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Deductively Closed
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 . Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of sentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a ''deductively closed theory'' to emphasize it is defined as a deductively closed set. Deductive closure 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky is blue" expresses the proposition that the sky is blue. Unlike sentences, propositions are not linguistic expressions, so the English sentence "Snow is white" and the German "Schnee ist weiß" denote the same proposition. Propositions also serve as the objects of belief and other propositional attitudes, such as when someone believes that the sky is blue. Formally, propositions are often modeled as functions which map a possible world to a truth value. For instance, the proposition that the sky is blue can be modeled as a function which would return the truth value T if given the actual world as input, but would return F if given some alternate world where the sky is green. However, a number of alternative formalizations have be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (logic), valid logical argument is one in which the Consequent, conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is logical truth, necessary and Formalism (philosophy of mathematics), formal, by wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and " Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is valid ''and'' all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concepts In Logic
A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, psychology, and philosophy, and these disciplines are interested in the logical and psychological structure of concepts, and how they are put together to form thoughts and sentences. The study of concepts has served as an important flagship of an emerging interdisciplinary approach, cognitive science. In contemporary philosophy, three understandings of a concept prevail: * mental representations, such that a concept is an entity that exists in the mind (a mental object) * abilities peculiar to cognitive agents (mental states) * Fregean senses, abstract objects rather than a mental object or a mental state Concepts are classified into a hierarchy, higher levels of which are termed "superordinate" and lower levels termed "subordinate". A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belief
A belief is a subjective Attitude (psychology), attitude that something is truth, true or a State of affairs (philosophy), state of affairs is the case. A subjective attitude is a mental state of having some Life stance, stance, take, or opinion about something. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either truth value, true or false. To believe something is to take it to be true; for instance, to believe that snow is white is comparable to accepting the truth of the proposition "snow is white". However, holding a belief does not require active introspection. For example, few individuals carefully consider whether or not the sun will rise tomorrow, simply assuming that it will. Moreover, beliefs need not be ''occurrent'' (e.g., a person actively thinking "snow is white"), but can instead be ''dispositional'' (e.g., a person who if asked about the color of snow would assert "snow is white"). There are various ways tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Justification
Justification (also called epistemic justification) is a property of beliefs that fulfill certain norms about what a person should believe. Epistemologists often identify justification as a component of knowledge distinguishing it from mere true opinion. They study the reasons why someone holds a belief. Epistemologists are concerned with various features of belief, which include the ideas of warrant (a proper justification for holding a belief), knowledge, rationality, and probability, among others. Debates surrounding epistemic justification often involve the ''structure'' of justification, including whether there are foundational justified beliefs or whether mere coherence is sufficient for a system of beliefs to qualify as justified. Another major subject of debate is the sources of justification, which might include perceptual experience (the evidence of the senses), reason, and authoritative testimony, among others. Justification and knowledge "Justification" involves the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knowledge
Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is often characterized as Truth, true belief that is distinct from opinion or guesswork by virtue of Justification (epistemology), justification. While there is wide agreement among philosophers that propositional knowledge is a form of true belief, many controversies focus on justification. This includes questions like how to understand justification, whether it is needed at all, and whether something else besides it is needed. These controversies intensified in the latter half of the 20th century due to a series of thought experiments called ''Gettier cases'' that provoked alternative definitions. Knowledge can be produced in many ways. The main source of empirical knowledge is perception, which involves the usage of the senses to learn about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epistemology
Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowledge in the form of skills, and knowledge by acquaintance as a familiarity through experience. Epistemologists study the concepts of belief, truth, and justification to understand the nature of knowledge. To discover how knowledge arises, they investigate sources of justification, such as perception, introspection, memory, reason, and testimony. The school of skepticism questions the human ability to attain knowledge while fallibilism says that knowledge is never certain. Empiricists hold that all knowledge comes from sense experience, whereas rationalists believe that some knowledge does not depend on it. Coherentists argue that a belief is justified if it coheres with other beliefs. Foundationalists, by contrast, maintain th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-formed Formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff". A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. Introduction A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Alth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
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 Truth value, 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 Logical conjunction, conjunction, Logical disjunction, disjunction, Material conditional, implication, Logical biconditional, 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 Quantifier (logic), quantifiers. However, all the machinery of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (logic), valid logical argument is one in which the Consequent, conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is logical truth, necessary and Formalism (philosophy of mathematics), formal, by wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
