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Destructive Dilemma
Destructive dilemmaMoore and Parker is the name of a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''Q'' is false or ''S'' is false, then either ''P'' or ''R'' must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. ''Destructive dilemma'' is the disjunctive version of ''modus tollens''. The disjunctive version of ''modus ponens'' is the constructive dilemma. The destructive dilemma rule can be stated: :\frac where the rule is that wherever instances of "P \to Q", "R \to S", and "\neg Q \lor \neg S" appear on lines of a proof, "\neg P \lor \neg R" can be placed on a subsequent line. Formal notation The ''destructive dilemma'' rule may be written in sequent notation: : (P \to Q), (R \to S), (\neg Q \lor \neg S) \vdash (\neg P \lor \neg R) where \vdash is a metalogical symbol meaning that \neg P \lor \neg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rule Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as ''modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Formal System
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Concepts A formal system has the following: * Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules). * Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal language. A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rules Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as '' modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argument forms involving lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logical Conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Disjunctive Syllogism
In classical logic, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises. An example in English: # I will choose soup or I will choose salad. # I will not choose soup. # Therefore, I will choose salad. Propositional logic In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid rule of inference. If it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. Equivalently, if ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Double Negation
In propositional logic, the double negation of a statement states that "it is not the case that the statement is not true". In classical logic, every statement is logically equivalent to its double negation, but this is not true in intuitionistic logic; this can be expressed by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign expresses negation. Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in ''Principia Mathematica'' as: :: \mathbf. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117. ::"This is the principle of double negation, ''i.e.'' a proposition is equivalent of the falsehood of its negation." Elimination and introduction Double negation elimination and double negation introduction a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
De Morgan's Law
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: * The negation of "A and B" is the same as "not A or not B". * The negation of "A or B" is the same as "not A and not B". or * The complement of the union of two sets is the same as the intersection of their complements * The complement of the intersection of two sets is the same as the union of their complements or * not (A or B) = (not A) and (not B) * not (A and B) = (not A) or (not B) where "A or B" is an " inclusive or" meaning ''at least'' one of A or B rather than an "exclusive or" that means ''exactly'' one of A or B. Another form of De Morgan's law is the following as see ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reductio Ad Absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called ''proof by contradiction''. In formal logic, this technique is captured by an axiom for "Reductio ad Absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see ''negation introduction''). Examples The "absurd" conclusion of a ''reductio ad absurdum'' argument can take a range of forms, as these examples show: * The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find peo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conditional Proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. Overview The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that ''if it were true'' it would lead to the consequent. Conditional proofs are of great importance in mathematics. Conditional proofs exist linking several otherwise unproven conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently. A famous network of conditional proofs is the NP-complete class of complexity theory. There is a large num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conjunction Introduction
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. It is the inference that if the proposition P is true, and the proposition Q is true, then the logical conjunction of the two propositions P and Q is true. For example, if it is true that "it is raining", and it is true that "the cat is inside", then it is true that "it is raining and the cat is inside". The rule can be stated: :\frac where the rule is that wherever an instance of "P" and "Q" appear on lines of a proof, a "P \land Q" can be placed on a subsequent line. Formal notation The ''conjunction introduction'' rule may be written in sequent notation: : P, Q \vdash P \land Q where P and Q are propositions expressed in some formal system, and \vdash is a metalogical symbol A symbol is a mark, Sign (semiotics), sign, or w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transposition (logic)
In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrapositive of a statement has its antecedent and consequent negated and swapped. Conditional statement P \rightarrow Q. In formulas: the contrapositive of P \rightarrow Q is \neg Q \rightarrow \neg P . If ''P'', Then ''Q''. — If not ''Q'', Then not ''P''. "If ''it is raining,'' then ''I wear my coat''." — "If ''I don't wear my coat,'' then ''it isn't raining''." The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. Contraposition ( \neg Q \rightarrow \neg P ) can be compared with three other operations: ; Inversion (the inverse), \neg P \rightarrow \neg Q:"If ''it is not raining,'' then ''I don't wear my coat''." Unlike the contrapositive, the inverse's truth value is not at all dependen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
