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The Hardest Logic Puzzle Ever
The Hardest Logic Puzzle Ever is a logic puzzle so called by American philosopher and logician George Boolos and published in ''The Harvard Review of Philosophy'' in 1996. Boolos' article includes multiple ways of solving the problem. A translation in Italian was published earlier in the newspaper ''La Repubblica'', under the title ''L'indovinello più difficile del mondo''. It is stated as follows: Boolos provides the following clarifications: a single god may be asked more than one question, questions are permitted to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a fair coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely. History Boolos credits the logician Raymond Smullyan as the originator of the puzzle and John McCarthy with adding the difficulty of not knowing what ''da'' and ''ja'' mean. Related puzzles can be found throughout Smullyan's writings. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Puzzle
A logic puzzle is a puzzle deriving from the mathematics, mathematical field of deductive reasoning, deduction. History The logic puzzle was first produced by Charles Lutwidge Dodgson, who is better known under his pen name Lewis Carroll, the author of ''Alice's Adventures in Wonderland''. In his book ''The Game of Logic'' he introduced a game to solve problems such as confirming the conclusion "Some greyhounds are not fat" from the statements "No fat creatures run well" and "Some greyhounds run well". Puzzles like this, where we are given a list of premises and asked what can be deduced from them, are known as syllogisms. Dodgson goes on to construct much more complex puzzles consisting of up to 8 premises. In the second half of the 20th century mathematician Raymond Smullyan, Raymond M. Smullyan continued and expanded the branch of logic puzzles with books such as ''The Lady or the Tiger?'', ''To Mock a Mockingbird'' and ''Alice in Puzzle-Land''. He popularized the "knights an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Labyrinth (1986 Film)
''Labyrinth'' is a 1986 musical fantasy film directed by Jim Henson with George Lucas as executive producer. Based on conceptual designs by Brian Froud, the film was written by Terry Jones, and many of its characters are played by puppets produced by Jim Henson's Creature Shop. The film stars Jennifer Connelly as 16-year-old Sarah and David Bowie as Jareth, The Goblin King. In ''Labyrinth'', Sarah embarks on a quest to reach the center of an enormous, otherworldly maze to rescue her infant half-brother Toby, whom she wished away to Jareth. The film started as a collaboration between Henson and Froud following their previous collaboration '' The Dark Crystal'' (1982). Terry Jones of Monty Python wrote the first draft of the film's script early in 1984, drawing on Froud's sketches for inspiration. Various other scriptwriters rewrote it and added to it, including Laura Phillips, Lucas, Dennis Lee, and Elaine May—although Jones received the film's sole screenwriting credit. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirghizia
Kyrgyzstan,, pronounced or the Kyrgyz Republic, is a landlocked country in Central Asia. Kyrgyzstan is bordered by Kazakhstan to the north, Uzbekistan to the west, Tajikistan to the south, and the People's Republic of China to the east. Its capital and largest city is Bishkek. Ethnic Kyrgyz make up the majority of the country's seven million people, followed by significant minorities of Uzbeks and Russians. The Kyrgyz language is closely related to other Turkic languages. Kyrgyzstan's history spans a variety of cultures and empires. Although geographically isolated by its highly mountainous terrain, Kyrgyzstan has been at the crossroads of several great civilizations as part of the Silk Road along with other commercial routes. Inhabited by a succession of tribes and clans, Kyrgyzstan has periodically fallen under larger domination. Turkic nomads, who trace their ancestry to many Turkic states. It was first established as the Yenisei Kyrgyz Khaganate later in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dushanbe
Dushanbe ( tg, Душанбе, ; ; russian: Душанбе) is the capital and largest city of Tajikistan. , Dushanbe had a population of 863,400 and that population was largely Tajik. Until 1929, the city was known in Russian as Dyushambe (russian: Дюшамбе, ''Dyushambe''), and from 1929 to 1961 as Stalinabad ( tg, Сталинобод, Stalinobod), after Joseph Stalin. Dushanbe is located in the Gissar Valley, bounded by the Gissar Range in the north and east and the Babatag, Aktau, Rangontau and Karatau mountains in the south, and has an elevation of 750–930 m. The city is divided into four districts, all named after historical figures: Ismail Samani, Avicenna, Ferdowsi, and Shah Mansur. In ancient times, what is now or is close to modern Dushanbe was settled by various empires and peoples, including Mousterian tool-users, various neolithic cultures, the Achaemenid Empire, Greco-Bactria, the Kushan Empire, and the Hephthalites. In the Middle Ages, more settlement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exclusive Disjunction
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , , , , , and . The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator ''excludes'' that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B". Since it is associative, it may be considered to be an ''n''-ary operator which is true if and only if an odd number of arguments are true. That is, ''a'' XOR ''b'' XOR ... may be treated as XOR(''a'',''b'',...). Truth table The truth table of A XOR B shows that it outputs true whenever the inputs differ: Equivalences, elimination, and introduct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Expression
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants true or false, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions. Boolean expressions correspond to propositional formulas in logic and are a special case of Boolean circuits. Boolean operators Most programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...s have the Boolean operators Logical disjunction, OR, Logical conjunction, AND and Negation, NOT; in C (programming language), C and some List of C-family programming languages, languages inspired by it, these are represented by ", , " ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Form
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language. The logical form of an argument is called the argument form of the argument. History The importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the ''Prior Analytics'', was probably the first to employ variable letters to represent valid inferences. Therefore, Jan Łukasiewicz claims that the introduction of variables was "one of Aristotle's greatest inventions." According ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterfactual Conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood. Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and 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), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as the '' antecedent'', and the ''consequent''. This is often abbreviated as " iff ". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔ or ⇔ may be represented in Unicode in various ways), a prefixed E "E''pq''" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡), or ''EQV''. It is logically equivalent to both (P \rightarrow Q) \land (Q \rightarrow P) and (P \land Q) \lor (\neg P \land \neg Q) , and the XNOR (exclusive nor) boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis is false but the conclusion is true. In this case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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