Predicate logic for instance does not allow to apply the quantifiers on function nor predicate names.
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Some authors refer to "predicate logic with identity" to emphasize this extension. See more about this below.
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In particular, the grammar of Lojban is carefully engineered to express such predicate logic in an unambiguous manner.
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Dynamic Predicate Logic models pronouns as first-order logic variables, but allows quantifiers in a formula to bind variables in other formulae.
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Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables.
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Fig 3. Empirical Model Modal predicate logic (a combination of modal logic and predicate logic) is used as the formal method of knowledge representation. The connectives from the language model are logically true (indicated by the "L" modal operator) and connective added at the knowledge elicitation stage are possibility true (indicated by the "M" modal operator). Before proceeding to stage 5, the models are expressed in logical formulae.
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Logico-linguistic modeling is a method for building knowledge-based systems with a learning capability using conceptual models from soft systems methodology, modal predicate logic, and the Prolog artificial intelligence language.
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Later Moshe Y. Vardi made a conjecture that a tree model would work for many modal style logics. The guarded fragment of first-order logic was first introduced by Hajnal Andréka, István Németi and Johan van Benthem in their article Modal languages and bounded fragments of predicate logic. They successfully transferred key properties of description, modal, and temporal logic to predicate logic. It was found that the robust decidability of guarded logic could be generalized with a tree model property.
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Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i.e., sentences that are true in every model).
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These results confirm the validity of the argument A Some arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truth tables forms. Consider the argument A1: > Some mortals are not Greeks > Some Greeks are not men > Not every man is a logician > Therefore Some mortals are not logicians > To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicate logic), negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid.
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In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic. The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory).
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In predicate logic, generalization (also universal generalization or universal introduction,Copi and CohenHurleyMoore and Parker GEN) is a valid inference rule. It states that if \vdash \\!P(x) has been derived, then \vdash \\!\forall x \, P(x) can be derived.
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Lambert's law is the major principle in any free definite description theory that says: For all x, x = the y (A) if and only if (A(x/y) & for all y (if A then y = x)). Free logic itself is an adjustment of a given standard predicate logic such as to relieve it of existential assumptions, and so make it a free logic. Taking Bertrand Russell's predicate logic in his Principia Mathematica as standard, one replaces universal instantiation, \forall x \,\phi x \rightarrow \phi y, with universal specification (\forall x \,\phi x \land E!y \,\phi y) \rightarrow \phi z.
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In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including Enderton (2001), Gamut (1990), and Kleene (1967) A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.
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Per Martin-Löf found a type theory that corresponded to predicate logic by introducing dependent types, which became known as intuitionistic type theory or Martin-Löf type theory. Martin-Löf's theory uses inductive types to represent unbounded data structures, such as natural numbers.
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In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something".
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Prolog programs describe relations, defined by means of clauses. Pure Prolog is restricted to Horn clauses, a Turing-complete subset of first- order predicate logic. There are two types of clauses: Facts and rules. A rule is of the form Head :- Body.
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An example of a rule that is not effective in this sense is the infinitary ω-rule. Popular rules of inference in propositional logic include modus ponens, modus tollens, and contraposition. First-order predicate logic uses rules of inference to deal with logical quantifiers.
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Parlog is a logic programming language designed for efficient utilization of parallel computer architectures. Its semantics is based on first order predicate logic. It expresses concurrency, interprocess communication, indeterminacy and synchronization within the declarative language framework.Andrew Cheese, "Parallel execution of Parlog", Springer, 1992, , 184 pp.
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For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.
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It was while Hasenjaeger was working at Westfälische Wilhelms-Universität University in Münster in the period between 1946 and 1953 that Hasenjaeger made a most amazing discovery - a proof of Kurt Gödel's Gödel's completeness theorem for full predicate logic with identity and function symbols. Gödel's proof of 1930 for predicate logic did not automatically establish a procedure for the general case. When he had solved the problem in late 1949, he was frustrated to find that a young American mathematician Leon Henkin, had also created a proof. Both construct from extension of a term model, which is then the model for the initial theory.
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Attributional calculus is a logic and representation system defined by Ryszard S. Michalski. It combines elements of predicate logic, propositional calculus, and multi-valued logic. Attributional calculus provides a formal language for natural induction, an inductive learning process whose results are in forms natural to people.
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In the late 19th century, the predicate logic of Gottlob Frege (1848–1925) overthrew Aristotelian logic (the dominant logic since its inception in Ancient Greece). This was the beginning of analytic philosophy. In the early part of the 20th century, a group of German and Austrian philosophers and scientists formed the Vienna Circle to promote scientific thought over Hegelian system-building, which they saw as a bad influence on intellectual thought. The group considered themselves logical positivists because they believed all knowledge is either derived through experience or arrived at through analytic statements, and they adopted the predicate logic of Frege, as well as the early work of Ludwig Wittgenstein (1889–1951) as foundations to their work.
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The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality. Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is the substitution property, :\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \implies \forall Y \, (A \in Y \iff B \in Y) \, ), and it becomes this axiom that is referred to as the axiom of extensionality in this context.
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Formulae in predicate logic translate easily into the Prolog artificial intelligence language. The modality is expressed by two different types of Prolog rules. Rules taken from the language creation stage of model building process are treated as incorrigible. While rules from the knowledge elicitation stage are marked as hypothetical rules.
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Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia.Boehner p. xiv The Stoics, especially Chrysippus, began the development of predicate logic.
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In machine learning, semantic analysis of a corpus is the task of building structures that approximate concepts from a large set of documents. It generally does not involve prior semantic understanding of the documents. A metalanguage based on predicate logic can analyze the speech of humans. Another strategy to understand the semantics of a text is symbol grounding.
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Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (extensions), by additional quantifiers that range over terms that may have such individuals as their value (intensions). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.
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These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal (resp. existential) quantifier in t-norm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula.
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Heinrich Behmann, 1930 at Jena Heinrich Behmann (10 January 1891, in Bremen- Aumund - 3 February 1970, in Bremen-Aumund) was a German mathematician. He performed research in the field of set theory and predicate logic. Behmann studied mathematics in Tübingen, Leipzig and Göttingen. During World War I, he was wounded and received the Iron Cross 2nd Class.
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Logical languages are meant to allow (or enforce) unambiguous statements. They are typically based on predicate logic but can also be based on any system of formal logic. The two best-known logical languages are the predicate languages Loglan and its successor Lojban. They both aim to eliminate syntactical ambiguity and reduce semantic ambiguity to a minimum.
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The predicate in traditional grammar is inspired by propositional logic of antiquity (as opposed to the more modern predicate logic). A predicate is seen as a property that a subject has or is characterized by. A predicate is therefore an expression that can be true of something. Thus, the expression "is moving" is true of anything that is moving.
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He thereby argued that the universe is the totality of actual states of affairs and that these states of affairs can be expressed by the language of first-order predicate logic. Thus a picture of the universe can be construed by means of expressing atomic facts in the form of atomic propositions, and linking them using logical operators.
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Logical connectives, along with quantifiers, are the two main types of logical constants used in formal systems (such as propositional logic and predicate logic). Semantics of a logical connective is often (but not always) presented as a truth function. A logical connective is similar to, but not equivalent to, a syntax commonly used in programming languages called a conditional operator.
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In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers)Johannes Stern, Toward Predicate Approaches to Modality, Springer, 2015, p. 11. that operate on terms to yield terms.
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Premise (1) states that "dog" is a subkind of the kind "mammal". Premise (4) is a (universal negative) claim about the kind "mammal". Statement (5) concludes that what is denied of the kind "mammal" is denied of the subkind "dog". Each of these two principles is an instance of a valid argument form known as universal hypothetical syllogism in first-order predicate logic.
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Before the use of guarded logic there were two major terms used to interpret modal logic. Mathematical logic and database theory (Artificial Intelligence) were first- order predicate logic. Both terms found sub-classes of first-class logic and efficiently used in solvable languages which can be used for research. But neither could explain powerful fixed-point extensions to modal style logics.
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Gisbert F. R. Hasenjaeger (June 1, 1919 – September 2, 2006) was a German mathematical logician. Independently and simultaneously with Leon Henkin in 1949, he developed a new proof of the completeness theorem of Kurt Gödel for predicate logic. He worked as an assistant to Heinrich Scholz at Section IVa of Oberkommando der Wehrmacht Chiffrierabteilung, and was responsible for the security of the Enigma machine.
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Czechoslovak Academy of Sciences, 1973, pp. 105–118. Kowalski, on the other hand, developed SLD resolution,Robert Kowalski, "Predicate Logic as a Programming Language", Memo 70, Department of Artificial Intelligence, Edinburgh University, 1973. Also in Proceedings IFIP Congress, Stockholm, North Holland Publishing Co., 1974, pp. 569–574. a variant of SL-resolution,Robert Kowalski and Donald and Kuehner, "Linear Resolution with Selection Function", Artificial Intelligence, Vol.
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An existential clause is a clause that refers to the existence or presence of something. Examples in English include the sentences "There is a God" and "There are boys in the yard". The use of such clauses can be considered analogous to existential quantification in predicate logic (often expressed with the phrase "There exist(s)..."). Different languages have different ways of forming and using existential clauses.
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Non-classical logic is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.Theodore Sider, (2010).
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The term reasoning system can be used to apply to just about any kind of sophisticated decision support system as illustrated by the specific areas described below. However, the most common use of the term reasoning system implies the computer representation of logic. Various implementations demonstrate significant variation in terms of systems of logic and formality. Most reasoning systems implement variations of propositional and symbolic (predicate) logic.
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Pure inductive logic (PIL) is the area of mathematical logic concerned with the philosophical and mathematical foundations of probabilistic inductive reasoning. It combines classical predicate logic and probability theory (Bayesian inference). Probability values are assigned to sentences of a first order relational language to represent degrees of belief that should be held by a rational agent. Conditional probability values represent degrees of belief based on the assumption of some received evidence.
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The interpretations of propositional logic and predicate logic described above are not the only possible interpretations. In particular, there are other types of interpretations that are used in the study of non-classical logic (such as intuitionistic logic), and in the study of modal logic. Interpretations used to study non-classical logic include topological models, Boolean-valued models, and Kripke models. Modal logic is also studied using Kripke models.
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The name "SLD resolution" was given by Maarten van Emden for the unnamed inference rule introduced by Robert Kowalski.Robert Kowalski Predicate Logic as a Programming Language Memo 70, Department of Artificial Intelligence, University of Edinburgh. 1973. Also in Proceedings IFIP Congress, Stockholm, North Holland Publishing Co., 1974, pp. 569-574. Its name is derived from SL resolution,Robert Kowalski and Donald Kuehner, Linear Resolution with Selection Function Artificial Intelligence, Vol.
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SBVR has the greatest expressivity of any OMG modeling language. The logics supported by SBVR are typed first order predicate logic with equality, restricted higher order logic (Henkin semantics), restricted deontic and alethic modal logic, set theory with bag comprehension, and mathematics. SBVR also includes projections, to support definitions and answers to queries, and questions, for formulating queries. Interpretation of SBVR semantic formulations is based on model theory.
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The closely related concept in set theory (see: projection (set theory)) differs from that of relational algebra in that, in set theory, one projects onto ordered components, not onto attributes. For instance, projecting (3,7) onto the second component yields 7. Projection is relational algebra's counterpart of existential quantification in predicate logic. The attributes not included correspond to existentially quantified variables in the predicate whose extension the operand relation represents.
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A knowledge representation language may be sufficiently expressive to describe nuances of meaning in well understood fields. There are at least five levels of complexity of these. For general semi-structured data one may use a general purpose language such as XML.XML as a tool for Semantic Interoperability Semantic Interoperability on the Web, Jeff Heflin and James Hendler Languages with the full power of first- order predicate logic may be required for many tasks.
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If, however, the variable y is not mentioned elsewhere (i.e. it can still be chosen freely, without influencing the other formulae), then one may assume, that A[y/x] holds for any value of y. The other rules should then be pretty straightforward. Instead of viewing the rules as descriptions for legal derivations in predicate logic, one may also consider them as instructions for the construction of a proof for a given statement.
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This theory of deductive reasoning – also known as term logic – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic. Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).
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These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.
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Example: Codechart modelling the Composite pattern in LePUS3 LePUS3 is a language for modelling and visualizing object-oriented (Java, C++, C#) programs and design patterns. It is defined as a formal specification language, formulated as an axiomatized subset of First-order predicate logic. A diagram in LePUS3 is also called a Codechart.Codechart, formal definition LePUS, the name of the first version of the language, is an abbreviation for Language for Pattern Uniform Specification.
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In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system. An example of deductive system is first order predicate logic.
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A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
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These type systems do not have decidable type inference and are difficult to understand and program with. But dependent types can express arbitrary propositions in predicate logic. Through the Curry–Howard isomorphism, then, well-typed programs in these languages become a means of writing formal mathematical proofs from which a compiler can generate certified code. While these languages are mainly of interest in academic research (including in formalized mathematics), they have begun to be used in engineering as well.
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Transaction Logic is an extension of predicate logic that accounts in a clean and declarative way for the phenomenon of state changes in logic programs and databases. This extension adds connectives specifically designed for combining simple actions into complex transactions and for providing control over their execution. The logic has a natural model theory and a sound and complete proof theory. Transaction Logic has a Horn clause subset, which has a procedural as well as a declarative semantics.
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In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
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2004 The symbol ∀ has the same shape as a capital turned A, sans- serif. It is used to represent universal quantification in predicate logic. It was first used in this way by Gerhard Gentzen in 1935, by analogy with Giuseppe Peano's turned E notation for existential quantification and the later use of Peano's notation by Bertrand Russell. In traffic engineering it is used to represent flow, the number of units (vehicles) passing a point in a unit of time.
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Define Naive Set Theory (NST) as the theory of predicate logic with a binary predicate \in and the following axiom schema of unrestricted comprehension: : \exists y \forall x (x \in y \iff \varphi(x)) for any formula \varphi with only the variable x free. Substitute x otin x for \varphi(x). Then by existential instantiation (reusing the symbol y) and universal instantiation we have :y \in y \iff y otin y a contradiction. Therefore, NST is inconsistent.
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This article is concerned only with this traditional use. The syllogism was at the core of traditional deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations. Within an academic context, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879). However, syllogisms remain useful in some circumstances, and for general-audience introductions to logic.
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CMDB schematic structures, also known as database schemas, take on multiple forms. Two of the most common forms are those of a relational data model and a semantic data model. Relational data models are based on first-order predicate logic and all data is represented in terms of tuples that are grouped into relations. In the relational model, related records are linked together with a "key", where the key is unique to an entry's data type definition.
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Additional axioms result in the Zermelo–Fraenkel set theory, which is much more handy in his class-logical representation than in the usual predicate logical representation.Gegenüberstellung von ZFC in klassenlogischer und prädikatenlogischer Form [Comparison of ZFC in class logic vs. predicate logic form], in: Oberschelp, Allgemeine Mengenlehre, 1994, p. 261 In 1962 he gave a lecture as an invited speaker at the International Congress of Mathematicians in Stockholm on classes as "primal elements" in set theory.
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Russell became an advocate of logical atomism. Wittgenstein developed a comprehensive system of logical atomism in his Tractatus Logico-Philosophicus (, 1921). He thereby argued that the universe is the totality of actual states of affairs and that these states of affairs can be expressed by the language of first-order predicate logic. Thus a picture of the universe can be construed using expressing atomic facts in the form of atomic propositions and linking them using logical operators.
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Critics have suggested that Cosmides and Tooby use untested evolutionary assumptions to eliminate rival reasoning theories and that their conclusions contain inferential errors. Davies et al., for example, have argued that Cosmides and Tooby did not succeed in eliminating the general- purpose theory because the adapted Wason selection task they used tested only one specific aspect of deductive reasoning and failed to examine other general-purpose reasoning mechanisms (e.g., reasoning based on syllogistic logic, predicate logic, modal logic, and inductive logic etc.).
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An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.
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A formal language for higher-order predicate logic looks much the same as a formal language for first-order logic. The difference is that there are now many different types of variables. Some variables correspond to elements of the domain, as in first-order logic. Other variables correspond to objects of higher type: subsets of the domain, functions from the domain, functions that take a subset of the domain and return a function from the domain to subsets of the domain, etc.
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Semantics is the branch of linguistics that examines the meaning of natural language, the notion of reference and denotation, and the concept of possible worlds. One concept used in the study of semantics is predicate logic, which is a system that uses symbols and alphabet letters to represent the overall meaning of a sentence. Quantifiers in semantics – such as the quantifier in the antecedent of a bound variable pronoun – can be expressed in two ways. There is an existential quantifier, ∃, meaning some.
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In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes checked within the code by an assertion call. Knowing its invariant(s) is essential in understanding the effect of a loop. In formal program verification, particularly the Floyd-Hoare approach, loop invariants are expressed by formal predicate logic and used to prove properties of loops and by extension algorithms that employ loops (usually correctness properties).
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While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics. Frege's Begriffsschrift (1879) introduced both a complete propositional calculus and what is essentially modern predicate logic. His Foundations of Arithmetic, published 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Russell and Whitehead in their influential Principia Mathematica, first published 1910–1913, and with a revised second edition in 1927.
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Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable). The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic.
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Their proofs demonstrate a connection between the unsolvability of the decision problem for first-order logic and the unsolvability of the halting problem. There are systems weaker than full first-order logic for which the logical consequence relation is decidable. These include propositional logic and monadic predicate logic, which is first-order logic restricted to unary predicate symbols and no function symbols. Other logics with no function symbols which are decidable are the guarded fragment of first-order logic, as well as two-variable logic.
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Loglan (an abbreviation for "logical language") was created to investigate whether people speaking a "logical language" would in some way think more logically, as the Sapir-Whorf hypothesis might predict. The language's grammar is based on predicate logic. The grammar was intended to be small enough to be teachable and manageable, yet complex enough to allow people to think and converse in the language. Brown intended Loglan to be as culturally neutral as possible, and metaphysically parsimonious, which means that obligatory categories are kept to a minimum.
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In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.
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Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.Logic for philosophy, Theodore Sider Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well.
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Association for Computational Linguistics, 2006. Advanced applications of natural-language understanding also attempt to incorporate logical inference within their framework. This is generally achieved by mapping the derived meaning into a set of assertions in predicate logic, then using logical deduction to arrive at conclusions. Therefore, systems based on functional languages such as Lisp need to include a subsystem to represent logical assertions, while logic-oriented systems such as those using the language Prolog generally rely on an extension of the built-in logical representation framework.
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An interesting consequence is that proofs become mathematical objects that can be examined, compared, and manipulated. Intuitionistic type theory's type constructors were built to follow a one-to-one correspondence with logical connectives. For example, the logical connective called implication (A \implies B) corresponds to the type of a function (A \to B). This correspondence is called the Curry–Howard isomorphism. Previous type theories had also followed this isomorphism, but Martin-Löf's was the first to extend it to predicate logic by introducing dependent types.
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An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.
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The derivation in Kripke's 'Identity and Necessity' is in three steps: :(1) ∀x nec(x = x) :(2) ∀x∀y(x = y → (nec(x = x) → nec(x = y))) :(3) ∀x∀y(x = y → nec(x = y)) The first premiss is simply postulated: every object is identical to itself. The second is an application of the principle of substitutivity: if a = b, then a has all the properties b has, thus from Fa, infer Fb, where F is 'nec(a = --)'. The third follows by elementary predicate logic.
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This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is allotted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is: :Some As are Bs :All Cs are Ds which is clearly invalid. The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift (1879), the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings.
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A semantic reasoner, reasoning engine, rules engine, or simply a reasoner, is a piece of software able to infer logical consequences from a set of asserted facts or axioms. The notion of a semantic reasoner generalizes that of an inference engine, by providing a richer set of mechanisms to work with. The inference rules are commonly specified by means of an ontology language, and often a description logic language. Many reasoners use first-order predicate logic to perform reasoning; inference commonly proceeds by forward chaining and backward chaining.
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In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) is true, or P(x) is not true. This is not trivially true constructively. For a decidable predicate P over the natural numbers, Markov's principle then reads: :(\forall n (P(n) \vee eg P(n)) \wedge eg \forall n\, eg P(n)) \rightarrow \exists n\, P(n) That is, if P cannot be false for all natural numbers n, then it is true for some n.
