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Barcan Formula
In quantified modal logic, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulas) (i) syntactically state principles of interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulas were introduced as axioms by Ruth Barcan Marcus, in the first extensions of modal propositional logic to include quantification.Journal of Symbolic Logic (1946),11 and (1947), 12 under Ruth C. Barcan Related formulas include the Buridan formula. The Barcan formula The Barcan formula is: :\forall x \Box Fx \rightarrow \Box \forall x Fx. In English, the schema reads: If every x is necessarily F, then it is necessary that every x is F. It is equivalent to :\Diamond\exists xFx\to\exists x\Diamond Fx. The Barcan formula has generated some controversy because—in terms of possible world semantics—it implies that all objects which exist in any possible world (accessible to the actual wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a ''possible world''. A formula's truth value at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruth Barcan Marcus
Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quantified modal logic and in so doing introduced the schema or principle known as the Barcan formula. (She would also introduce the now standard "box" operator for necessity in the process.) Marcus, who originally published as Ruth C. Barcan, was, as Don Garrett notes "one of the twentieth century's most important and influential philosopher-logicians". Timothy Williamson, in a 2008 celebration of Marcus' long career, states that many of her "main ideas are not just original, and clever, and beautiful, and fascinating, and influential, and way ahead of their time, but actually – I believe – ''true''". Academic career and service Ruth Barcan (as she was known before marrying the physicist Jules Alexander Marcus in 1942 Gendler, T. S."Ruth B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buridan Formula
In quantified modal logic, the Buridan formula and the converse Buridan formula (more accurately, schemata rather than formulas) (i) syntactically state principles of interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulas are named in honor of the medieval philosopher Jean Buridan by analogy with the Barcan formula and the converse Barcan formula introduced as axioms by Ruth Barcan Marcus. The Buridan formula The Buridan formula is: :\Diamond \forall x Fx \rightarrow \forall x\Diamond Fx. In English, the schema reads: If possibly everything is F, then everything is possibly F. It is equivalent in a classical modal logic In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators \Diamond A \leftrightarrow \lnot\Box\lnot A that is also closed under the rule \frac. Alternatively, one can gi ... (but not necessarily in other form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
English Language
English is a West Germanic language of the Indo-European language family, with its earliest forms spoken by the inhabitants of early medieval England. It is named after the Angles, one of the ancient Germanic peoples that migrated to the island of Great Britain. Existing on a dialect continuum with Scots, and then closest related to the Low Saxon and Frisian languages, English is genealogically West Germanic. However, its vocabulary is also distinctively influenced by dialects of France (about 29% of Modern English words) and Latin (also about 29%), plus some grammar and a small amount of core vocabulary influenced by Old Norse (a North Germanic language). Speakers of English are called Anglophones. The earliest forms of English, collectively known as Old English, evolved from a group of West Germanic (Ingvaeonic) dialects brought to Great Britain by Anglo-Saxon settlers in the 5th century and further mutated by Norse-speaking Viking settlers starting in the 8th and 9th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actualism
In analytic philosophy, actualism is the view that everything there ''is'' (i.e., everything that has ''being'', in the broadest sense) is actual. Another phrasing of the thesis is that the domain of unrestricted quantification ranges over all and only actual existents. The denial of actualism is possibilism, the thesis that there are some entities that are ''merely possible'': these entities have being but are not actual and, hence, enjoy a "less robust" sort of being than do actually existing things. An important, but significantly different notion of possibilism known as '' modal realism'' was developed by the philosopher David Lewis. On Lewis's account, the actual world is identified with the physical universe of which we are all a part. Other possible worlds exist in exactly the same sense as the actual world; they are simply spatio-temporally unrelated to our world, and to each other. Hence, for Lewis, "merely possible" entities—entities that exist in other possible 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] |
Commutative Property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
