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# Topic: Predicate logic

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 Logic - Wikipedia, the free encyclopedia The ambiguity is that "formal logic" is very often used with the alternate meaning of symbolic logic as we have defined it, with informal logic meaning any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory". The discovery of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Theoretical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of the predicate logic allowed the formalisation of mathematics, and drove the investigation of set theory, allowed the development of Alfred Tarski's approach to model theory; it is no exaggeration to say that it is the foundation of modern mathematical logic. en.wikipedia.org /wiki/Logic   (3434 words)

 Predicate calculus - Wikipedia, the free encyclopedia In mathematical logic the predicate calculus, predicate logic or calculus of propositional functions is a formal system used to describe mathematical theories. The predicate calculus is an extension of propositional calculus, which is inadequate for describing more complex mathematical structures. A subject is a name for a member of a given group of individuals (a set) and a predicate is a relation on this group. en.wikipedia.org /wiki/Predicate_logic   (109 words)

 First-order logic - Wikipedia, the free encyclopedia First-order logic is mathematical logic that is distinguished from higher-order logic in that it does not allow quantification over properties; i.e. Nevertheless, first-order logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. The predicate calculus is an extension of the propositional calculus. www.wikipedia.org /wiki/First-order_predicate_calculus   (791 words)

 Predicate Logic   (Site not responding. Last check: 2007-11-07) Predicate logic is a mathematical model for reasoning with predicates (just as propositional logic is an algebra for reasoning about the truth of logical expressions). Predicates are atomic operands in the logical expressions of predicate logic. A logical expression in predicate logic has much the same form as a logical expression in propositional logic, with the addition of atomic formulae (ie., predicates), and the universal and existential quantifiers. www.cs.rochester.edu /u/leblanc/csc173.94/predlogic.html   (3023 words)

 Syntactic and Semantic Foundations of Predicate Logic Whereas, semantically, a 2-place predicate says something about a relationship between two elements of a set, a 0-place predicate just ``says something'' (which may be true or false, in a given state), but says nothing about the elements of the set in question. In propositional logic, one has the ``standard models'' for interpreting the uninterpreted strings of symbols that we recursively defined as sentences (we have the notion of truth value assignments, of states, of the truth table definitions of the connectives, etc.). One can go further and introduce ``predicate logic with equality and function symbols'', ``typed predicate logic'', ``theories'', in which certain predicate symbols and function symbols are singled out for special roles in supplemental axioms of the logic, etc. We may not have time for much beyond pure predicate logic. www.math.yorku.ca /Courses/9798/Math2090/predsyntax/predsyntax.html   (518 words)

 First-Order Predicate Logic A predicate may be thought of as a kind of function which applies to individuals (which would not usually themselves be propositions) and yields a proposition. Analysing the predicate structure of sentences permits us to make use of the internal structure of atomic sentences, and to understand the structure of arguments which cannot be accounted for by propositional logic alone. Unlike propositional logics, in which specific propositional operators are identified and treated, predicate logic uses arbitrary names for predicates and relations which have no specific meaning (until an attempt may be made to apply the logic). www.rbjones.com /rbjpub/logic/log019.htm   (521 words)

 Predicate Logic   (Site not responding. Last check: 2007-11-07) Predicate logic is the study of why the above is funny. Actually, predicate logic is a mathematical model for reasoning with predicates (just as propositional logic is an algebra for reasoning about the truth of logical expressions). As in propositional logic, we can create logical expressions containing predicates, manipulate those expressions according to the algebraic laws of predicate logic, and construct proofs using rules of inference to deduce new facts from axioms. www.cs.rochester.edu /users/faculty/nelson/courses/csc_173/predlogic   (145 words)

 Peter Suber, "Glossary of First-Order Logic" A wff A of propositional logic created from a wff B of predicate logic by (1) removing the quantifiers from B, and (2) replacing each predicate symbol (and its arguments) in B with a propositional symbol. In wffs of predicate logic in which all quantifiers are clustered together at the left side, the section to the right of the quantifiers. In predicate logic wffs in which all quantifiers are clustered at the left, the section of quantifiers. www.earlham.edu /~peters/courses/logsys/glossary.htm   (9715 words)

 predicate logic   (Site not responding. Last check: 2007-11-07) (Or "predicate calculus") An extension of propositional logic with separate symbols for predicates, subjects, and quantifiers. For example, where propositional logic might assign a single symbol P to the proposition "All men are mortal", predicate logic can define the predicate M(x) which asserts that the subject, x, is mortal and bind x with the universal quantifier ("For all"): Higher-order predicate logic allows predicates to be the subjects of other predicates. www.linuxguruz.com /foldoc/foldoc.php?predicate+logic   (101 words)

 Intuitionistic Logic Intuitionistic logic encompasses the principles of logical reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics. While identity can of course be added to intuitionistic logic, for applications (e.g., to arithmetic) the equality symbol is generally treated as a distinguished predicate constant satisfying nonlogical axioms (e.g., the primitive recursive definitions of addition and multiplication) in addition to reflexivity, symmetry and transitivity. plato.stanford.edu /entries/logic-intuitionistic   (6042 words)

