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If And Only If
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning
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Jan Łukasiewicz
Jan Łukasiewicz
Jan Łukasiewicz
(Polish: [ˈjan wukaˈɕɛvʲitʂ]; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lwów, which, before the Polish partitions, was in Poland, Galicia, then Austria-Hungary. His work centred on philosophical logic, mathematical logic, and history of logic. He thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Łukasiewicz of a revolutionary paradigm
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Exportation (logic)
Exportation[1][2][3][4] is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs
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Mathematical Logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic
Logic
in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics
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Distributive Property
In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs. For example, in arithmetic:2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards. Because these give the same final answer (8), it is said that multiplication by 2 distributes over addition of 1 and 3
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Double Negation
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.[1] Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ∗ 4 ⋅ 13 .     ⊢ .   p   ≡   ∼ ( ∼ p ) displaystyle mathbf *4cdot 13 . vdash
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XOR Gate
The X OR gate
OR gate
(sometimes EOR gate, or EX OR gate
OR gate
and pronounced as Exclusive OR gate) is a digital logic gate that gives a true (1 or HIGH) output when the number of true inputs is odd. An XOR gate implements an exclusive or; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "one or the other but not both". XOR can also be viewed as addition modulo 2. As a result, XOR gates are used to implement binary addition in computers. A half adder consists of an X OR gate
OR gate
and an AND gate
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Bidirectional Traffic
In transportation infrastructure, a bidirectional traffic system divides travelers into two streams of traffic that flow in opposite directions.[1] In the design and construction of tunnels, bidirectional traffic can markedly affect ventilation considerations.[2] Microscopic traffic flow models have been proposed for bidirectional automobile, pedestrian, and railway traffic.[3] Bidirectional traffic can be observed in ant trails[4] and this has been researched for insight into human traffic models.[5] In a macroscopic theory proposed by Laval, the interaction between fast and slow vehicles conforms to the Newell kinematic wave model of moving bottlenecks.[6] In air traffic control traffic is normally separated by elevation, with east bound flights at odd thousand feet elevations and west bound flights at even thousand feet elevations (1000 ft ≈ 305m). Above 28,000 ft (~8.5 km) only odd flight levels are used, with FL 290, 330, 370, etc., for eastbound flights and FL 31
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Material Implication (rule Of Inference)
In propositional logic, material implication [1][2] is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated
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Tautology (rule Of Inference)
In propositional logic, tautology is one of two commonly used rules of replacement.[1][2][3] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs
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Logical Connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective. Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic
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Negation Introduction
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.[1] [2] Formal notation[edit] This can be written as: ( P → Q ) ∧ ( P → ¬ Q ) ↔ ¬ P displaystyle (Prightarrow Q)land (Prightarrow neg Q)leftrightarrow neg P An example of its use would be an attempt to prove two contradictory statements from a single fact
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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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Universal Generalization
In predicate logic, generalization (also universal generalization or universal introduction,[1][2][3] GEN) is a valid inference rule. It states that if ⊢ P ( x ) displaystyle vdash P(x) has been derived, then ⊢ ∀ x P ( x ) displaystyle vdash forall x,P(x) can be derived.Contents1 Generalization with hypotheses 2 Example of a proof 3 See also 4 ReferencesGeneralization with hypotheses[edit] The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume Γ is a set of formulas, φ displaystyle varphi a formula, and Γ ⊢ φ ( y ) displaystyle Gamma vdash varphi (y) has been derived
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Universal Instantiation
In predicate logic universal instantiation[1][2][3] (UI, also called universal specification or universal elimination, and sometimes confused with Dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog
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Existential Generalization
In predicate logic, existential generalization[1][2] (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (∃) in formal proofs. Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail." In the Fitch-style calculus: Q ( a ) →   ∃ x Q ( x ) displaystyle Q(a)to exists x ,Q(x) Where a replaces all free instances of x within Q(x).[3] Quine[edit] Universal instantiation and Existential Generalization are two aspects of a single principle, for instead of saying that "∀x x=x" implies "Socrates=Socrates", we could as well say that the denial "Socrates≠Socrates"' implies "∃x x≠x"
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