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Monadic First-order Logic
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form P(x), where P is a relation symbol and x is a variable. Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments. Expressiveness The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains). Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül," ''Mathematische Annalen'' 76: 447-470. Translated as "On possibilities in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for thirty years, Peirce made major contributions to logic, a subject that, for him, encompassed much of what is now called epistemology and the philosophy of science. He saw logic as the formal branch of semiotics, of which he is a founder, which foreshadowed the debate among logical positivists and proponents of philosophy of language that dominated 20th-century Western philosophy. Additionally, he defined the concept of abductive reasoning, as well as rigorously formulated mathematical induction and deductive reasoning. As early as 1886, he saw that logic gate, logical operations could be carried out by electrical switching circuits. The same idea was used decades later to produce digital computers. See Also In 1934, the philosopher Paul W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Variables
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unary Operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation: :3 − −2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expression i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In logic and philosophy, it is also called adicity and degree. In linguistics, it is usually named valency. Examples The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example: * A nullary function takes no arguments. ** Example: f()=2 * A unary function takes one argument. ** Example: f(x)=2x * A binary function takes two arguments. ** Examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence \forall P\,\forall x (Px \lor \neg Px) says that for every formula ''P'', and every individual ''x'', either ''Px'' is true or not(''Px'') is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problem Of Multiple Generality
The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if: :''Some cat is feared by every mouse'' then it follows logically that: :''All mice are afraid of at least one cat''. The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, w ... [...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] |
Syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BCE book '' Prior Analytics''), a syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \implies q \equiv \neg p \vee q :#p \implies q \equiv \neg q \implies \neg p :#p \vee q \equiv \neg p \implies q :#p \wedge q \equiv \neg (p \implies \neg q) :#\neg (p \implies q) \equiv p \wedge \neg q :#(p \implies q) \wedge (p \implie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it, so to understand the newer systems, one must be acquainted with the earlier one. Aristotle's system Aristotle's logical work is collected in the six texts that are collectively known as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
