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Propositional Logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-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 representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below. 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 logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Atomic Formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a first-order logic#Formation rules, term. In model theory, atomic formulas are merely string (computer science), strings of symbols with a given signature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Stoics
Stoicism is a school of Hellenistic philosophy that flourished in ancient Greece and Rome. The Stoics believed that the universe operated according to reason, ''i.e.'' by a God which is immersed in nature itself. Of all the schools of ancient philosophy, Stoicism made the greatest claim to being utterly systematic. The Stoics provided a unified account of the world, constructed from ideals of logic, monistic physics, and naturalistic ethics. These three ideals constitute virtue which is necessary for 'living a well reasoned life', seeing as they are all parts of a logos, or philosophical discourse, which includes the mind's rational dialogue with itself. Stoicism was founded in the ancient Agora of Athens by Zeno of Citium around 300 BC, and flourished throughout the Greco-Roman world until the 3rd century AD, and among its adherents was Roman Emperor Marcus Aurelius. Along with Aristotelian term logic, the system of propositional logic developed by the Stoics was one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chrysippus
Chrysippus of Soli (; , ; ) was a Ancient Greece, Greek Stoicism, Stoic Philosophy, philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of the Stoic philosopher Cleanthes. When Cleanthes died, around 230 BC, Chrysippus became the third head of the Stoic school. A prolific writer, Chrysippus expanded the fundamental doctrines of Cleanthes' mentor Zeno of Citium, the founder and first head of the school, which earned him the title of the Second Founder of Stoicism. Chrysippus excelled in logic, the theory of knowledge, ethics, and Stoic physics, physics. He created an original system of propositional logic in order to better understand the workings of the universe and role of humanity within it. He adhered to a fatalistic view of Destiny, fate, but nevertheless sought a role for personal agency (philosophy), agency in thought and action. Ethics, he thought, depended on understanding the nature of the universe, and he taught ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Principle Of Bivalence
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic. In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent. The principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined truth value. Sentences that predict events in the future, and sentences that seem open to interpretation, are particularly difficult for philosophers who hold that the principle of bivalence applies to all declarative natural-language statements. Many-valued logics formalize ideas that a realistic characterization of the notion of conseq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in computing as well as various types of logic. Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called falsy and truthy. For example, in Lisp, nil, the empty list, is treated as false, and all other values are treated as true. In C, the number 0 or 0.0 is false, and all other values are treated as true. In JavaScript, the empty string (""), null, undefined, NaN, +0, −0 and false are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Exclusive Or
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR ''excludes'' that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B". It is symbolized by the prefix operator J Translated as and by the infix operators XOR (, , or ), EOR, EXOR, \dot, \overline, \underline, , \oplus, \nleftrightarrow, and \not\equiv. Definition The truth table of A\nleftrightarrow B shows that it outputs true whenever the inputs differ: Equivalences, elimination, and introduction Exclusive disjunction essentially ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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XNOR Gate
The XNOR gate (sometimes ENOR, EXNOR, NXOR, XAND and pronounced as exclusive NOR) is a digital logic gate whose function is the logical complement of the exclusive OR ( XOR) gate. It is equivalent to the logical connective (\leftrightarrow) from mathematical logic, also known as the material biconditional. The two-input version implements logical equality, behaving according to the truth table to the right, and hence the gate is sometimes called an "equivalence gate". A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results. The algebraic notation used to represent the XNOR operation is S = A \odot B. The algebraic expressions (A + \overline) \cdot (\overline + B) and A \cdot B + \overline A \cdot \overline B both represent the XNOR gate with inputs ''A'' and ''B''. Symbols There are two symbols for XNOR gates: one with distinctive shape and one with rectangular shape and label. Both symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Negation
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary operation, unary logical connective. It may furthermore be applied not only to propositions, but also to notion (philosophy), notions, truth values, or interpretation (logic), semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logical NOR
In Boolean logic, logical NOR, non-disjunction, or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (''p'' NOR ''q'') is true precisely when neither ''p'' nor ''q'' is true—i.e. when both ''p'' and ''q'' are ''false''. It is logically equivalent to \neg(p \lor q) and \neg p \land \neg q, where the symbol \neg signifies logical negation, \lor signifies OR, and \land signifies AND. Non-disjunction is usually denoted as \downarrow or \overline or X (prefix) or \operatorname. As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either \uparrow, \mid or /), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sheffer Stroke
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, alternative denial (since it says in effect that at least one of its operands is false), or NAND ("not and"). In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as \mid or as \uparrow or as \overline or as Dpq in Polish notation by Łukasiewicz (but not as , , , often used to represent disjunction). Its dual is the NOR operator (also known as the Peirce arrow, Quine dagger or Webb operator). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete). This property makes the NAND gate crucial to modern digital electronics, including its use in computer processor design. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |