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List Of Rules Of Inference
This is a list of rules of inference, logical laws that relate to mathematical formulae. Introduction Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. ''Discharge rules'' permit inference from a subderivation based on a temporary assumption. Below, the notation : \varphi \vdash \psi indicates such a subderivation from the temporary assumption \varphi to \psi. Rules for classical sentential calculus Sentential calculus is also known as propositional calculus. Rules for negations ;Reductio ad absurdum (or ''Negation Introduction''): : \varphi \vdash \psi : \underline : \lnot \varphi ;Reductio ad absurdum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called '' modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Dilemma
Constructive dilemmaCopi and Cohen is a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then either ''Q or S'' has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. ''Constructive dilemma'' is the disjunctive version of modus ponens, whereas, destructive dilemma is the disjunctive version of ''modus tollens''. The constructive dilemma rule can be stated: :\frac where the rule is that whenever instances of "P \to Q", "R \to S", and "P \lor R" appear on lines of a proof, "Q \lor S" can be placed on a subsequent line. Formal notation The ''constructive dilemma'' rule may be written in sequent notation: : (P \to Q), (R \to S), (P \lor R) \vdash (Q \lor S) where \vdash is a metalogical symbol meaning that Q \lor S is a syntactic consequence of P \to Q, R \to S, and P \lor R in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-deleting Theorem
In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L. Braunstein proved this theorem. The no-deleting theorem, together with the no-cloning theorem, underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger symmetric monoidal category. This formulation, known as categorical quantum mechanics, in turn allows a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in exact analogy to classical logic being founded on Cartesia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotency Of Entailment
Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. This property can be captured by a structural rule called contraction, and in such systems one may say that entailment is idempotent if and only if contraction is an admissible rule. Rule of contraction: from :''A'',''C'',''C'' → ''B'' is derived :''A'',''C'' → ''B''. Or in sequent calculus notation, :\frac In linear and affine logic, entailment is not idempotent. See also * No-deleting theorem In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no ... Logical consequence Theorems in propositional logic {{logic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-cloning Theorem
In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek as well as Dieks the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity Of Entailment
Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called ''monotonic logics'' in order to differentiate them from non-monotonic logics. Weakening rule To illustrate, consider the natural deduction sequent: Γ \vdash C That is, on the basis of a list of assumptions Γ, one can prove C. Weakening, by adding an assumption A, allows one to conclude: Γ, A \vdash C For example, the syllogism "All men are mortal. Socrates is a man. Therefore Socrates is mortal." can be weakened by adding a premise: "All men are mortal. Socrates is a man. Cows produce milk. Therefore Socrates is mortal." The validity of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "tru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substructural Logic
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic. Examples In a sequent calculus, one writes each line of a proof as :\Gamma\vdash\Sigma. Here the structural rules are rules for rewriting the LHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is as conjunction: we expect to read :\mathcal A,\mathcal B \vdash\mathcal C as the sequent notation for :(''A'' and ''B'') implies ''C''. Here we are taking the RHS Σ to be a single proposition ''C'' (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol \vdash. Since conjunction is a commutative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Instantiation
In predicate logic, existential instantiation (also called existential elimination)Moore and Parker is a rule of inference which says that, given a formula of the form (\exists x) \phi(x), one may infer \phi(c) for a new constant symbol ''c''. The rule has the restrictions that the constant ''c'' introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof. It is also necessary that every instance of x which is bound to \exists x must be uniformly replaced by ''c''. This is implied by the notation P\left(\right), but its explicit statement is often left out of explanations. In one formal notation, the rule may be denoted by :\exists x P \left(\right) \implies P \left(\right) where ''a'' is a new constant symbol that has not appeared in the proof. See also * Existential fallacy * Universal instantiation * List of rules of inference References Rules of inference Predicate logic {{logic-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Generalization
In predicate logic, existential generalization (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 (\exists) in formal proofs. Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail." Example: "Alice made herself a cup of tea. Therefore, Alice made someone a cup of tea." Example: "Alice made herself a cup of tea. Therefore, someone made someone a cup of tea." In the Fitch-style calculus: : Q(a) \to\ \exists\, Q(x) , where Q(a) is obtained from Q(x) by replacing all its free occurrences of x (or some of them) by a. Quine According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that \forall x \, x=x implies \text= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Instantiation
In predicate logic, universal instantiation (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 schema. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal." Formally, the rule as an axiom schema is given as : \forall x \, A \Rightarrow A\, for every formula ''A'' and every term ''a'', where A\ is the result of substituting ''a'' for each ''free'' occurrence of ''x'' in ''A''. \, A\ is an instance of \forall x \, A. And as a rule of inference it is :from \vdash \forall x A infer \vdash A \ . Irving Copi noted that universal instantiation "... follows from variants of rules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Generalization
In predicate logic, generalization (also universal generalization or universal introduction,Moore 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. Generalization with hypotheses The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume \Gamma is a set of formulas, \varphi a formula, and \Gamma \vdash \varphi(y) has been derived. The generalization rule states that \Gamma \vdash \forall x \, \varphi(x) can be derived if y is not mentioned in \Gamma and x does not occur in \varphi. These restrictions are necessary for soundness. Without the first restriction, one could conclude \forall x P(x) from the hypothesis P(y). Without the second restriction, one could make the following deduction: #\exists z \, \exists w \, ( z \not = w) (Hypothesis) #\exists w \, (y \not = w) (Existential instantiation) #y \not = x (Existential instantiation) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |