A History of Quantification
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Handbook Vatican 2014
HANDBOOK OF THE WORLD CONGRESS ON THE SQUARE OF OPPOSITION IV Pontifical Lateran University, Vatican May 5-9, 2014 Edited by Jean-Yves Béziau and Katarzyna Gan-Krzywoszyńska www.square-of-opposition.org 1 2 Contents 1. Fourth World Congress on the Square of Opposition ..................................................... 7 1.1. The Square : a Central Object for Thought ..................................................................................7 1.2. Aim of the Congress.....................................................................................................................8 1.3. Scientific Committee ...................................................................................................................9 1.4. Organizing Committee .............................................................................................................. 10 2. Plenary Lectures ................................................................................................... 11 Gianfranco Basti "Scientia una contrariorum": Paraconsistency, Induction, and Formal Ontology ..... 11 Jean-Yves Béziau Square of Opposition: Past, Present and Future ....................................................... 12 Manuel Correia Machuca The Didactic and Theoretical Expositions of the Square of Opposition in Aristotelian logic ....................................................................................................................... 13 Rusty Jones Bivalence and Contradictory Pairs in Aristotle’s De Interpretatione ................................ -
Categorize" a Person, Place, Time, Thing, Or Situation Is to Characterize It As a Member of a Class of Similar Things
CHAPTER 2 CATEGORICAL PROPOSITIONS A. The Structure of Categorical Propositions To "categorize" a person, place, time, thing, or situation is to characterize it as a member of a class of similar things. One does not consider the thing in question from its purely individual point of view, that is, in terms of the qualities it has without relationship to any other things. Upon being categorized, an individual thing is known by properties that it has by virtue of its being a member of the class of things referred to by that category. All propositions expressed in the Aristotelian system of logic are called categorical propositions because they are constructed using two categories of things: the subject category (or class) and the predicate category (or class). The copula, or connector between the subject and predicate, of a categorical proposition indicates how its subject category is related to its predicate category. The copula specifies whether members of the subject category are also members of the predicate category or whether members of the subject category are not members of the predicate category. Illustration: Subject Copula Predicate Quality of Copula (1) Strawberries are red. Affirmative (2) Strawberries are not red. Negative However, this is not enough. (1) declares that members of the class of strawberries are also members of the class of red things, but (1) is ambiguous because it does not specify the quantity of the subject class that is included in the predicate class. Thus, Ms. A might take (1) to mean “all strawberries are red,” while Mr. B might take (1) to mean “some strawberries are red.” Ms. -
1 UNIT 4 PROPOSITIONS Contents 4.0 Objectives 4.1 Introduction 4.2
UNIT 4 PROPOSITIONS Contents 4.0 Objectives 4.1 Introduction 4.2 History of Logic and Proposition 4.3 Propositions and Sentences 4.4 Propositions and Judgments 4.5 Types of Proposition 4.6 Quality and Quantity 4.7 Let Us Sum Up 4.8 Key Words 4.9 Further Readings and References 4.10 Answers to Check Your Progress 4.0 OBJECTIVES As we know inference is the main subject matter of logic. The term refers to the argument in which a proposition is arrived at and affirmed or denied on the basis of one or more other propositions accepted as the starting point of the process. To determine whether or not an inference is correct the logician examines the propositions that are the initial and end points of that argument and the relationships between them. This clearly denotes the significance of propositions in the study of logic. In this unit you are expected to study: • the nature • the definition • the types and forms of propositions • the difference between propositions and sentences and judgments • the description of various types of propositions viewed from different standpoints like, composition, generality, relation, quantity, quality, and modality. 4.1 INTRODUCTION Classical logic concerns itself with forms and classifications of propositions. We shall begin with the standard definition of proposition. A proposition is a declarative sentence which is either true or false but not both. Also a proposition cannot be neither true nor false. A proposition is always expressed with the help of a sentence. For example - the same proposition “It is raining” can be expressed in English, Hindi, and Sanskrit and so on. -
Two Squares of Opposition: the Conceptual Square Underwrites the Alethic Square’S Validity
1 TWO SQUARES OF OPPOSITION: THE CONCEPTUAL SQUARE UNDERWRITES THE ALETHIC SQUARE’S VALIDITY ABSTRACT I have made explicit an implicit conceptual logic that exists in the English lang- uage and others; its evaluation terms of propositions are coherent|incoherent. This con- trasts with the usual alethic true|false evaluations for statement logic. I call conceptual logic a basement logic; it’s more basic than statement logic. I demonstrate this by show- ing how the long-standard, ever-present Alethic Square of Opposition’s AEIO state- ments’ truth value and the validity of their immediate inferences rests on the Conceptual Square’s (a) coherent propositions and (b) coherent emplacements of substantives and tropes into, respectively, AEIOs’ subjects and predicates. ConceptFunctor operations bear the leutic [Enjoined] modal that governs the coherence of de dicto and de jure grounded propositions. The FactFunctor operation bears the [Allowed] modal that govern coherent emplacement of the world’s substantives and tropes into sentences’ subjects and predicates and, subsequent to that, the truth value of their allied statements. My strategy is (1) to construct a Conceptual Square of Opposition by rewriting the Aletic Square’s AEIO statements as conceptual propositions. (2) If the propositions are coherent by virtue of coherent emplacements of substantives and tropes into their subjects and predicates that turns them into conceptualized substantives and tropes, then, by the Coherence Account of Truth, they entail the truth of the Alethic Squares’ statements and the validity of its traditionally accepted inferences. (3) This is enough to prove conceptual logic is the basement ground for the traditional Alethic Square, and on which any alethic logic rests; and it supports the claims made in (2) above.1 KEY WORDS: Coherence, [Emplace], [Functor], ConceptFunctor, FactFunctor, [Any], collective, distributive, and singular (uses of [Any]), grammatical and semantic subjects and predicates, Concepttual and Alethic Square (of opposition), leutic modal- ities, [Enjoin], [Allow]. -
Logical Extensions of Aristotle's Square
Logical Extensions of Aristotle’s Square Dominique Luzeaux, Jean Sallantin, Christopher Dartnell To cite this version: Dominique Luzeaux, Jean Sallantin, Christopher Dartnell. Logical Extensions of Aristotle’s Square. Logica Universalis, Springer Verlag, 2008, 2, pp.167-187. 10.1007/s11787-007-0022-y. lirmm- 00274314 HAL Id: lirmm-00274314 https://hal-lirmm.ccsd.cnrs.fr/lirmm-00274314 Submitted on 17 Apr 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Logical extensions of Aristotle's square Dominique Luzeaux, Jean Sallantin and Christopher Dartnell Abstract. We start from the geometrical-logical extension of Aristotle's square in [Bla66], [Pel06] and [Mor04], and study them from both syntactic and semantic points of view. Recall that Aristotle's square under its modal form has the following four vertices: A is α, E is :α, I is ::α and O is :α, where α is a logical formula and is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether : is involutive or not) modal logic. [B´ez03]has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. -
Alice in the Wonderful Land of Logical Notions)
The Mystery of the Fifth Logical Notion (Alice in the Wonderful Land of Logical Notions) Jean-Yves Beziau University of Brazil, Rio de Janeiro, Brazilian Research Council and Brazilian Academy of Philosophy [email protected] Abstract We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski- Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a non-logical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first- order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. -
Immediate Inference
Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P. -
Saving the Square of Opposition∗
HISTORY AND PHILOSOPHY OF LOGIC, 2021 https://doi.org/10.1080/01445340.2020.1865782 Saving the Square of Opposition∗ PIETER A. M. SEUREN Max Planck Institute for Psycholinguistics, Nijmegen, the Netherlands Received 19 April 2020 Accepted 15 December 2020 Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cogni- tive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leak- ing O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute -
Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics
axioms Article Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics Lorenz Demey 1,2 1 Center for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, Belgium; [email protected] 2 KU Leuven Institute for Artificial Intelligence, KU Leuven, 3000 Leuven, Belgium Abstract: Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distin- guish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic. Keywords: Aristotelian diagram; non-normal modal logic; square of opposition; logical geometry; neighborhood semantics; bitstring semantics MSC: 03B45; 03A05 Citation: Demey, L. Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. Axioms 2021, 10, 128. https://doi.org/ 1. Introduction 10.3390/axioms10030128 Aristotelian diagrams, such as the so-called square of opposition, visualize a number Academic Editor: Radko Mesiar of formulas from some logical system, as well as certain logical relations holding between them. -
The Peripatetic Program in Categorical Logic: Leibniz on Propositional Terms
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 65 THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS MARKO MALINK Department of Philosophy, New York University and ANUBAV VASUDEVAN Department of Philosophy, University of Chicago Abstract. Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such as reductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titled Specimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule of reductio ad absurdum in a purely categor- ical calculus in which every proposition is of the form AisB. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule of reductio ad absurdum,but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic ¬ known as RMI→ . From its beginnings in antiquity up until the late 19th century, the study of formal logic was shaped by two distinct approaches to the subject. The first approach was primarily concerned with simple propositions expressing predicative relations between terms. -
Testing Validity Using Venn's Diagrams
TESTING VALIDITY USING VENN'S DIAGRAMS To test the validity of a categorical syllogism, one can use the method of Venn diagrams. Since a categorical syllogism has three terms, we need a Venn diagram using three intersecting circles, one representing each of the three terms in a categorical syllogism. A three term diagram has eight regions (the number of regions being 2n where n is the number of terms). The following chart gives the extension of the predicates in the various regions of the diagram. Region S (Minor Term) P (Major Term) M (Middle Term) 1 yes no no 2 yes yes no 3 no yes no 4 yes no yes 5 yes yes yes 6 no yes yes 7 no no yes 8 no no no In order to use a Venn diagram to test a syllogism, the diagram must be filled in to reflect the contents of the premises. Remember, shading an area means that that area is empty, the term represented has no extension in that area. What one is looking for in a Venn diagram test for validity is an accurate diagram of the conclusion of the argument that logically follows from a diagram of the premises. Since each of the premises of a categorical syllogism is a categorical proposition, diagram the premise sentences independently and then see whether the conclusion has already been diagramed. If so, the argument is valid. If not, then it is not. Remember, one definition of validity is that the propositional (informational) content of the conclusion is already expressed in the premises. -
MICHAEL FREDE Michael Johannes Frede 1940–2007
MICHAEL FREDE Michael Johannes Frede 1940–2007 WHEN MICHAEL JOHANNES FREDE died (11 August 2007), by drowning in the Corinthian Gulf at a beach near Itea below Delphi, Greece, the world of ancient philosophy lost the most accomplished philosopher and scholar, and one of the most distinguished and influential teachers, of the generation of specialists in this field internationally who began their careers in the 1960s and 1970s. He was born 31 May 1940, in Kreuzberg, a working-class district of Berlin, the first child of Roman Catholic parents (a second, Stefanie, was born several years later). He was brought up in Germany and educated there through the Ph.D. and beyond, emigrating later, first to the United States in 1971, then to the United Kingdom, where he was Professor of the History of Philosophy at Oxford University and Fellow of Keble College from 1991 until he chose to retire two years before required, in 2005. In 1943, at the age of two, the apartment building where his family lived was reduced to rubble in an Allied bombing attack while his mother had taken him out for a walk in his carriage, so the family had to move, first to Lippstadt in Westphalia and then in the early 1950s to Hamburg, where Frede attended the Sankt Ansgar-Gymnasium, a Jesuit school, studying classics and the classical languages (Abitur 1959). At university, he studied first in Munich (1959–60) then back in Hamburg (1960–2), where he became a student of Günther Patzig; he rejoined Patzig at Göttingen in 1963 to complete his Ph.D., after a momentous year (1962–3) at Oxford, on Patzig’s suggestion and with his assistance, as a visiting research student, working on his dissertation on Plato’s Sophist, in consul- tation with G.