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Inverse (logic)
In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form P \rightarrow Q , the inverse refers to the sentence \neg P \rightarrow \neg Q . Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other. For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition :"If it's raining, then Sam will meet Jack at the movies." would be :"If it's not raining, then Sam will not meet Jack at the movies." The inverse of the inverse, that is, the inverse of \neg P \rightarrow \neg Q , is \neg \neg P \rightarrow \neg \neg Q , and since the double negation of any statement is equivalent to the original statement in classical logic, the inverse of the inverse is logically equivalent to the original conditional P \rightarrow Q . Thus it is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Sentence
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''conditional'' on the dependent clause. A full conditional thus contains two clauses: a dependent clause called the ''antecedent'' (or ''protasis'' or ''if-clause''), which expresses the condition, and a main clause called the ''consequent'' (or ''apodosis'' or ''then-clause'') expressing the result. Languages use a variety of grammatical forms and constructions in conditional sentences. The forms of verbs used in the antecedent and consequent are often subject to particular rules as regards their tense, aspect, and mood. Many languages have a specialized type of verb form called the conditional mood – broadly equivalent in meaning to the English "would (do something)" – for use in some types of conditional sentences. Types of conditiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immediate Inference
An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" (Obverse). There are a number of ''immediate inferences'' which can validly be made using logical operations, the result of which is a logically equivalent statement form to the given statement. There are also invalid immediate inferences which are syllogistic fallacies. Valid immediate inferences Converse *Given a type E statement, "No ''S'' are ''P''.", one can make the ''immediate inference'' that "No ''P'' are ''S''" which is the converse of the given statement. *Given a type I statement, "Some ''S'' are ''P''.", one can make the ''immediate inference'' that "Some ''P'' are ''S''" which is the converse of the given statement. Obverse *Given a type A statement, "All ''S'' are ''P''.", one can make the ''immediate inference'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraposition
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The contrapositive of a statement has its Antecedent (logic), antecedent and consequent Inverse (logic), inverted and Conversion (logic), flipped. Material conditional, Conditional statement P \rightarrow Q. In Logical connective, 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. The contrapositive ( \neg Q \rightarrow \neg P ) can be compared with three other statements: ;Inverse (logic), Inversion (the inverse), \neg P \rightarrow \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the converse of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 are two valid rules of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traditional Logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it, so to understand the newer systems, one must be acquainted with the earlier one. Aristotle's system Aristotle's logical work is collected in the six texts that are collectively known as the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Propositions
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are: *All ''S'' are ''P''. (''A'' form, \forall _\rightarrow P_xequiv \forall neg S_\lor P_x/math>) *No ''S'' are ''P''. (''E'' form, \forall _\rightarrow \neg P_xequiv \forall neg S_\lor \neg P_x/math>) *Some ''S'' are ''P''. (''I'' form, \exists _\land P_x/math>) *Some ''S'' are not ''P''. (''O'' form, \ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obversion
In traditional logic, obversion is a "type of immediate inference in which from a given proposition another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original proposition's quality was negative and vice versa". The quality of the inferred categorical proposition is changed but the truth value is the same to the original proposition. The immediately inferred proposition is termed the "obverse" of the original proposition, and is a valid form of inference for all types (A, E, I, O) of categorical propositions. In a universal affirmative and a universal negative proposition the subject term and the predicate term are both replaced by their negated counterparts: The universal affirmative ("A" proposition) is obverted to a universal negative ("E" proposition). :''"All S are P"'' and ''"No S are non-P"'' :''"All cats are animals"'' and ''"No cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transposition (logic)
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "''A'' implies ''B''" to the truth of "Not-''B'' implies not-''A''", and conversely. It is very closely related to the rule of inference modus tollens In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' .... It is the rule that (P \to Q) \Leftrightarrow (\neg Q \to \neg P) where "\Leftrightarrow" is a metalogical Symbol (formal), symbol representing "can be replaced in a proof with". Formal notation The ''transposition'' rule may be expressed as a sequent: :(P \to Q) \vdash (\neg Q \to \neg P) where \vdash is a metalogical symbol mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
