This presentation of Aristotle's naturaldeduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".
We argue that the need for commentary in commonly used linear calculi of naturaldeduction is connected to the “deletion” of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the (...) original proof. We then present a linear naturaldeduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed – thus showing that formalizing illocutionary expressions can eliminate the need for commentary. (shrink)
Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised naturaldeduction calculus, i.e. a naturaldeduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of naturaldeduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of (...) our German paper "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (shrink)
We present a sound and complete Fitch-style naturaldeduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over (...) some point on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)
Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of naturaldeduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic (...) versions of the connectives in question. (shrink)
Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a naturaldeduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal (...) scope. I prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation. (shrink)
PARC is an "appended numeral" system of naturaldeduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system can (...) be used with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of naturaldeduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)
This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.
In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermedi- ate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, naturaldeduction sys- tems are built only for strong Kleene’s logic both (...) with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide naturaldeduction systems for weak and intermediate regular logics both with one and two designated values. (shrink)
This paper presents two systems of naturaldeduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two (...) systems of naturaldeduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)
Some logic students falter at the transition from the mechanical method of truth tables to the less-mechanical method of naturaldeduction. This short paper introduces a word game intended to ease that transition.
A construction principle for naturaldeduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for naturaldeduction systems.
my goal here is to provide a detailed analysis of the methods of inference that are employed in De prospectiva pingendi. For this purpose, a method of naturaldeduction is proposed. the treatise by Piero della Francesca is a manifestation of a union between the ne arts and the mathematical sciences of arithmetic and geometry. He de nes painting as a part of perspective and, speaking precisely, as a branch of geometry, which is why we nd advanced geometrical (...) exercises here. (shrink)
Future Logic is an original, and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities. Traditional and Modern logic have covered in detail only formal deduction from actual categoricals, or from logical conditionals (conjunctives, hypotheticals, and disjunctives). Deduction from modal categoricals has also been considered, though very vaguely and roughly; whereas deduction from natural, temporal and extensional forms of conditioning (...) has been all but totally ignored. As for induction, apart from the elucidation of adductive processes (the scientific method), almost no formal work has been done. This is the first work ever to strictly formalize the inductive processes of generalization and particularization, through the novel methods of factorial analysis, factor selection and formula revision. This is the first work ever to develop a formal logic of the natural, temporal and extensional types of conditioning (as distinct from logical conditioning), including their production from modal categorical premises. Future Logic contains a great many other new discoveries, organized into a unified, consistent and empirical system, with precise definitions of the various categories and types of modality (including logical modality), and full awareness of the epistemological and ontological issues involved. Though strictly formal, it uses ordinary language, wherever symbols can be avoided. Among its other contributions: a full list of the valid modal syllogisms (which is more restrictive than previous lists); the main formalities of the logic of change (which introduces a dynamic instead of merely static approach to classification); the first formal definitions of the modal types of causality; a new theory of class logic, free of the Russell Paradox; as well as a critical review of modern metalogic. But it is impossible to list briefly all the innovations in logical science — and therefore, epistemology and ontology — this book presents; it has to be read for its scope to be appreciated. (shrink)
This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding (...) of natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)
In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . (...) We propose a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules. (shrink)
In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a naturaldeduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system (...) is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)
This presentation includes a complete bibliography of John Corcoran’s publications relevant on Aristotle’s logic. The Sections I, II, III, and IV list respectively 23 articles, 44 abstracts, 3 books, and 11 reviews. Section I starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article—from Corcoran’s Philadelphia period that antedates his discovery of Aristotle’s naturaldeduction system—and the Journal of Symbolic Logic article—from his Buffalo period first reporting his original results. It ends with works published (...) in 2015. Some items are annotated as listed or with endnotes connecting them with other work and pointing out passages that, in retrospect, are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 10 items published in the 1970s, 24 in the 1980s, 42 in the 1990s, 60 in the 2000s, and 70 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally in some cases, even of the Łukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably George Boger, Charles Kahn, John Mulhern, Mary Mulhern, Anthony Preus, Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. (shrink)
Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised naturaldeduction calculus, i.e., a naturaldeduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of naturaldeduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs.
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, naturaldeduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued (...) first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and naturaldeduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)
As the 19th century drew to a close, logicians formalized an ideal notion of proof. They were driven by nothing other than an abiding interest in truth, and their proofs were as ethereal as the mind of God. Yet within decades these mathematical abstractions were realized by the hand of man, in the digital stored-program computer. How it came to be recognized that proofs and programs are the same thing is a story that spans a century, a chase with as (...) many twists and turns as a thriller. At the end of the story is a new principle for designing programming languages that will guide computers into the 21st century. -/- For my money, Gentzen’s naturaldeduction and Church’s lambda calculus are on a par with Einstein’s relativity and Dirac’s quantum physics for elegance and insight. And the maths are a lot simpler. I want to show you the essence of these ideas. I’ll need a few symbols, but not too many, and I’ll explain as I go along. -/- To simplify, I’ll present the story as we understand it now, with some asides to fill in the history. First, I’ll introduce Gentzen’s naturaldeduction, a formalism for proofs. Next, I’ll introduce Church’s lambda calculus, a formalism for programs. Then I’ll explain why proofs and programs are really the same thing, and how simplifying a proof corresponds to executing a program. Finally, I’ll conclude with a look at how these principles are being applied to design a new generation of programming languages, particularly mobile code for the Internet. (shrink)
The vision of natural kinds that is most common in the modern philosophy of biology, particularly with respect to the question whether species and other taxa are natural kinds, is based on a revision of the notion by Mill in A System of Logic. However, there was another conception that Whewell had previously captured well, which taxonomists have always employed, of kinds as being types that need not have necessary and sufficient characters and properties, or essences. These competing (...) views employ different approaches to scientific methodologies: Mill’s class-kinds are not formed by induction but by deduction, while Whewell’s type-kinds are inductive. More recently, phylogenetic kinds (clades, or monophyletic-kinds) are inductively projectible, and escape Mill’s strictures. Mill’s version represents a shift in the notions of kinds from the biological to the physical sciences. (shrink)
In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include both (...) the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)
Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism is (...) a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading. (shrink)
Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)
JUNE 2015 UPDATE: A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015 By John Corcoran -/- This presentation includes a complete bibliography of John Corcoran’s publications relevant to his research on Aristotle’s logic. Sections I, II, III, and IV list 21 articles, 44 abstracts, 3 books, and 11 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article from Corcoran’s Philadelphia period that antedates his Aristotle studies and the Journal of Symbolic Logic article from his (...) Buffalo period first reporting his original results; it ends with works published in 2015. A few of the items are annotated as listed or with endnotes connecting them with other work and pointing out passages that in-retrospect are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 23 in the 1980s, 42 in the 1990s, 56 in the 2000s, and 69 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally, of the Lukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. -/- REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. Also, let me know of passages I should comment on. (shrink)
In this paper, I'll present a general way of "reading off" introduction/elimination rules from elimination/introduction rules, and define notions of harmony and stability on the basis of it.
This paper is about teaching elementary logic to blind or visually impaired students. The targeted audience are teachers who all of sudden have a blind or visually impaired student in their introduction to logic class, find limited help from disability centers in their institution, and have no idea what to do. We provide simple techniques that allow direct communication between a teacher and a visually impaired student. We show how the use of what is known as Polish notation simplifies communication, (...) and pedagogically is a great notation for a Braille reader. (shrink)
This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting them with (...) other work. In addition, Section V “Discussions” is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 22 in the 1980s, 39 in the 1990s, 56 in the 2000s, and 65 in the current decade. The secondary bibliography is annotated with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. As is evident from the Acknowledgements sections, all of Corcoran’s publications benefited from correspondence with other scholars, most notably Timothy Smiley, Michael Scanlan, and Kevin Tracy. All of Corcoran’s Greek translations were done in consultation with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. (shrink)
1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement of (...) mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)
This review places this translation and commentary on Book A of Prior Analytics in historical, logical, and philosophical perspective. In particular, it details the author’s positions on current controversies. The author of this translation and commentary is a prolific and respected scholar, a leading figure in a large and still rapidly growing area of scholarship: Prior Analytics studies PAS. PAS treats many aspects of Aristotle’s Prior Analytics: historical context, previous writings that influenced it, preservation and transmission of its manuscripts, editions (...) of its manuscripts, interpretations, commentaries, translations, and its influence on subsequent logic, philosophy, and mathematics. All this attention is warranted because Prior Analytics marks the origin of logic: the field that, among other things, asks of a given proposition whether it follows from a given set of propositions; and, if it follows, how we determine that it follows; and, if it does not follow, how we determine that it does not follow. This translation and commentary is not suitable for use in an undergraduate course. It has too many quirks that the teacher would want to warn against. A copy editor should have dealt with these things and with other matters such as incorrect punctuation and improper end-of-line divisions. The prose is heavily laden with glaring clichés. The one-page preface contains “longer than I care to remember”, “more than I can possibly list here”, “first and foremost”, and “last and by no means least”—a sentence later is devoted to thanking the “incredibly meticulous and helpful copy-editor”. A few pages later the translator reveals the need “to find a path between the Scylla … and the Charybdis …”. Moreover, the index is far from meeting the needs of undergraduate students. The attention to scholarly detail is not what one hoped for from Oxford University Press. At 26b10-15, this translation reads “let swan and white be chosen as white things” for what Smith correctly translates “let swan and snow be selected from among those white things”. At 41b16, “angles AB and CD” should read “angles AC and BD”. Despite this book’s flaws, it will be found useful if not indispensable for those currently engaged in Prior Analytics studies. The alternatives suggested to Robin Smith’s translation choices are often worth consideration. It is to be emphasized, however, that this book is unsuitable for those entering Prior Analytics studies. (shrink)
Articles by Ian Mueller, Ronald Zirin, Norman Kretzmann, John Corcoran, John Mulhern, Mary Mulhern,Josiah Gould, and others. Topics: Aristotle's Syllogistic, Stoic Logic, Modern Research in Ancient Logic.
For centuries humans have considered just two main possibilities for what awaits us at death: a “nothingness” like that of our before-life or some type of supernatural afterlife. The theory of a natural afterlife defines a vastly different, real possibility. The natural afterlife embodies all of the sensory perceptions, thoughts, and emotions present in the final moment of a near-death, dreamlike experience. With death this moment becomes timeless and everlasting to the dying person—essentially, a never-ending experience. The relativeness (...) and timelessness of the natural afterlife must be clearly understood to appreciate why it’s not supernatural yet indeed an afterlife and potentially the optimal heaven. The theory of a natural afterlife is now only a hypothesis; however, science, human experience, and logical deduction suggest that it’s extremely plausible and advances in science and technology could someday make it a scientific theory. This paper states the theory, describes the unconventional afterlife it defines, extensively analyzes its validity, and briefly addresses how it can significantly impact how people view death. Analytical tools, typically used for system modeling and language definition, are applied here to present an abstract model of a lifetime within time eternal. The model is used to support and explain the theory. (shrink)
This chapter considers Kant's relation to Hume as Kant himself understood it when he wrote the Critique of Pure Reason and the Prolegomena. It first seeks to refine the question of Kant's relation to Hume's skepticism, and it then considers the evidence for Kant's attitude toward Hume in three works: the A Critique, Prolegomena, and B Critique. It argues that in the A Critique Kant viewed skepticism positively, as a necessary reaction to dogmatism and a spur toward critique. In his (...) initial statement of the critical philosophy Kant treated Hume as an ally in curbing dogmatism, but one who stopped short of what was really needed: a full critique of reason, to establish the boundaries of metaphysical cognition. Kant found fault with Hume's analyses of cognition and experience, and specifically his failure to see the crucial importance of synthetic a priori cognition in metaphysics. In particular, he held that Hume's empiricist account of cognition could neither explain the synthetic a priori cognition actually found in mathematics and natural science, nor provide a principled account of the limits on what can be known--and what can be thought--through the pure concepts of the understanding. According to Kant, Hume therefore failed in his attempt to determine the limits of metaphysics, whereas he was able to succeed because his transcendental philosophy provided a thorough account of cognition, its structure and limits. In the Prolegomena and the B Critique Kant distinguished his position more sharply from Hume's. He also adopted a more negative attitude toward "skeptical idealism" than before; but he attributed such skepticism to Descartes, not Hume. Prior to the B Critique Kant did not see Hume as attacking natural science or ordinary cognition. In none of the three works was Kant's main aim to "answer the skeptic." His primary aim was to firmly establish the boundary of metaphysics, by discovering the elements of human cognition and fixing its proper domain. His purported discoveries about the limits of metaphysical cognition meant that the traditional objects of metaphysical knowledge, God, the soul, and the world as it is in itself, are unknowable, hence that traditional metaphysics itself is impossible. Besides settling the possibility or impossibility of metaphysics, his findings would also prevent the illegitimate extension of principles of sensibility to God and the noumenal self, an extension that would threaten the metaphysics of morals by incorrectly denying the thinkability of noumenal freedom, and that might otherwise lead to "materialism, fatalism, atheism, and freethinking unbelief" (B xxxiv). (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)
Abstract. Aristotelian assertoric syllogistic, which is currently of growing interest, has attracted the attention of the founders of modern logic, who approached it in several (semantical and syntactical) ways. Further approaches were introduced later on. These approaches (with few exceptions) are here discussed, developed and interrelated. Among other things, di-erent facets of soundness, completeness, decidability and independence are investigated. Speci/cally arithmetization (Leibniz), algebraization (Leibniz and Boole), and Venn models (Euler and Venn) are closely examined. All proofs are simple. In particular (...) there is no recourse to maximal nor minimal conditions (with only one, dispensable, exception), which makes the long awaited deciphering of the enigmatic Leibniz characteristic numbers possible. The problem was how to look at matters from the right perspective. (shrink)
The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to naturaldeduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of (...) a specific sequence of reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (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 naturaldeduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)
ABSTRACT This part of the series has a dual purpose. In the first place we will discuss two kinds of theories of proof. The first kind will be called a theory of linear proof. The second has been called a theory of suppositional proof. The term "naturaldeduction" has often and correctly been used to refer to the second kind of theory, but I shall not do so here because many of the theories so-called are not of the (...) second kind--they must be thought of either as disguised linear theories or theories of a third kind (see postscript below). The second purpose of this part is 25 to develop some of the main ideas needed in constructing a comprehensive theory of proof. The reason for choosing the linear and suppositional theories for this purpose is because the linear theory includes only rules of a very simple nature, and the suppositional theory can be seen as the result of making the linear theory more comprehensive. CORRECTION: At the time these articles were written the word ‘proof’ especially in the phrase ‘proof from hypotheses’ was widely used to refer to what were earlier and are now called deductions. I ask your forgiveness. I have forgiven Church and Henkin who misled me. (shrink)
JOHN CORCORAN AND WAGNER SANZ, Disbelief Logic Complements Belief Logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 USA E-mail: corcoran@buffalo.edu Filosofia, Universidade Federal de Goiás, Goiás, GO 74001-970 Brazil E-mail: sanz@fchf.ufg.br -/- 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 naturaldeduction 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)
Proceeding on the Automatic Deduction System developped at the Philosophy Faculty of the UNAM at Mexico City. (Deduktor Mexican Group of Logics work under the direction of the professor Hugo Padilla Chacón). Conference presented at the mexican City of Guadalajara at the Universidad de Guadalajara, Jalisco, by invitation of the latinoamerican association of philosophy SOPHIA. Early stage of the deductional systems at 2-valued logic. This work embodies the implementation of the first whole and standalone arithmetization of bivalent Logic, the (...) theoretical framework of Hugo Padilla Chacón published in 1984. (shrink)
I propose that we approach the epistemology of modality by putting modal metaphysics first and, specifically, by investigating the metaphysics of essence. Following a prominent Neo-Aristotelian view, I hold that metaphysical necessity depends on the nature of things, namely their essences. I further clarify that essences are core properties having distinctive superexplanatory powers. In the case of natural kinds, which is my focus in the paper, superexplanatoriness is due to the fact that the essence of a kind is what (...) causes all the many properties and behaviors that are typically shared by all the instances of the kind. Accordingly, we know what is necessarily true of kinds by knowing what is essential to them in the sense of actually playing such causal-explanatory roles. Modal reasoning aimed at discovering metaphysical necessity thus proceeds via essentialist deduction: we move from essentialist truths to reach necessary truths. (shrink)
Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)
Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (shrink)
In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. (...) Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)
ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; (...) but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. -/- This 1974 paper uses the linear notation for naturaldeduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of naturaldeduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R -/- 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: NaturalDeduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253). (shrink)
The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic type interpretation (in arbitrary Henkin models). The domains (...) of derivations are collections of derivations in the associated “dedicated” natural-deduction proof-system, and functions therein (with no appeal to models, truth-values and elements of a domain). The compositionality of the semantics is analyzed. (shrink)
forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), translating (formalizing) English in TFL and FOL, and Fitch-style naturaldeduction proof systems for both TFL and FOL. It also deals with some advanced topics such as truth-functional completeness and modal logic. Exercises with solutions (...) are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics) and in LaTeX source code. (shrink)
This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of naturaldeduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the (...) meanings of modal operators in terms of rules of inference. (shrink)
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