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Topic: Categorical logic

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Martin Scorsese

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Category theory

	Categorical logic - Wikipedia, the free encyclopedia
		Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science.
		Categorical logic originated within sheaf theory, as a suitable version of Kripke semantics one can say with hindsight, and emerged as a theory with a character of its own only in shedding the necessary connection with sheaves.
		Kolmogorov was also a pioneer writer in the early 1920s on the formulation of intuitionistic logic, in a style entirely supported by the later categorical logic approach (again, one of the formulations, not the only one; the realizability concept of Stephen Kleene is also a serious contender here).
	en.wikipedia.org /wiki/Categorical_logic (766 words)

	Categorical logic -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-07)
		Categorical logic originated within (additional info and facts about sheaf theory) sheaf theory, as a suitable version of (additional info and facts about Kripke semantics) Kripke semantics one can say with hindsight, and emerged as a theory with a character of its own only in shedding the necessary connection with sheaves.
		Kolmogorov was also a pioneer writer in the early 1920s on the formulation of intuitionistic logic, in a style entirely supported by the later categorical logic approach (again, one of the formulations, not the only one; the realizability concept of (additional info and facts about Stephen Kleene) Stephen Kleene is also a serious contender here).
		Another route to categorical logic would therefore have been through Kolmogorov, and this is one way to explain the protean (additional info and facts about Curry-Howard isomorphism) Curry-Howard isomorphism.
	www.absoluteastronomy.com /encyclopedia/c/ca/categorical_logic.htm (792 words)

	[No title] (Site not responding. Last check: 2007-09-07)
		A proposition In categorical logic, an A proposition is a universal affirmative proposition.
		Aristotelian Interpretation of Categorical Logic The Aristotelian interpretation of categorical logic is an interpretation of categorical logic that ascribes existential import to universal propositions.
		Transposition In propositional logic, transposition is an equivalence falling under the rule of replacement that a conditional is logically equivalent to another statement in which the denial of the consequent of the first is the antecedent and the denial of the antecedent of the first is the consequent.
	www2.semo.edu /philosophy/courses/pl120/glossary.htm (7787 words)

	PL 120 Symbolic Logic I (Site not responding. Last check: 2007-09-07)
		Absorption (Abs.) In propositional logic, absorption is a rule of inference in which a conditional statement is given as a premise; you conclude a conditional statement with the same antecedent, and the consequent is a conjunction of the antecedent and consequent of the given proposition.
		Distribution (Dist.) In propositional logic, distribution is an equivalence falling under the Rule of Replacement that specifies the relationship between a statement conjoined to a disjunction and a conjunction of disjunctions, or a statement conjoined to a disjunction and a disjunction of conjunctions.
		Transposition (Trans.) In propositional logic, transposition is an equivalence falling under the rule of replacement that a conditional is logically equivalent to another statement in which the denial of the consequent of the first is the antecedent and the denial of the antecedent of the first is the consequent.
	cstl-cla.semo.edu /hill/pl120/glossary.htm (10152 words)

	Peter Suber, "Non-Standard Logics"
		Logics that include apparatus for signifying when two meanings (as opposed to two wffs, truth-values, sets, predicates, functions) are identical, and that analyzes inferences involving meanings.
		Logics in which the set of implications determined by a given group of premises does not necessarily grow, and can shrink, when new wffs are added to the set of premises.
		Logics in which the times at which propositions bear certain truth-values can be indicated, in which the "tense" of the assertion can be indicated, and in which truth-values can be affected by the passage of time.
	www.earlham.edu /~peters/courses/logsys/nonstbib.htm (2695 words)

