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Predicate Logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic
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" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming
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Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind– Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano
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Non-logical Symbols
In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants). The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates. A language of first-order logic is a formal language over the alphabet consisting of its non-logical symbols and its logical symbols. The latter include logical connectives, quantifiers, and variables that stand for statements. A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation
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Exclusive Or
but not is Venn diagram
Venn diagram
of A ⊕ B ⊕ C displaystyle scriptstyle Aoplus Boplus C   ⊕   displaystyle ~oplus ~   ⇔   displaystyle ~Leftrightarrow ~ Exclusive or
Exclusive or
or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).[1] It is symbolized by the prefix operator J[2] and by the infix operators XOR (/ˌɛks ˈɔːr/), EOR, EXOR, ⊻, ⩒, ⩛, ⊕, ↮, and ≢. The negation of XOR is logical biconditional, which outputs true only when both inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both"
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Sheffer Stroke
In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "" (see vertical bar, not to be confused with "" which is often used to represent disjunction), "Dpq", or "↑" (an upwards arrow), denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation. Like its dual, the NOR operator (also known as the Peirce arrow
Peirce arrow
or Quine dagger), NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete)
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Polish Notation
Infix notation Prefix notation ("Polish")v t e Polish notation
Polish notation
(PN), also known as normal Polish notation
Polish notation
(NPN),[1] Łukasiewicz notation, Warsaw
Warsaw
notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to reverse Polish notation
Polish notation
(RPN) in which operators follow their operands
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Wedge (∧) is a symbol that looks similar to an in-line caret (^). It is used to represent various operations. In Unicode, the symbol is encoded U+2227 ∧ Logical and (HTML ∧ · ∧) and by wedge and land in TeX. The opposite symbol (∨) is called a vel, or sometimes a (descending) wedge
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Logical Biconditional
In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "p if and only if q", where p is an antecedent and q is a consequent.[1] This is often abbreviated "p iff q". The operator is denoted using a doubleheaded arrow (↔), a prefixed E (Epq), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to (p → q) ∧ (q → p). It is also logically equivalent to (p ∧ q) ∨ (¬p ∧ ¬q) (or the XNOR (exclusive nor) boolean operator), meaning "both or neither". The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false. In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the sets a and b coincide: they are identical
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Semantics
Semantics (from Ancient Greek: σημαντικός sēmantikos, "significant")[1][2] is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics. It is concerned with the relationship between signifiers—like words, phrases, signs, and symbols—and what they stand for, their denotation. In international scientific vocabulary semantics is also called semasiology. The word semantics was first used by Michel Bréal, a French philologist.[3] It denotes a range of ideas—from the popular to the highly technical. It is often used in ordinary language for denoting a problem of understanding that comes down to word selection or connotation. This problem of understanding has been the subject of many formal enquiries, over a long period of time, especially in the field of formal semantics
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Syntax
In linguistics, syntax (/ˈsɪntæks/[1][2]) is the set of rules, principles, and processes that govern the structure of sentences in a given language, usually including word order. The term syntax is also used to refer to the study of such principles and processes.[3] The goal of many syntacticians is to discover the syntactic rules common to all languages. In mathematics, syntax refers to the rules governing the behavior of mathematical systems, such as formal languages used in logic
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Republic (Plato)
A republic (Latin: res publica) is a form of government in which the country is considered a "public matter", not the private concern or property of the rulers. The primary positions of power within a republic are not inherited. It is a form of government under which the head of state is not a monarch.[1][2][3] In American English, the definition of a republic refers specifically to a form of government in which elected individuals represent the citizen body[2] and exercise power according to the rule of law under a constitution, including separation of powers with an elected head of state, referred to as a constitutional republic[4][5][6][7] or representative democracy. [8] As of 2017[update], 159 of the world's 206 sovereign states use the word "republic" as part of their official names – not all of these are republics in the sense of having elected governments, nor is the word "republic" used in the names of all nations with elected governments
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Material Conditional
The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form p → q (termed a conditional statement) which is read as "if p then q". Unlike the English construction "if... then...", the material conditional statement p → q does not specify a causal relationship between p and q
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Charles Sanders Peirce
CDPT: Commens Dictionary of Peirce's Terms CP x.y: Collected Papers, volume x, paragraph y EP x:y: The Essential Peirce, volume x, page y W x:y Writings of Charles S. Peirce, volume x, page yv t e Charles Sanders Peirce
Charles Sanders Peirce
(/pɜːrs/,[9] like "purse"; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism". He was educated as a chemist and employed as a scientist for 30 years. Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism. An innovator in mathematics, statistics, philosophy, research methodology, and various sciences, Peirce considered himself, first and foremost, a logician. He made major contributions to logic, but logic for him encompassed much of that which is now called epistemology and philosophy of science
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Categorical Theory
In model theory, a branch of mathematical logic, a theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965), which states that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ.Contents1 History and motivation 2 Examples 3 See also 4 ReferencesHistory and motivation[edit] Oswald Veblen
Oswald Veblen
in 1904 defined a theory to be categorical if all of its models are isomorphic
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Natural Number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country")
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