that subalternation holds. It continues to discuss the development of the logic since the time of its founder and address traditional (...) class='Hi'>logic as it existed in the twentieth century. The primary consideration of the book is the inclusion of negative terms - obversion, contraposition etc. – within traditional logic by addressing three questions, of systematization, the rules, and the interpretation. (shrink)

The paper uses the theory of generalized quantifiers to discuss existential import and its implications for Aristotelian logic, namely the square of opposition, conversions and the assertoric syllogistic, as well as for more recent generalizations to intermediate quantifiers like “most”. While this is a systematic discussion of the semantic background one should assume in order to obtain the inferences and oppositions Aristotle proposed, it also sheds some light on the interpretation of his writings. Moreover by applying tools from modern (...) formal semantics to the investigation of classical Aristotelian logic and its extensions, we combine different approaches to the logic of quantification. We will present variants of quantifiers that are associated to the four corners of the square of opposition with and without existential import and discuss their role for the logical square, conversions and the syllogistic. It will turn out that there is no way to ascribe existential import that validates all inferences and relations which one is willing to hold in Aristotelian logic. Two options, however, provide reasonable results. Existential import should either be ascribed only to affirmative statements or only be ascribed to universal quantification. The former option is preferable for a mere reconstruction of the classical Aristotelian logic while the latter option is more attractive if Aristotelian logic is generalized to intermediate quantifiers. (shrink)

JOHN CORCORAN AND WAGNER SANZ, Disbelief Logic Complements Belief Logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 USA E-mail: [email protected] Filosofia, Universidade Federal de Goiás, Goiás, GO 74001-970 Brazil E-mail: [email protected] -/- Consider two doxastic states belief and disbelief. Belief is taking a proposition to be true and disbelief taking it to be false. Judging also dichotomizes: accepting a proposition results in belief and rejecting in disbelief. Stating follows suit: asserting a proposition conveys belief and denying conveys disbelief. (...) Traditional logic implicitly focused on logical relations and processes needed in expanding and organizing systems of beliefs. Deducing a conclusion from beliefs results in belief of the conclusion. Deduction presupposes consequence: one proposition is a consequence of a set of a propositions if the latter logically implies the former. The role of consequence depends on its being truth-preserving: every consequence of a set of truths is true. This paper, which builds on previous work by the second author, explores roles of logic in expanding and organizing systems of disbeliefs. Aducing a conclusion from disbeliefs results in disbelief of the conclusion. Aduction presupposes contrequence: one proposition is a contrequence of a set of propositions if the set of negations or contradictory opposites of the latter logically implies that of the former. The role of contrequence depends on its being falsity-preserving: every contrequence of a set of falsehoods is false. A system of aductions that includes, for every contrequence of a given set, an aduction of the contrequence from the set is said to be complete. Historical and philosophical discussion is illustrated and enriched by presenting complete systems of aductions constructed by the second author. One such, a natural aduction system for Aristotelian categorical propositions, is based on a natural deduction system attributed to Aristotle by the first author and others. ADDED NOTE: Wagner Sanz reconstructed Aristotle’s logic the way it would have been had Aristole focused on constructing “anti-sciences” instead of sciences: more generally, on systems of disbeliefs. (shrink)

In this paper, we study multiplicative extensions of propositional many-place sequent calculi for finitely-valued logics arising from those introduced in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) through their translation by means of singularity determinants for logics and restriction of the original many-place sequent language. Our generalized approach, first of all, covers, on a uniform formal basis, both the one developed in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) for singular (...) finitely-valued logics (when singularity determinants consist of a variable alone) and conventional Gentzen-style (i.e., two-place sequent) calculi suggested in Pynko (Bull Sect Logic 33(1):23–32, 2004) for finitely-valued logics with equality determinant. In addition, it provides a universal method of constructing Tait-style (i.e., one-place sequent) calculi for finitely-valued logics with singularity determinant (in particular, for Łukasiewicz finitely-valued logics) that fits the well-known Tait calculus (Lecture Notes in Mathematics, Springer, Berlin, 1968) for the classical logic. We properly extend main results of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) and explore calculi under consideration within the framework of Sect. 7 of Pynko (Arch Math Logic 45:267–305, 2006), generalizing the results obtained in Sect. 7.5 of Pynko (Arch Math Logic 45:267–305 2006) for two-place sequent calculi associated with finitely-valued logics with equality determinant according to Pynko (Bull Sect Logic 33(1):23–32, 2004). We also exemplify our universal elaboration by applying it to some denumerable families of well-known finitely-valued logics. (shrink)

Envy is almost universally condemned. But is its reputation warranted? Sara Protasi argues envy is multifaceted and sometimes even virtuous.

This paper addresses the question of what existential assumptions are needed for the Aristotelian interpretation of the relationships between the four categorical propositions. The particular relationships in question are those unique to the Aristotelian logic, namely, contrariety, subcontrariety, subaltemation, conversion by limitation, and contraposition by limitation. The views of several recent authors of logic textbooks are surveyed. While most construe the Aristotelian logic as capable of being preserved by assuming that the subject class has a member, Irving (...) Copi construes that logic as requiring that four assumptions about class membership be made. These are that the subject, predicate, complement of subject, and complement of predicate classes all have members. It is argued that only three assumptions about class membership are needed, viz., that subject, predicate, and complement of predicate classes have members. (shrink)

This paper addresses the question of what existential assumptions are needed for the Aristotelian interpretation of the relationships between the four categorical propositions. The particular relationships in question are those unique to the Aristotelian logic, namely, contrariety, subcontrariety, subaltemation, conversion by limitation, and contraposition by limitation. The views of several recent authors of logic textbooks are surveyed. While most construe the Aristotelian logic as capable of being preserved by assuming that the subject class has a member, Irving (...) Copi construes that logic as requiring that four assumptions about class membership be made. These are that the subject, predicate, complement of subject, and complement of predicate classes all have members. It is argued that only three assumptions about class membership are needed, viz., that subject, predicate, and complement of predicate classes have members. (shrink)

A universally free logic is a system of quantification theory, with or without identity, whose theses remain logically true if the domain of quantification is empty and some of the singular terms present in the language do not denote existing objects. In the West, logics satisfying and ones satisfying were developed starting in the 1950s. But Stanisław Jaśkowski preceded all this work by some twenty years: his paper “On the Rules of Supposition in Formal Logic” of 1934 can (...) be regarded as containing, at least implicitly, the first universally free logic. The system proposed there is an inclusive logic as it is, and a straightforward extension of it is also free. (shrink)

The word pro-tasis is etymologically a near equivalent of pre-mise, pro-position, and ante-cedent—all having positional, relational connotations now totally absent in contemporary use of proposition. Taking protasis for premise, Aristotle’s statement (24a16) -/- A protasis is a sentence affirming or denying something of something…. -/- is not a definition of premise—intensionally: the relational feature is absent. Likewise, it is not a general definition of proposition—extensionally: it is too narrow. This paper explores recent literature on these issues.

"The role of logic in the Middle Ages. Regarding the role of logic within the framework of arts and sciences during the Middle Ages, we have to distinguish two related aspects, one institutional and the other scientific. As to the first aspect, we have to remember that the medieval educational system was based on the seven liberal arts, which were divided into the trivium, i.e., three arts of language, and the quadrivium, i.e., four mathematical arts. The so-called trivial (...) arts were grammar, rhetoric, and logic, and during a period of several centuries virtually every educated person, at least every university graduate, received a training in these matters, especially in logic. Students in the medieval faculty of arts probably spent more time studying logic than any other discipline. This first -- institutional -- aspect concerning the role of logic is explained by the second -- scientific -- aspect. The trivial disciplines provided techniques of analysis and a technical vocabulary that permeate philosophical, scientific and theological writings. Logic, as mentioned before, was referred to and was generally regarded as the art of arts and the science of sciences. The increasing cultural dominance of the universities with their obligatory disputationes and their hierarchy of examinations on the one hand and the outstanding status of logic on the other were corresponding features of the educational world of the 13th century. The core of the logic curriculum from the 12th century onwards was provided by the logical works of Aristotle. These represented the material for the study of types of predication, the analysis of simple propositions or statements (2) and their relations of inference and equivalence, the analysis of modal propositions, of the structure and the types of the syllogism, dialectical topics, fallacies and scientific reasoning as based on the demonstrative syllogism. Medieval logicians, however, realized that there were other, non-Aristotelian, approaches to logical subjects, questions and methods that could be investigated.. (shrink)

