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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 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. This distinguishes it from propositional logic , which does not use quantifiers or relations. 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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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 y * v * t * e CHARLES SANDERS PEIRCE (/ˈpɜːrs/ , 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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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. It is merely to be understood to mean "if p is true, then q is also true" such that the statement p→q is false only when p is true and q is false.  The material conditional only states that q is true when (but not necessarily only when) p is true, and makes no claim that p causes q
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Republic (Plato)
The REPUBLIC (Greek : Πολιτεία, Politeia ; Latin
Latin
: Res Publica ) is a Socratic dialogue , written by Plato
Plato
around 380 BC, concerning justice (δικαιοσύνη), the order and character of the just city-state and the just man. It is Plato's best-known work, and has proven to be one of the world's most influential works of philosophy and political theory , both intellectually and historically. In the book's dialogue, Socrates
Socrates
discusses the meaning of justice and whether or not the just man is happier than the unjust man with various Athenians and foreigners. They consider the natures of existing regimes and then propose a series of different, hypothetical cities in comparison. This culminates in the discussion of Kallipolis (Καλλίπολις), a hypothetical city-state ruled by a philosopher king
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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. CONTENTS * 1 History and motivation * 2 Examples * 3 See also * 4 References HISTORY AND MOTIVATIONOswald 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"). In common language, words used for counting are "cardinal numbers " and words used for ordering are "ordinal numbers ". Some definitions, including the standard ISO 80000-2 , begin the natural numbers with 0 , corresponding to the NON-NEGATIVE INTEGERS 0, 1, 2, 3, …, whereas others start with 1, corresponding to the POSITIVE INTEGERS 1 , 2 , 3 , …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the WHOLE NUMBERS, but in other writings, that term is used instead for the integers (including negative integers)
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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 . These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete . The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann , who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction . In 1881, Charles Sanders Peirce
Charles Sanders Peirce
provided an axiomatization of natural-number arithmetic
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Number Theory
NUMBER THEORY or, in older usage, ARITHMETIC is a branch of pure mathematics devoted primarily to the study of the integers . It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number
Number
theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers ) or defined as generalizations of the integers (e.g., algebraic integers ). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry ). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function
Riemann zeta function
) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory )
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Syntax
In linguistics , SYNTAX (/ˈsɪntæks/ ) is the set of rules, principles, and processes that govern the structure of sentences in a given language , specifically word order and punctuation. The term syntax is also used to refer to the study of such principles and processes. 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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Semantics
SEMANTICS (from Ancient Greek
Ancient Greek
: σημαντικός sēmantikos, "significant") 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. 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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Exclusive Or
but not is Venn diagram
Venn diagram
of A B C {displaystyle scriptstyle Aoplus Boplus C} {displaystyle ~oplus ~} {displaystyle ~Leftrightarrow ~} EXCLUSIVE OR or EXCLUSIVE DISJUNCTION is a logical operation that outputs true only when inputs differ (one is true, the other is false). It is symbolized by the prefix operator J 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". This could be written as "A or B, but not, A and B". More generally, XOR is true only when an odd number of inputs are true
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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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Signature (mathematical 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 . They are rarely made explicit in more philosophical treatments of logic
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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 or Quine dagger), NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete ). This property makes the NAND gate
NAND gate
crucial to modern digital electronics , including its use in computer processor design
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Polish Notation
Infix notation
Infix notation
Prefix notation ("Polish") * v * t * e POLISH NOTATION (PN), also known as NORMAL POLISH NOTATION (NPN), ŁUKASIEWICZ NOTATION, 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. It does not need any parentheses as long as each operator has a fixed number of operands . The description "Polish" refers to the nationality of logician Jan Łukasiewicz , who invented Polish notation
Polish notation
in 1924. The term Polish notation
Polish notation
is sometimes taken (as the opposite of infix notation ) to also include reverse Polish notation
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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 . 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 and q) or (not p and not 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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