	Book Review: "Categorical Logic and Type Theory" by B. Jacobs (Site not responding. Last check: 2007-09-07)
		logic over dependant types or over polymorphic types, may be dealt with by treating the logic as a "module" over the appropriate type theory.
		He introduces the notion of functorial semantics (the insight that a model is a structure preserving functor, for the appropriate notion of structure for the logic being considered), and extends it to the fibrational context so he can handle the various logics and type theories being considered.
		For example, he says that traditional logic is single sorted, unlike the multisorted logic necessary for applications in computer science (and for categorical applications in general).
	www.math.mcgill.ca /rags/jacobs.html (2583 words)

	catlog.html
		The essentials of categorical logic fit into just a few pages, but they can be used to focus attention in some very effective ways and can provide the means to analyze a wide variety of claims, arguments, and situations.
		A categorical claim always refers to two classes, which by convention are called S (for the subject) and P (for the predicate) in discussions of the general principles of categorical logic.
		Categorical logic gives "some" an indefinitely wide range of reference, so in a standard-form categorical claim, "some" can refer to as few as one of a class and as many as all the members of the class.
	www.csuchico.edu /phil/sdobra_mat/catlog.html (3566 words)

	Chap 8 Notes (Site not responding. Last check: 2007-09-07)
		In the present chapter, we use the logic of categorical statements, which is a powerful tool for handling one large group of arguments.
		Logical Reasoning questions include questions about a passage's main idea or conclusion, about assumptions made in the passage or inferences that can be drawn from it, and about fallacies or other flaws exhibited in the passage.
		This branch of logic specifies the logical relationships among claims that can be expressed in the forms "All Xs are Ys," "No Xs are Ys," "Some Xs are Ys," and "Some Xs are not Ys." Developed by Aristotle in the fourth century B.C.E., categorical logic is also known as Aristotelian or traditional logic.
	www.humboldt.edu /~campbell/p100lectch8.htm (4663 words)

	Sophia on the web: Logic Page ~ Categorical Diagraming
		For example the proposition that "All humans are mortal" is a categorical proposition, and asserts (roughly) that all members of the class of humans are members of the class of mortals.
		The description of a categorical syllogism in terms of mood (AAA) and figure (1) is called the form (AAA- 1) of the syllogism.
		The mood represents the categorical type of the major premise, minor premise, and conclusion of a categorical syllogism.
	members.aol.com /moresophia/topic10.html (488 words)

	Categorical Logic Review Question Set (Site not responding. Last check: 2007-09-07)
		This set of questions is designed to help you review your understanding of categorical logic generally and the Venn diagram technique of determining validity in particular.
		Restate this sentence as a standard form categorical claim: In the event payment is not received by 9 AM of the due date, a late fee of 1.5% will be added to the amount due.
		Remember, validity is a characteristic of the logic of an argument; finding that an argument is valid or invalid doesn't say anything about whether the premises or conclusion of an argument are, in fact, true.
	www.csuchico.edu /phil/gtropea_mat/catqs2.html (330 words)

	Categorical Logic (Site not responding. Last check: 2007-09-07)
		A leading idea is functorial semantics, according to which a model of a logical theory is a set-valued functor on a structured category determined by the theory.
		Similarly higher-order logic is modelled by the categorical notion of a topos.
		Lambek, J. and Scott, P.: Introduction to Higher-Order Categorical Logic.
	www.andrew.cmu.edu /user/awodey/catlog (271 words)

	Introduction to Categorical Logic
		This is why, in spite of the the sometimes obscure symbolism connected with logic, it's still perfectly reasonable to expect the practical application of logic to provide useful assistance in keeping our beliefs about ourselves and our world in some coherent order.
		Categorical logic has been serving just these purposes for over two thousand years with remarkably little modification.
		While the essentials of categorical logic fit into just a few pages, they can provide the means to analyze a wide variety of claims, arguments, and situations and can be used to focus attention in some very effective ways.
	www.csuchico.edu /phil/gtropea_mat/catintro.html (1864 words)

	Categorical logic - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-09-07)
		Categorical Logic and Type Theory (Studies in Logic and the Foundations of Mathematics)
		Logical Aspects of Computational Linguistics : Third International Conference, LACL'98 Grenoble, France, December 14-16, 1998 Selected Papers (Lecture...
		Logical Aspects of Computational Linguistics : 5th International Conference, LACL 2005, Bordeaux, France, April 28-30, 2005, Proceedings (Lecture Notes...
	encyclopedia.worldsearch.com /categorical_logic.htm (107 words)

	Traditional Logic - categorical form (Site not responding. Last check: 2007-09-07)
		Traditional Logic is not capable of analyzing the structure of complex terms, with the one exception of negative terms.
		All terms, regardless of their complexity, are symbolized as a single capital letter, except for negative terms, which are symbolized by applying the operator "non-" to a captial letter.
		Luckily the logic of Traditional Logic is quite a bit simpler than this.
	www.luc.edu /faculty/avande1/logic/ch2-categorical-form.htm (473 words)

	That Logic Blog
		One of the salient features of logical formalisms such as sequent calculi, tableaux and natural deduction is that the logical inference rules attack the main connective of a formula, either introducing it or eliminating it.
		For instance, sticking with boolean algebras, the associated logic is consistent iff I is a proper ideal.
		Taking logic to be the study of deduction, it incorporates aspects of mathematics, computer science, philosophy and linguistics and should not be seen as lying completely in any one of these.
	thatlogicblog.blogspot.com (1441 words)

	Deductive Arguments I: Categorical Logic
		In the present chapter, we use the logic of categorical statements, which dates back to Aristotle, and is a powerful tool for handling one large group of arguments.
		Categorical logic studies the relations among classes or categories of things.
		Categorical logic begins with the task of translating claims into standard-form categorical claims.
	highered.mcgraw-hill.com /sites/0072818816/student_view0/chapter8 (2912 words)

	Philosophical Dictionary: Caird-Catharsis
		A logical argument consisting of exactly three categorical propositions, two premises and the conclusion, with a total of exactly three categorical terms, each used in only two of the propositions.
		Each categorical term divides the world into two parts: the original class and its complement; the things to which the term applies and those to which it does not.
		In Aristotle's logic specifically, the categories are the ten general modes of being (substance, quantity, quality, relation, place, time, position, possession, doing, and undergoing) by reference to which any individual thing may be described.
	www.philosophypages.com /dy/c.htm (1327 words)

	Categorical Syllogism
		A categorical syllogism is an argument consisting of exactly three categorical propositions (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.
		Since one of the premises of the syllogism must be a categorical proposition that affirms some relation between its middle and major terms, we call that the major premise of the syllogism.
		This method of differentiating syllogisms is significant because the validity of a categorical syllogism depends solely upon its logical form.
	www.philosophypages.com /lg/e08a.htm (1504 words)

	Categorical Logic with Partial Elements - Coniglio (ResearchIndex) (Site not responding. Last check: 2007-09-07)
		Abstract: In this paper we extend the usual categorical semantics by rede ning the interpretation of the constant symbols.
		Fibring Logics with Topos Semantics - Coniglio, Sernadas, Sernadas (2002)
		Proceedings of the XI Brazilian Conference on Mathematical Logic.
	citeseer.ist.psu.edu /coniglio99categorical.html (464 words)

	Phil 174 Logical Terms Chapter 2 (Site not responding. Last check: 2007-09-07)
		If the term occurs on the "subject side" of a categorical sentence, then it is also called the subject term of that sentence, and it occurs on the "predicate side," it is called the predicate term.
		When the four types of categorical sentences are arranged as a square (the A and E forms at the upper left and right corners, and I and O forms at the lower left and right corners), some important logical relationships are displayed, including that opposite corners are negation opposites, that is, contradictory sentences.
		All the sentences are categorical sentences, and have a total of three terms, each of which is used (twice) in two different sentences.
	www.luc.edu /faculty/avande1/logic/defs-chap2.htm (1395 words)