This chapter has two aims. The first aim is to present an overview of Bolzano's universe from the point of view of his metaphysics and its relationship to logic, relying fundamentally on Bolzano's Wissenschaftslehre. The author's preferred reading of Bolzano is one according to which he is a 'platonistic nominalist': a platonist about propositions and a nominalist about properties. Bolzano's nominalistic tendencies are particularly conspicuous in his mereological analyses, which play a major role in every aspect of his philosophy. (...) The author's second aim is to answer the open question of whether in Bolzano there is any 'ontology of truth'. This chapter argues for a negative answer. Bolzano does not make room for any special object which plays the role of the counterpart of a truth-bearer, or at least his propositions do not connect semantically in a direct way to anything that would count as such a special object. (shrink)

Aristotelian causal theories incorporate some philosophically important features of the concept of cause, including necessity and essential character. The proposed formalization is restricted to one-place predicates and a finite domain of attributes (without individuals). Semantics is based on a labeled tree structure, with truth defined by means of tree paths. A relatively simple causal prefixing mechanism is defined, by means of which causes of propositions and reasoning with causes are made explicit. The distinction of causal and factual explanation are elaborated, (...) and examples of cyclic and convergent causation are given. Soundness and completeness proofs are sketched. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Logic Pamphlets of Charles Lutwidge Dodgson and Related PiecesIrving H. AnellisFrancine F. Abeles, editor. The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces. The Pamphlets of Lewis Carroll, 4. New York-Charlottesville-London: Lewis Carroll Society of North America-University Press of Virginia, 2010. Pp. xx + 271. Cloth, $75.00.Until William Bartley’s rediscovery and reconstruction of Dodgson’s lost Part II of Symbolic Logic, Lewis Carroll’s reputation (...) in logic, when taken seriously, rested upon his few published articles on logical paradoxes and puzzles, while his published books, The Game of Logic and Symbolic Logic, were regarded as suitable perhaps as pedagogical tools, but dismissed otherwise as amusements, on a par with the Alice works. Richard Braithwaite asserted (“Lewis Carroll as Logician”) that “Carroll regarded formal and symbolic logic not as a corpus of systematic knowledge about valid thought nor yet as an art for teaching a person to think correctly.” He suggested that Dodgson used his pseudonym for his popular and whimsical writings, reserving his legal name only for his serious mathematical products. Philosophers practicing linguistic analysis in their assault on nonsense deriving from misuse of language meanwhile developed an interest in Carroll (e.g. George Pitcher, “Wittgenstein, Nonsense, and Lewis Carroll”). Abeles, on the contrary, stresses the seriousness of Dodgson’s pedagogical and popularizing mission (195–99).Bartley’s discovery opened the door for reevaluating Dodgson as a serious logician, and Abeles has been in the forefront of that reevaluation. Her general introduction and introductions to the main divisions of this collection of Carroll’s publications (and additional material from his Nachlass and a handful of publications of others, composed in response to Carroll’s published articles) place his serious logical work and its significance in historical perspective, providing a deeper, more sophisticated view than found in Bartley’s “Editor’s Introduction” to Lewis Carroll’s Symbolic Logic. Abeles also offers brief introductions to the individual entries, which describe their physical characteristics and circumstances of composition.Dodgson aligned with the majority of those British logicians of his day, starting with George Boole, who understood formal logic to be Aristotelian syllogistic, and symbolic logic to be syllogistic logic expressed algebraically. His notational innovation was employing indices to terms so that, independently of the arrangement of terms, propositions could easily be read; thus, “xy0” can be read as “no x are y” or “no y are x” and “x1y0” as “all x are not y” or “no x are y”. In dealing with the classical elimination problem in the class calculus—to determine the maximum information, without duplications, obtainable from a given set of premises—Dodgson in his later work and unpublished letters anticipated several concepts of automated theorem proving.Central to both geometry and to formal logic are the rules of logical inference that establish the validity of arguments and guarantee that conclusions inferred from true [End Page 506] premises are true. The modern analytic tableaux, or tree method, is a fundamental tool in this regard, both for deriving theorems and for determining the validity or invalidity of the proofs for propositional calculus and first-order predicate calculus.Introducing his reconstruction of Part II of Symbolic Logic, Bartley noted that, along with the logic diagrams that Carroll devised, he also devised a tableau method that strongly resembled Beth’s semantic tableaux. Abeles went more deeply and convincingly into that resemblance in her earlier research. Beth’s tableaux, together with Hintikka’s model sets, were the direct ancestors of Smullyan’s analytic tableaux, which in turn are the theoretical basis for the Robinson resolution method that continues to be central to much work in programming logic.Abeles has also shown that the work in Studies in Logic by Charles Peirce and his logic students at Johns Hopkins University served as a source of inspiration for Dodgson. More critically, she demonstrated that Dodgson devised and employed the tree method for carrying out proofs of syllogisms and chains of syllogisms, or soriteses, in Part II of Symbolic Logic (see her “Lewis Carroll’s Method of Trees: Its Origins in Studies in Logic,” Modern Logic 1 [1990]: 25–34). The tree... (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

A comunidade de filósofos da Metafísica Analítica Contemporânea não consegue entender-se com respeito ao estatuto ontológico das “Propriedades”. Uma parte defende que as propriedades existem, mas, têm grandes dificuldades em explicar a sua existência. Outra parte acredita que as propriedades não existem, mas, não conseguem justificar o problema ontológico da sua predicação. Esta divisão é o tema central deste livro que vos apresentamos, intitulado, “Properties and Propositions: The Metaphysics of Higher-Order Logic” da autoria de Robert Trueman e publicado pela (...) Cambridge University Press. (shrink)

Some of the most interesting recent work in philosophy of language and metaphysics is focused on questions about propositions, the abstract, truth-bearing contents of sentences and beliefs. The aim of this guide is to give instructors and students a road map for some significant work on propositions since the mid-1990s. This work falls roughly into two areas: challenges to the existence of propositions and theories about the nature and structure of propositions. The former includes both a widely discussed puzzle about (...) propositional designators as well as direct and indirect arguments against the existence of propositions. The latter is dominated by what is currently the central debate about the metaphysics of propositions, i.e. whether they are structured, composite entities or unstructured ontological simples. This issue has eclipsed older debates about whether propositions can be identified with sets of possible worlds or other kinds of sentence intensions. Author Recommends 1. Soames, Scott. 'Direct Reference, Propositional Attitudes, and Semantic Content.' Philosophical Topics 15 (1987): 47–87. Reprinted in Propositions and Attitudes . Eds. N. Salmon and S. Soames. Oxford: Oxford University Press, 1988. 197–239. Essential groundwork for more recent work on propositions. Soames gives a careful and exacting presentation of the case against identifying propositions with sets of possible worlds or other truth-supporting circumstances. Also contains a detailed statement of the Russellian conception of propositions on which propositions are ordered sets of objects, properties and relations. 2. King, Jeffrey. 'Designating Propositions.' The Philosophical Review 111 (2002): 341–71. Sometimes substituting a definite description for a corresponding 'that'-clause can lead to bizarre changes in truth-conditions: compare 'Bill fears that Hillary will be president' with 'Bill fears the proposition that Hillary will be president'. This puzzle about propositional designators threatens the relational analysis of propositional attitude reports, the view that 'believes' expresses a relation to the proposition designated by its 'that'-clause, and thereby poses an indirect threat to the existence of propositions. King's solution posits an ambiguity in verbs like 'fear' that embed both 'that'-clauses and definite descriptions. 3. Jubien, Michael. 'Propositions and the Objects of Thought.' Philosophical Studies 104 (2001): 47–62. A direct attack on the existence of propositions. Jubien deploys an analogue of the problem that Paul Benacerraf raised for set-theoretical reductions of numbers against metaphysical reductions of propositions. Just as numbers can be reduced to sets in many different ways, any reduction of propositions brings with it equally good variants, thus making any such reduction arbitrary and unmotivated. The only alternative is to treat propositions as abstract metaphysical primitives. As Jubien argues, however, abstract primitive entities are incapable of doing what propositions must do, i.e. represent objects and states of affairs on their own, without the input of thinking subjects. The upshot is the propositions cannot be reduced and they cannot be primitive, and so they must not exist. 4. Hanks, Peter. 'How Wittgenstein Defeated Russell's Multiple Relation Theory of Judgment.' Synthese 154 (2007): 121–46. Scepticism about propositions has recently led some philosophers, Jubien included, to resuscitate Russell's multiple relation theory of judgment, the idea that judgment is a many-place relation to objects, properties and relations. This paper explains why Russell himself abandoned that theory, and why the theory is still refuted by an objection due to Wittgenstein. 5. Hofweber, Thomas. 'Inexpressible Properties and Propositions.' Oxford Studies in Metaphysics . 2 vols. Ed. D. Zimmerman. Oxford: Oxford University Press, 2006. 