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Doxastic Logic
Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation \mathcalx to mean "It is believed that x is the case", and the set \mathbb : \left \ denotes a set of beliefs. In doxastic logic, belief is treated as a modal operator. There is complete parallelism between a person who believes propositions and a formal system that derives propositions. Using doxastic logic, one can express the epistemic counterpart of Gödel's incompleteness theorem of metalogic, as well as Löb's theorem, and other metalogical results in terms of belief. Smullyan, Raymond M., (1986''Logicians who reason about themselves'' Proceedings of the 1986 conference on Theoretical aspects of reasoning about knowledge, Monterey (CA), Morgan Kaufmann Publishers Inc., San Francisco (CA), pp. 341â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Types Of 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Increasing Levels Of Rationality
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self Awareness
In philosophy of self, self-awareness is the experience of one's own personality or individuality. It is not to be confused with consciousness in the sense of qualia. While consciousness is being aware of one's environment and body and lifestyle, self-awareness is the recognition of that awareness. Self-awareness is how an individual consciously knows and understands their own character, feelings, motives, and desires. Neurobiological basis Introduction There are questions regarding what part of the brain allows us to be self-aware and how we are biologically programmed to be self-aware. V.S. Ramachandran has speculated that mirror neurons may provide the neurological basis of human self-awareness. In an essay written for the Edge Foundation in 2009, Ramachandran gave the following explanation of his theory: "... I also speculated that these neurons can not only help simulate other people's behavior but can be turned 'inward'—as it were—to create second-order representa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lottery Paradox
The lottery paradoxKyburg, H. E. (1961). ''Probability and the Logic of Rational Belief'', Middletown, CT: Wesleyan University Press, p. 197. arises from Henry E. Kyburg Jr. considering a fair 1,000-ticket lottery that has exactly one winning ticket. If that much is known about the execution of the lottery, it is then rational to accept that some ticket will win. Suppose that an event is very likely only if the probability of it occurring is greater than 0.99. On those grounds, it is presumed to be rational to accept the proposition that ticket 1 of the lottery will not win. Since the lottery is fair, it is rational to accept that ticket 2 will not win either. Indeed, it is rational to accept for any individual ticket ''i'' of the lottery that ticket ''i'' will not win. However, accepting that ticket 1 will not win, accepting that ticket 2 will not win, and so on until accepting that ticket 1,000 will not win entails that it is rational to accept that ''no'' ticket will win, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P implies Q.'' ''P'' is true. Therefore ''Q'' must also be true." ''Modus ponens'' is closely related to another valid form of argument, ''modus tollens''. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of ''modus ponens''. Hypothetical syllogism is closely related to ''modus ponens'' and sometimes thought of as "double ''modus ponens''." The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with ''modus tollens'', is one of the standard patterns of inference that can be applied to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Closure
In mathematical logic, a set of logical formulae is deductively closed if it contains every formula that can be logically deduced from , formally: if always implies . If is a set of formulae, the deductive closure of is its smallest superset that is deductively closed. The deductive closure of a theory is often denoted or . This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of is exactly the closure of with respect to the operation of logical consequence (). Examples In propositional logic, the set of all true propositions is deductively closed. This is to say that only true statements are derivable from other true statements. Epistemic closure In epistemology, many philosophers have and continue to debate whether particular subsets of propositions—especially ones ascribing knowledge or justification of a belief A belief is an attitude that something is the case, or that some propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Tables
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each input variable (for example, P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, P XOR Q). Each row of the truth table contains one possible configuration of the input variables (for instance, P=true Q=false), and the result of the operation for those values. See the examples below for further clarification. Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Löb's Theorem
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula ''P'', if it is provable in PA that "if ''P'' is provable in PA then ''P'' is true", then ''P'' is provable in PA. If Prov(''P'') means that the formula ''P'' is provable, we may express this more formally as :If :PA\,\vdash\, :then :PA\,\vdash\,P An immediate corollary (the contrapositive) of Löb's theorem is that, if ''P'' is not provable in PA, then "if ''P'' is provable in PA, then ''P'' is true" is not provable in PA. For example, "If 1+1=3 is provable in PA, then 1+1=3" is not provable in PA. Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox. Löb's theorem in provability logic Provability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of \phi in the given system in the language of modal logic, by means of the moda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kripke Semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Semantics of modal logic The language of propositional modal logic consists of a countable set, countably infinite set of propositional variables, a set of truth-functional Logical connective, connectives (in this article \to and \neg), and the modal operator \Box ("necessarily"). The modal operator \Diamond ("possibly") is (classically) the duality (mathematics)#Duality in log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Order
In mathematics, a partial order or total order < on a is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are . Example The s as a linearly ordered set are a densely o ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