	Categorical Logic - Pitts (ResearchIndex) (Site not responding. Last check: 2007-09-07)
		Equational Logic of Polynomial Coalgebras - Goldblatt (2003)
		30 The theory of constructions: categorical semantics and topos..
		1 the unity of algebra and logic (context) - Lambek - 1988
	citeseer.ist.psu.edu /pitts01categorical.html (1006 words)

	Propositional Logic Applet
		Categorical Logic: Three chapters in Copi address categorical logic, one on categorical propositions, one on syllogisms, and one on translation.
		The introduction of the ideas of predicate logic and quantification is good, but there is no discussion of relational predicates.
		This theorem is the basis of reasoning in propositional logic.
	www.wu.ece.ufl.edu /books/CS/TheoreticalCS/logic.html (1382 words)

	[No title] (Site not responding. Last check: 2007-09-07)
		We will take a close look at recent developments in symbolic logic, but we will not ignore the ancient and still helpful Aristotel ian account of the syllogism from which all later developments came.
		We will discuss the sense in which logic involves analytical reasoning skills, as well as the strengths and limitations of logic.
		The categorical proposition and th e categorical syllogism, the "Square of Opposition," Venn Diagrams, and the rules of syllogistic inference will be among the topics covered.
	mason.gmu.edu /~dgregory/173word.htm (574 words)

	Square of Opposition [Internet Encyclopedia of Philosophy]
		The square of opposition is a chart that was introduced within classical (categorical) logic to represent the logical relationships holding between certain propositions in virtue of their form.
		Modern logic deals with uninstantiated terms such as "unicorn" and "ether flow" the same as it does other terms such as "apple" and "orangutan".
		Consider, e.g., "all unicorns have horns" and "no unicorns have horns." Within contemporary logic, these are both regarded as true, so strictly speaking, they cannot be contrary, despite the former's status as an A proposition and the latter's status as an E proposition.
	www.iep.utm.edu /s/sqr-opp.htm (691 words)

	Stephen Downes : Logical Fallacies: References (Site not responding. Last check: 2007-09-07)
		This is a classical logic text with numbered paragraphs and a focus on the universal and the particular.
		Instead of describing categorical logic, it insteads treats the subject (more accurately, in my view) as a branch of set theory and the logic of properties and relations.
		While it focusses entirely on deductive logic, its crisp definitions and theorems supplement a traditional (truth table) method of derivation along with truth trees.
	www.datanation.com /fallacies/refernce.htm (1355 words)

	Amazon.com: Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics): Books: J. ... (Site not responding. Last check: 2007-09-07)
		Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory.
		Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given.
		Categorical Logic and Type Theory (Studies in Logic and the Foundations of Mathematics) by B.
	www.amazon.com /exec/obidos/tg/detail/-/0521356539?v=glance (884 words)

	Deductive Logic Final Exam Review Sheet
		Be able to give some examples of things that are logically and/or physically impossible.
		Also, make sure that you can identify the subject and predicate terms (always nouns or noun phrases) in categorical propositions, as well as their distributions, and practice translating everyday English claims into those of the four standard forms.
		The basic idea behind refutation by logical analogy remains the same in modern symbolic logic, though you should note that our analysis of the logical form of arguments in modern symbolic logic is different from the (mood-figure) analysis of categorical logic.
	www.medaille.edu /gerion/ub/phi215mfc/examreview/finalreview.html (868 words)

	Categorical Logic (Site not responding. Last check: 2007-09-07)
		Once you know whether a given categorical statement is true or false, you also know, without the addition of more information, the truth value of certain corresponding categorical claims.
		Immediate Inference and Venn Diagrams: Although Venn diagrams are primarily used to evaluate cases of mediate inference, i.e., categorical arguments having two premises and a conclusion, they may also be used to evaluate immediate inference, where the conclusion is drawn from only one premise.
		Note that categorical arguments may be precisely identified by specifying two characteristics, referred to as Mood and Figure.
	a-s.clayton.edu /jackson/deduct.html (1966 words)

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