 Predicate Logic, Inc.   (Site not responding. Last check: 2007-11-07) Predicate Logic is committed to protecting your consumer privacy while online at www.predicate.com. If you apply for a job at Predicate Logic, we request information about you, but this will be done only with your permission and consent, except where required by law. Predicate Logic reserves the right to change this policy at any time and will post any changes to this policy as soon as they go into effect. www.predicate.com /privacy.shtml   (290 words)

 Peter Suber, "Predicate Logic Terms and Symbols" Grammatically, the argument to a one-place predicate is the subject of the sentence. The arguments to a many-place predicate are the subject of the sentence and objects of the verb. Hence, monadic predicate logic is sometimes called the logic of attributes, and polyadic predicate logic is sometimes called the logic of relations. www.earlham.edu /~peters/courses/log/terms3.htm   (1736 words)

 Introduction to Logic   (Site not responding. Last check: 2007-11-07) The foundation of the logic we are going to learn here was laid down by a British mathematician George Boole in the middle of the 19th century, and it was further developed and used in an attempt to derive all of mathematics by Gottlob Frege, a German mathematician, towards the end of the 19th century. In logic we are interested in true or false of statements, and how the truth/falsehood of a statement can be determined from other statements. There are various types of logic such as logic of sentences (propositional logic), logic of objects (predicate logic), logic involving uncertainties, logic dealing with fuzziness, temporal logic etc. Here we are going to be concerned with propositional logic and predicate logic, which are fundamental to all types of logic. www.cs.odu.edu /~toida/nerzic/content/logic/intr_to_logic.html   (262 words)

 Modal Logic Modal logic is, strictly speaking, the study of the deductive behavior of the expressions ‘it is necessary that’ and ‘it is possible that’. For this reason, there is no one modal logic, but rather a whole family of systems built around M. The relationship between these systems is diagrammed in Section 8, and their application to different uses of ‘necessarily’ and ‘possibly’ can be more deeply understood by studying their possible world semantics in Section 6. Deontic logics introduce the primitive symbol O for ‘it is obligatory that’, from which symbols P for ‘it is permitted that’ and F for ‘it is forbidden that’ are defined: PA = ~O~A and FA = O~A. The deontic analog of the modal axiom (M): OA→A is clearly not appropriate for deontic logic. plato.stanford.edu /entries/logic-modal   (7308 words)

 Temporal Logic Tense Logic was introduced by Arthur Prior (1957, 1967, 1969) as a result of an interest in the relationship between tense and modality attributed to the Megarian philosopher Diodorus Cronus (ca. This kind of manoeuvre lies at the heart of hybrid temporal logics in which the standard apparatus of propositions and tense operators is supplemented by propositions which are true at unique instants, thereby effectively naming those instants without invoking philosophically dubious reification. Prior's motivation for inventing Tense Logic was largely philosophical, his idea being that the precision and clarity afforded by a formal logical notation was indispensible for the careful formulation and resolution of philosophical issues concerning time. plato.stanford.edu /entries/logic-temporal   (3508 words)

 symbolic logic -> The Predicate Calculus on Encyclopedia.com 2002   (Site not responding. Last check: 2007-11-07) Socrates is a man. Therefore Socrates is mortal.” The syllogism and many other more complicated arguments are the subject of the predicate calculus, or quantification theory, which is based on the calculus of classes. The predicate calculus of monadic (one-variable) predicates, also called uniform quantification theory, has been shown to be complete and has a decision procedure, analogous to truth tables for truth-functional analysis, whereby the validity or invalidity of any statement can be determined. The general predicate calculus, or quantification theory, was also shown to be complete by Kurt Gödel, but Alonso Church subsequently proved (1936) that it has no possible decision procedure. www.encyclopedia.com /html/section/symbolic_thepredicatecalculus.asp   (384 words)

 Predication I: Simple Predications Predicate logic is an extension of sentential logic which studies why those additional arguments are valid. You might put it this way: a predicate expression is something that becomes a sentence when its blank space (or spaces) is (or are) filled in with names; a name is a word that can be used to fill in a blank in a predicate expression. That's important in defining a wff in predicate logic: an n-place predicate letter followed by exactly n names is a wff, but if it is followed by more or fewer than n names it's not a wff. aristotle.tamu.edu /~rasmith/Courses/Logic/predication.1.html   (1464 words)

 Formal Specification of First-Order Predicate Logic in ML The system is a classical first-order predicate logic with two propositional connectives (not and implies) and a universal quantifier, presented as a "Hilbert style" axiom system in which there are two inference rules, modus ponens and generalisation, and six axiom schemata. The second component named map assigns to each predicate letter a set of tuples of elements from the domain of interpretation for which the predicate is true under the interpretation. The logic is boolean and therefore the predicate is to be considered false at all other points. www.rbjones.com /rbjpub/logic/log021.htm   (1419 words)

