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Drinker Paradox
The drinker paradox (also known as the drinker's theorem, the drinker's principle, or the drinking principle) is a theorem of classical predicate logic that can be stated as "There is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking." It was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book ''What Is the Name of this Book?'' The apparently paradoxical nature of the statement comes from the way it is usually stated in natural language. It seems counterintuitive both that there could be a person who is ''causing'' the others to drink, or that there could be a person such that all through the night that one person were always the ''last'' to drink. The first objection comes from confusing formal "if then" statements with causation (see Correlation does not imply causation or Relevance logic for logics that demand relevant relationships between premise and consequent, unlik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuous Truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she does not own a cell phone" will imply that the statement "all of her cell phones are turned off" will be assigned a truth value. Also, the statement "all of her cell phones are turned ''on''" would also be vacuously true, as would the conjunction of the two: "all of her cell phones are turned on ''and'' turned off", which would otherwise be incoherent and false. For that reason, it is sometimes said that a statement is vacuously true because it is meaningless. More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temporal Logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I eat something"). It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians. Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that ''whenever'' a request is made, access to a resource is ''eventually'' granted, but it is ''never'' granted to two requestors simultaneously. Such a statement can conveniently be expressed in a temporal logic. Motivation Consider the statement "I am hungry". Though its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reification (linguistics)
Reification in natural language processing refers to where a natural language statement is transformed so actions and events in it become quantifiable variables. For example "John chased the duck furiously" can be transformed into something like :(Exists e)(chasing(e) & past_tense(e) & actor(e,John) & furiously(e) & patient(e,duck)). Another example would be "Sally said John is mean", which could be expressed as something like :(Exists u,v)(saying(u) & past_tense(u) & actor(u,Sally) & that(u,v) & is(v) & actor(v,John) & mean(v)). Such representations allow one to use the tools of classical first-order predicate calculus even for statements which, due to their use of tense, modality, adverbial constructions, propositional arguments (''e.g.'' "Sally said that X"), etc., would have seemed intractable. This is an advantage because predicate calculus is better understood and simpler than the more complex alternatives (higher-order logics, modal logics, temporal logics, etc.), and ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. Although considered paradoxes, some of these are simply based on fallacious reasoning ( falsidical), or an unintuitive solution (veridical). Informally, the term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream perception of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called ''antinomy,'' point out genuine problems in our understanding of the ideas of truth and description. Logic * : The supposition that, 'if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved' leads to paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistants
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Reasoning
In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer programs that allow computers to reason completely, or nearly completely, automatically. Although automated reasoning is considered a sub-field of artificial intelligence, it also has connections with theoretical computer science and philosophy. The most developed subareas of automated reasoning are automated theorem proving (and the less automated but more pragmatic subfield of interactive theorem proving) and automated proof checking (viewed as guaranteed correct reasoning under fixed assumptions). Extensive work has also been done in reasoning by analogy using induction and abduction. Other important topics include reasoning under uncertainty and non-monotonic reasoning. An important part of the uncertainty field is that of argumentation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. Notation In l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witness (mathematics)
In mathematical logic, a witness is a specific value ''t'' to be substituted for variable ''x'' of an existential statement of the form ∃''x'' ''φ''(''x'') such that ''φ''(''t'') is true. Examples For example, a theory ''T'' of arithmetic is said to be inconsistent if there exists a proof in ''T'' of the formula "0 = 1". The formula I(''T''), which says that ''T'' is inconsistent, is thus an existential formula. A witness for the inconsistency of ''T'' is a particular proof of "0 = 1" in ''T''. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which ''S'' is an ''n''-place relation on natural numbers, ''R'' is an ''(n+1)''-place recursive relation, and ↔ indicates logical equivalence (if and only if): :: ''S''(''x''1, ..., ''x''''n'') ↔ ∃''y'' ''R''(''x''1, . . ., ''x''''n'', ''y'') :"A ''y'' such that ''R'' holds of the ''xi'' may be called a 'witness' to the relation ''S'' holding of the ''xi'' (provided w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antecedent (logic)
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the ''protasis''. Examples: * If P, then Q. This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q. In an implication, if \phi implies \psi then \phi is called the antecedent and \psi is called the consequent.Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics, 3rd ed., 2004 Antecedent and consequent are connected via logical connective to form a proposition. * If X is a man, then X is mortal. "X is a man" is the antecedent for this proposition. * If men have walked on the moon, then I am the king of France. Here, "men have walked on the moon" is the antecedent. Let y=x+1. If x=1 then y=2 See also * Consequent * Affirming the consequent (fallacy) * Denying the antecedent (fallacy) * Necessity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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