155–206. An indirect attack on the existence of propositions. Hofweber argues that sentences like 'Bill believes something that Hillary asserted' do not commit us to the existence of propositions. His view is that propositional quantification is an instance of what he calls 'internal' or 'inferential role' quantification, a kind of quantification that carries no ontological implications. 6. Schiffer, Stephen. The Things We Mean . Oxford: Oxford University Press, 2003. esp. chs 1–2. Schiffer defends his theory of pleonastic propositions, on which propositions are unstructured, have no parts, and are very finely grained. 7. Bealer, George. 'Propositions.' Mind 107 (1998): 1–32. Bealer defends his algebraic theory of propositions, which, like Schiffer's pleonastic account, treats propositions as unstructured metaphysical simples. 8. King, Jeffrey. The Nature of and Structure of Content . Oxford: Oxford University Press, 2007. The best developed current theory of the structure in structured propositions. King identifies propositions with certain kinds of facts in which objects, properties and relations are bound together by amalgams of syntactic and semantic relations. 9. Hanks, Peter. 'Recent Work on Propositions.' Philosophy Compass 4 (2009): 1–18. A survey of work on propositions since the mid-1990s that complements this teaching and learning guide. Contains responses to Jubien's and Hofweber's arguments against propositions and critical discussions of Schiffer's pleonastic propositions and King's theory of propositional structure. Online Resources 1. http://plato.stanford.edu/entries/propositions/ Propositions (Matthew McGrath) 2. http://plato.stanford.edu/entries/propositions-structured/ Structured Propositions (Jeffrey King) 3. http://plato.stanford.edu/entries/propositions-singular/ Singular Propositions (Greg Fitch) Sample Partial Syllabus The following partial syllabus can be used as a unit on recent work on propositions in graduate level courses in philosophy of language or metaphysics. Week 1: A Substitution Puzzle About Propositional Designators King, Jeffrey. 'Designating Propositions'. Moltmann, Friederike. 'Propositional Attitudes Without Propositions.' Synthese 135 (2003): 77–118. Week 2: The Benacerraf Problem and Propositional Representation Benacerraf, Paul. 'What Numbers Could Not Be.' Philosophical Review 74 (1965): 47–73. Jubien, Michael. 'Propositions and the Objects of Thought.' Week 3: Propositional Quantification Hofweber, Thomas. 'Inexpressible Properties and Propositions'. Hofweber, Thomas. 'A Puzzle about Ontology.' Noûs 39 (2005): 256–83. Week 4: Schiffer on Pleonastic Propositions Schiffer, Stephen. 'Language-Created Language-Independent Entities.' Philosophical Topics 24 (1996): 149–67. Schiffer, Stephen. The Things We Mean , chs 1–2. Week 5: King on Structured Propositions King, Jeffrey. 'Structured Propositions and Complex Predicates.' Noûs , 29 (1995): 516–35. King, Jeffrey. The Nature and Structure of Content , chs 1–3. Focus Questions 1. Why does identifying propositions with sentence intensions, e.g. sets of possible worlds, 'require the attitudes to have a particular sort of closure under logical consequence, which they clearly don't have' (Mark Richard)? 2. How does the difference between (a) and (b) pose a threat to the existence of propositions? (a) Bill fears that Hillary will be president. (b) Bill fears the proposition that Hillary will be president. 3. What is the Benacerraf problem for metaphysical reductions of propositions? 4. Why must a proposition represent 'on its own cuff' (Michael Jubien)? Why is this a problem for the view that propositions are primitive abstract entities? 5. What does it mean to say that propositions are structured ? Give two different accounts of what propositional structure might be. (shrink)

By a pure modal logic of names we mean a quantifier-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic :271–284, 1989; Thomason in J Philos Logic 22:111–128, 1993 and J Philos (...) class='Hi'>Logic 26:129–141, 1997. In both cases we use Johnson-like models. We also analyze different kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the different versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidability” in the following sense: for every inference starting with a finite set of premises we can specify a finite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference. (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)