 The lower predicate calculus (from formal logic) --  Encyclopædia Britannica   (Site not responding. Last check: 2007-11-07) A predicate calculus in which the only variables that occur in quantifiers are individual variables is known as a lower (or first-order) predicate calculus. in logic and mathematics, abstract, theoretical organization of terms and implicit relationships that is used as a tool for the analysis of the concept of deduction. Discusses the Aristotelean logic, predicate calculus, geometry of Euclid, formal theories of mathematics, and Plato and Aristotle’s philosophy of mathematics. www.britannica.com /eb/article-65843   (887 words)

 Mechanized Reasoning Systems   (Site not responding. Last check: 2007-11-07) E is a theorem prover for clausal logic with equality. MONA is an implementation of decision procedures for weak second-order logics of successors. Nqthm is a prover for quantifier free logic for recursive functions over the integers and other finitely generated structures, combining rewriting, heuristics for induction, and other techniques. www-formal.stanford.edu /clt/ARS/systems.html   (1545 words)

 Frege's Logic, Theorem, and Foundations for Arithmetic Again, these are essentially the same as the rules for the first-order predicate calculus, except for the addition of new rules for the second-order quantifiers that correspond to the generalization and instantiation rules (i.e., introduction and elimination rules) for the first-order quantifiers. Thus, Frege's second-order logic and theory of extensions together required the impossible situation in which the domain of concepts has to be strictly larger than the domain of extensions while at the same time the domain of extensions has to be as large as the domain of concepts. A properly reformulated theory of ‘logical’ objects should have: (1) a separate non-logical comprehension principle which explicitly asserts the existence of logical objects, and (2) a separate identity principle which asserts the conditions under which logical objects are identical. plato.stanford.edu /entries/frege-logic   (15095 words)

 Copi & Cohen: Textbook   (Site not responding. Last check: 2007-11-07) Categorical Logic: Three chapters in Copi address categorical logic, one on categorical propositions, one on syllogisms, and one on translation. Predicate Logic: A single chapter introduces predicate logic and extends the proof techniques to them. The introduction of the ideas of predicate logic and quantification is good, but there is no discussion of relational predicates. mbyron.philosophy.kent.edu /Logic/textbooks/copi.html   (357 words)

 A Note on Fuzzy Predicate Logic (ResearchIndex)   (Site not responding. Last check: 2007-11-07) The set of all formulas of predicate logic that are tautologies with respect to all continuous t-norms is shown to be heavily non-recursive (\Pi 2 -hard). 6 Basic fuzzy logic is the logic of continuous t-norms and the.. Trakhtenbrot Theorem and Fuzzy Logic - Hajek (1996) citeseer.ist.psu.edu /302387.html   (250 words)

 Brainstorms: William A. Dembski: Random Predicate Logic I: A Probabilistic Approach to Vagueness Since Bart Kosko has shown that fuzzy logic cannot be subsumed under Bayesian techniques, and since fuzzy logic is strictly subsumable under random predicate logic, I intend, as it were, to transcend Bayesian techniques by moving to a completely different formalism. Any fuzzy (or crisp for that matter) logic combination operaton set can only approximate refined description of relationships in the real world with some degree of a "music of the spheres" ad-hoc approximation. Now that is probably the very point intended in Fuzzy Logic theory, that one can arbitrarily approximate what is needed, and do so with less symbology than with crisp logic. www.iscid.org /boards/ubb-get_topic-f-6-t-000133.html   (577 words)

 Predicate Logic Statement logic is very good at what it does. However, because it cannot make the very important distinction between "all" and "some," it is severely restricted in the kind and complexity of arguments with which it can effectively deal. Predicate logic expands upon statement logic, using all of the same concepts and rules, and with the addition of two more symbols and four more rules.. home.att.net /~tangents/issue/think/predicat.htm   (175 words)

 Introduction to Predicate Logic   (Site not responding. Last check: 2007-11-07) The propositional logic is not powerful enough to represent all types of assertions that are used in computer science and mathematics, or to express certain types of relationship between propositions such as equivalence. For example, the assertion "x is greater than 1", where x is a variable, is not a proposition because you can not tell whether it is true or false unless you know the value of x. The predicate logic is one of such logic and it addresses these issues among others. www.cs.odu.edu /~toida/nerzic/content/logic/pred_logic/intr_to_pred_logic.html   (191 words)

 Translation Strategies for Monadic Predicate Logic Statements in predicate logic will either be a singular statement (or a truth functional combination of singular statements) or a quantified statement (or, at worst, a truth functional statement that contains a quantified statement as a component). For example, the singular statement 'Jane is married' might use the predicate letter 'M' for the attribute of being married and the individual constant 'j' for Jane. General statements with two predicates will be one of the four traditional categorical statements or a variation on one of them. www.hu.mtu.edu /~wsewell/hu250/monadic_tips.htm   (1626 words)

 03: Mathematical logic and foundations   (Site not responding. Last check: 2007-11-07) Mathematical Logic is the study of the processes used in mathematical deduction. In second-order logic, the quantifiers are allowed to apply to relations and functions -- to subsets as well as elements of a set. There are some elementary exercises in first-order and predicate logic available for students on a separate math teaching page. www.math.niu.edu /~rusin/known-math/index/03-XX.html   (2050 words)

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