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According to a tradition associated with predicate logic and dependency grammars, the subject is the most prominent overt argument of the predicate. By this position all languages with arguments have subjects, though there is no way to define this consistently for all languages.See Tesnière (1969:103-105) for the alternative concept of sentence structure that puts the subject and the object on more equal footing since they can both be dependents of a (finite) verb. From a functional perspective, a subject is a phrase that conflates nominative case with the topic.
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Tableaux are extended to first order predicate logic by two rules for dealing with universal and existential quantifiers, respectively. Two different sets of rules can be used; both employ a form of Skolemization for handling existential quantifiers, but differ on the handling of universal quantifiers. The set of formulae to check for validity is here supposed to contain no free variables; this is not a limitation as free variables are implicitly universally quantified, so universal quantifiers over these variables can be added, resulting in a formula with no free variables.
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In second-order intuitionistic logic, the second-order polymorphic lambda calculus (F2) was discovered by Girard (1972) and independently by Reynolds (1974). Girard proved the Representation Theorem: that in second-order intuitionistic predicate logic (P2), functions from the natural numbers to the natural numbers that can be proved total, form a projection from P2 into F2. Reynolds proved the Abstraction Theorem: that every term in F2 satisfies a logical relation, which can be embedded into the logical relations P2. Reynolds proved that a Girard projection followed by a Reynolds embedding form the identity, i.e.
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Kurt Gödel (1932) stated without proof that intuitionistic propositional logic (with no additional axioms) has the disjunction property; this result was proven and extended to intuitionistic predicate logic by Gerhard Gentzen (1934, 1935). Stephen Cole Kleene (1945) proved that Heyting arithmetic has the disjunction property and the existence property. Kleene's method introduced the technique of realizability, which is now one of the main methods in the study of constructive theories (Kohlenbach 2008; Troelstra 1973). While the earliest results were for constructive theories of arithmetic, many results are also known for constructive set theories (Rathjen 2005).
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Schönfinkel developed a formal system that avoided the use of bound variables. His system was essentially equivalent to a combinatory logic based upon the combinators B, C, I, K, and S. Schönfinkel was able to show that the system could be reduced to just K and S and outlined a proof that a version of this system had the same power as predicate logic. His paper also showed that functions of two or more arguments could be replaced by functions taking a single argument. (Reprinted lecture notes from 1967.)Kenneth Slonneger and Barry L. Kurtz.
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Z is based on the standard mathematical notation used in axiomatic set theory, lambda calculus, and first-order predicate logic. All expressions in Z notation are typed, thereby avoiding some of the paradoxes of naive set theory. Z contains a standardized catalogue (called the mathematical toolkit) of commonly used mathematical functions and predicates, defined using Z itself. Because Z notation (just like the APL language, long before it) uses many non-ASCII symbols, the specification includes suggestions for rendering the Z notation symbols in ASCII and in LaTeX.
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In this sense, along with that of Foucault's in the previous section, the analysis of a discourse examines and determines the connections among language and structure and agency. Moreover, because a discourse is a body of text meant to communicate specific data, information, and knowledge, there exist internal relations in the content of a given discourse, as well as external relations among discourses. As such, a discourse does not exist per se (in itself), but is related to other discourses, by way of inter-discursive practices. In formal semantics, discourse representation theory describes the formal semantics of a sentence using predicate logic.
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He regarded such relations as (real) qualities of things (Leibniz admitted unary predicates only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of De Morgan, Peirce, Schröder and Russell himself, now standard in predicate logic. Notably, Leibniz also declared space and time to be inherently relational. Leibniz's 1690 discovery of his algebra of conceptsLeibniz: Die philosophischen Schriften VII, 1890, pp. 236–247; translated as "A Study in the Calculus of Real Addition" (1690) by G. H. R. Parkinson, Leibniz: Logical Papers – A Selection, Oxford 1966, pp. 131–144.
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In response to criticism of his approach, emanating from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution,Kowalski, R. Predicate Logic as a Programming Language Memo 70, Department of Artificial Intelligence, Edinburgh University. 1973. Also in Proceedings IFIP Congress, Stockholm, North Holland Publishing Co., 1974, pp. 569–574. which solves problems by problem decomposition. He has advocated logic for both computer and human problem solvingKowalski, R., Logic for Problem Solving, North Holland, Elsevier, 1979 and computational logic to improve human thinkingKowalski, R., Computational Logic and Human Thinking: How to be Artificially Intelligent, Cambridge University Press, 2011.
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Instead of any unique formalization, though, he simply adjusts the axioms of a standard predicate logic such as that found in Willard Van Orman Quine's Methods of Logic. Instead of an axiom like \forall x\,Px \Rightarrow \exists x\,Px he uses ( \forall x\,Px \land \exists x\,(x = a)) \Rightarrow \exists x\,Px; this will naturally be true if the existential claim of the antecedent is false. If a name fails to refer, then an atomic sentence containing it, that is not an identity statement, can be assigned a truth value arbitrarily. Free logic is proved to be complete under this interpretation.
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The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, due to the intellectual environment at the time in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study. This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.
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Moreover, the proof of the relationship is entirely constructive, giving a way to transform a proof of in Peano arithmetic into a proof of in Heyting arithmetic. (By combining the double-negation translation with the Friedman translation, it is in fact possible to prove that Peano arithmetic is Π02-conservative over Heyting arithmetic.) The propositional mapping of φ to ¬¬φ does not extend to a sound translation of first-order logic, because is not a theorem of intuitionistic predicate logic. This explains why φN has to be defined in a more complicated way in the first-order case.
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In predicate logic, what is described in layman's terms as "something" can more specifically be regarded as existential quantification, that is, the predication of a property or relation to at least one member of the domain. It is a type of quantifier, a logical constant which is interpreted as "there exists," "there is at least one," or "for some." It expresses that a propositional function can be satisfied by at least one member of a domain of discourse. In other terms, it is the predication of a property or relation to at least one member of the domain.
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There is a big difference between the kinds of formulas seen in traditional term logic and the predicate calculus that is the fundamental advance of modern logic. The formula A(P,Q) (all Ps are Qs) of traditional logic corresponds to the more complex formula \forall x (P(x) \rightarrow Q(x)) in predicate logic, involving the logical connectives for universal quantification and implication rather than just the predicate letter A and using variable arguments P(x) where traditional logic uses just the term letter P. With the complexity comes power, and the advent of the predicate calculus inaugurated revolutionary growth of the subject.
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In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus. The concept of explicit substitutions has become notorious (despite a large number of published calculi of explicit substitutions in the literature with quite different characteristics) because the notion often turns up (implicitly and explicitly) in formal descriptions and implementation of all the mathematical forms of substitution involving variables such as in abstract machines, predicate logic, and symbolic computation.
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A related problem is when identical sentences have the same truth-value, yet express different propositions. The sentence “I am a philosopher” could have been spoken by both Socrates and Plato. In both instances, the statement is true, but means something different. These problems are addressed in predicate logic by using a variable for the problematic term, so that “X is a philosopher” can have Socrates or Plato substituted for X, illustrating that “Socrates is a philosopher” and “Plato is a philosopher” are different propositions. Similarly, “I am Spartacus” becomes “X is Spartacus”, where X is replaced with terms representing the individuals Spartacus and John Smith.
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Instead of attempting to derive the conclusion from the premises proceed as follows. To test the validity of an argument (a) translate, as necessary, each premise and the conclusion into sentential or predicate logic sentences (b) construct from these the negation of the corresponding conditional (c) see if from it a contradiction can be derived (or if feasible construct a truth table for it and see if it comes out false on every row.) Alternatively construct a truth tree and see if every branch is closed. Success proves the validity of the original argument. In case of the difficulty in trying to derive a contradiction, one should proceed as follows.
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Many treatments of predicate logic don't allow functional predicates, only relational predicates. This is useful, for example, in the context of proving metalogical theorems (such as Gödel's incompleteness theorems), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter). But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result. Specifically, if F has domain type T and codomain type U, then it can be replaced with a predicate P of type (T,U).
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A formal system is syntactically complete or deductively complete or maximally complete if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of . This is also called negation completeness, and is stronger than semantic completeness. In another sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation).
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The T-schema ("truth schema"; not to be confused with 'Convention T') is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Michael Dummett. The T-schema is often expressed in natural language, but it can be formalized in many-sorted predicate logic or modal logic; such a formalisation is called a "T-theory." T-theories form the basis of much fundamental work in philosophical logic, where they are applied in several important controversies in analytic philosophy.
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In philosophy and mathematical logic, mereology (from the Greek μέρος meros (root: μερε- mere-, "part") and the suffix -logy "study, discussion, science") is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets. Mereology has been explored in various ways as applications of predicate logic to formal ontology, in each of which mereology is an important part. Each of these fields provides its own axiomatic definition of mereology.
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The desire for a language more perfect than any natural language had been expressed before Leibniz by John Wilkins in his An Essay towards a Real Character and a Philosophical Language in 1668. Leibniz attempts to work out the possible connections between algebra, infinitesimal calculus, and universal character in an incomplete treatise titled "Mathesis Universalis" in 1695. Predicate logic could be seen as a modern system with some of these universal characteristics, at least as far as mathematics and computer science are concerned. More generally, mathesis universalis, along with perhaps François Viète's algebra, represents one of the earliest attempts to construct a formal system.
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More generally, game semantics may be applied to predicate logic; the new rules allow a dominant quantifier to be removed by its "owner" (the Verifier for existential quantifiers and the Falsifier for universal quantifiers) and its bound variable replaced at all occurrences by an object of the owner's choosing, drawn from the domain of quantification. Note that a single counterexample falsifies a universally quantified statement, and a single example suffices to verify an existentially quantified one. Assuming the axiom of choice, the game-theoretical semantics for classical first-order logic agree with the usual model-based (Tarskian) semantics. For classical first- order logic the winning strategy for the Verifier essentially consists of finding adequate Skolem functions and witnesses.
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In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (logical constants) and the non-logical symbols. The idea behind this terminology is that logical symbols have the same meaning regardless of the subject matter being studied, while non-logical symbols change in meaning depending on the area of investigation. Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for logical connectives ∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =.
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Predicate logic was introduced to the mathematical community by C. S. Peirce, who coined the term second-order logic and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of Frege, who published his work several years prior to Peirce but whose works remained less known until Bertrand Russell and Alfred North Whitehead made them famous. Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of Russell's paradox it was realized that something was wrong with his system.
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PLN begins with a term logic foundation, and then adds on elements of probabilistic and combinatory logic, as well as some aspects of predicate logic and autoepistemic logic, to form a complete inference system, tailored for easy integration with software components embodying other (not explicitly logical) aspects of intelligence. PLN represents truth values as intervals, but with different semantics than in Imprecise Probability Theory. In addition to the interpretation of truth in a probabilistic fashion, a truth value in PLN also has an associated amount of certainty. This generalizes the notion of truth values used in autoepistemic logic, where truth values are either known or unknown, and when known, are either true or false.
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A logical system or language (not be confused with the kind of "formal language" discussed above which is described by a formal grammar), is a deductive system (see section above; most commonly first order predicate logic) together with additional (non-logical) axioms and a semantics. According to model-theoretic interpretation, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. A structure that satisfies all the axioms of the formal system is known as a model of the logical system. A logical system is sound if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
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Kleene's recursive realizability splits proofs of intuitionistic arithmetic into the pair of a recursive function and of a proof of a formula expressing that the recursive function "realizes", i.e. correctly instantiates the disjunctions and existential quantifiers of the initial formula so that the formula gets true. Kreisel's modified realizability applies to intuitionistic higher-order predicate logic and shows that the simply typed lambda term inductively extracted from the proof realizes the initial formula. In the case of propositional logic, it coincides with Howard's statement: the extracted lambda term is the proof itself (seen as an untyped lambda term) and the realizability statement is a paraphrase of the fact that the extracted lambda term has the type that the formula means (seen as a type).
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Indeed, he had already become known by the Scholastics (medieval Christian scholars) as "The Philosopher", due to the influence he had upon medieval theology and philosophy. His influence continued into the Early Modern period and Organon was the basis of school philosophy even in the beginning of 18th century. Since the logical innovations of the 19th century, particularly the formulation of modern predicate logic, Aristotelian logic had for a time fallen out of favor among many analytic philosophers. However the logic historian John Corcoran and others have shown that the works of George Boole and Gottlob Frege—which laid the groundwork for modern mathematical logic—each represent a continuation and extension to Aristote's logic and in no way contradict or displace it.
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If "predicate variables" are only allowed to be bound to predicate letters of zero arity (which have no arguments), where such letters represent propositions, then such variables are propositional variables, and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate calculus, or second-order logic. If predicate variables are also allowed to be bound to predicate letters which are unary or have higher arity, and when such letters represent propositional functions, such that the domain of the arguments is mapped to a range of different propositions, and when such variables can be bound by quantifiers to such sets of propositions, then the result is a higher-order predicate calculus, or higher-order logic.
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Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by "categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory.
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Right Euclidean property: solid and dashed arrows indicate antecedents and consequents, respectively. A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c.. To write this in predicate logic: :\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c). Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b is related to c: :\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c).
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In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimal element. Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relation defined on A. Then (A, R) is a foundational relation if any nonempty subset in A has an R-minimal element. In predicate logic, :(\forall S)\left(S \subseteq A \land S ot= \emptyset \Rightarrow (\exists x \in S)(\forall y \in S)(\lnot y R x)\right), See Definition 6.21 in in which \emptyset denotes the empty set. Here x is an R-minimal element in the subset S, since none of its R-predecessors is in S.
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First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable.Hodgson, Dr. J. P. E., "First Order Logic", Saint Joseph's University, Philadelphia, 1995. This distinguishes it from propositional logic, which does not use quantifiers or relations;Hughes, G. E., & Cresswell, M. J., A New Introduction to Modal Logic (London: Routledge, 1996), p.161.
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The drinker paradox (also known as the drinker's theorem, the drinker's principle, or the drinking principle) is a theorem of classical predicate logic which can be stated as "There is someone in the pub such that, if he is drinking, then everyone in the pub is drinking." It was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book What Is the Name of this Book? The apparently paradoxical nature of the statement comes from the way it is usually stated in natural language. It seems counterintuitive both that there could be a person who is causing the others to drink, or that there could be a person such that all through the night that one person were always the last to drink.
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Kleene (1967:33) observes that "logic" can be "founded" in two ways, first as a "model theory", or second by a formal "proof" or "axiomatic theory"; "the two formulations, that of model theory and that of proof theory, give equivalent results"(Kleene 1967:33). This foundational choice, and their equivalence also applies to predicate logic (Kleene 1967:318). In his introduction to Post 1921, van Heijenoort observes that both the "truth-table and the axiomatic approaches are clearly presented".van Heijenoort:264 This matter of a proof of consistency both ways (by a model theory, by axiomatic proof theory) comes up in the more-congenial version of Post's consistency proof that can be found in Nagel and Newman 1958 in their chapter V "An Example of a Successful Absolute Proof of Consistency".
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In 1930, Gödel's completeness theorem showed that first-order predicate logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms (such as those of Principia Mathematica) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms. Gödel's incompleteness theorems cast unexpected light on these two related questions. Gödel's first incompleteness theorem showed that no recursive extension of Principia could be both consistent and complete for arithmetic statements.
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Wayman (1977) proffers that the Catuṣkoṭi may be employed in different ways and often these are not clearly stated in discussion nor the tradition. Wayman (1977) holds that the Catuṣkoṭi may be applied in suite, that is all are applicable to a given topic forming a paradoxical matrix; or they may be applied like trains running on tracks (or employing another metaphor, four mercury switches where only certain functions or switches are employed at particular times). This difference in particular establishes a distinction with the Greek tradition of the Tetralemma. Also, predicate logic has been applied to the Dharmic Tradition, and though this in some quarters has established interesting correlates and extension of the logico-mathematical traditions of the Greeks, it has also obscured the logico- grammatical traditions of the Dharmic Traditions of Catuṣkoṭi within modern English discourse.
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The most famous concept in Nishida's philosophy is the logic of basho (Japanese: 場所; usually translated as "place" or "topos"), a non- dualistic concrete logic, meant to overcome the inadequacy of the subject–object distinction essential to the subject logic of Aristotle and the predicate logic of Immanuel Kant, through the affirmation of what he calls the "absolutely contradictory self-identity", a dynamic tension of opposites that, unlike the dialectical logic of Georg Wilhelm Friedrich Hegel, does not resolve in a synthesis. Rather, it defines its proper subject by maintaining the tension between affirmation and negation as opposite poles or perspectives. When David A. Dilworth wrote about Nishida's work, he did not mention the debut book in his useful classification. In his book Zen no kenkyū (An Inquiry into the Good), Nishida writes about experience, reality, good and religion.
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Luckily, one of the more common patterns of code that normally relies on branching has a more elegant solution. Consider the following pseudocode: if condition {dosomething} else {dosomethingelse}; On a system that uses conditional branching, this might translate to machine instructions looking similar to: branch-if-condition to label1 dosomethingelse branch-to label2 label1: dosomething label2: ... With predication, all possible branch paths are coded inline, but some instructions execute while others do not. The basic idea is that each instruction is associated with a predicate (the word here used similarly to its usage in predicate logic) and that the instruction will only be executed if the predicate is true. The machine code for the above example using predication might look something like this: (condition) dosomething (not condition) dosomethingelse Besides eliminating branches, less code is needed in total, provided the architecture provides predicated instructions.
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The system of ω-logic includes all axioms and rules of the usual first-order predicate logic, together with, for each T-formula P(x) with a specified free variable x, an infinitary ω-rule of the form: :From P(0),P(1),P(2),\ldots infer \forall x\,(N(x)\to P(x)). That is, if the theory asserts (i.e. proves) P(n) separately for each natural number n given by its specified name, then it also asserts P collectively for all natural numbers at once via the evident finite universally quantified counterpart of the infinitely many antecedents of the rule. For a theory of arithmetic, meaning one with intended domain the natural numbers such as Peano arithmetic, the predicate N is redundant and may be omitted from the language, with the consequent of the rule for each P simplifying to \forall x\,P(x).
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In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem. If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema. The formulation used in the axiom schema of replacement article only allows to construct the image F[a] when a is contained in the domain of the class function F; then the derivation of separation requires the axiom of empty set.
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The basic goal of PLN is to provide reasonably accurate probabilistic inference in a way that is compatible with both term logic and predicate logic, and scales up to operate in real time on large dynamic knowledge bases. The goal underlying the theoretical development of PLN has been the creation of practical software systems carrying out complex, useful inferences based on uncertain knowledge and drawing uncertain conclusions. PLN has been designed to allow basic probabilistic inference to interact with other kinds of inference such as intensional inference, fuzzy inference, and higher-order inference using quantifiers, variables, and combinators, and be a more convenient approach than Bayesian networks (or other conventional approaches) for the purpose of interfacing basic probabilistic inference with these other sorts of inference. In addition, the inference rules are formulated in such a way as to avoid the paradoxes of Dempster-Shafer theory.
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In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Given the function symbols F and G, one can introduce a new function symbol F ∘ G, the composition of F and G, satisfying (F ∘ G)(X) = F(G(X)), for all X. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain type of G, so this is required for the composition to be defined. One also gets certain function symbols automatically. In untyped logic, there is an identity predicate id that satisfies id(X) = X for all X. In typed logic, given any type T, there is an identity predicate idT with domain and codomain type T; it satisfies idT(X) = X for all X of type T. Similarly, if T is a subtype of U, then there is an inclusion predicate of domain type T and codomain type U that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.
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The aim of the AI techniques embedded in an intelligent decision support system is to enable these tasks to be performed by a computer, while emulating human capabilities as closely as possible. Many IDSS implementations are based on expert systems,Matsatsinis, N.F., Y. Siskos (1999), MARKEX: An intelligent decision support system for product development decisions, European Journal of Operational Research, vol. 113, no. 2, pp. 336-354 a well established type of KBS that encode knowledge and emulate the cognitive behaviours of human experts using predicate logic rules, and have been shown to perform better than the original human experts in some circumstances.Baron J.: Thinking and Deciding (1998) Cambridge University PressTurban E., Volonio L., McLean E. and Wetherbe J.: Information Technology for Management (2009) Wiley Expert systems emerged as practical applications in the 1980sJackson P.: Introduction to expert systems (1986) Addison-Wesley based on research in artificial intelligence performed during the late 1960s and early 1970s.Power, D.J. A Brief History of Decision Support Systems, DSSResources.COM, World Wide Web, version 4.0, March 10, 2007 They typically combine knowledge of a particular application domain with an inference capability to enable the system to propose decisions or diagnoses.
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