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Meta-logic
Metalogic is the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory. Overview Formal language A ''formal language'' is an organized set of symbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings of its expressions; it can exist before any interpretation is assigned to it—that is, befor ... [...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 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] |
Metatheory
A metatheory or meta-theory is a theory on a subject matter that is a theory in itself. Analyses or descriptions of an existing theory would be considered meta-theories. For mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. Meta-theoretical investigations are part of the philosophy of science. The topic of metascience is an attempt to use scientific knowledge to improve the practice of science itself. The study of metatheory became widespread during the 20th century after its application to various topics, including scientific linguistics and its concept of metalanguage. Examples of metatheories Metascience Metascience is the use of scientific method to study science itself. Metascience is an attempt to increase the quality of scientific research while reducing wasted activity; it uses research methods to study how research is done or can be improved. It has been described as "''research on research''", "''the sci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sense And Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the two ways he believed a singular term may have meaning. The reference (or "referent"; ''Bedeutung'') of a ''proper name'' is the object it means or indicates (''bedeuten''), whereas its sense (''Sinn'') is what the name expresses. The reference of a ''sentence'' is its truth value, whereas its sense is the thought that it expresses."On Sense and Reference" Über Sinn und Bedeutung" '' Zeitschrift für Philosophie und philosophische Kritik'', vol. 100 (1892), pp. 25–50, esp. p. 31. Frege justified the distinction in a number of ways. #Sense is something possessed by a name, whether or not it has a reference. For example, the name "Odysseus" is intelligible, and therefore has a sense, even though there is no individual object (its refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as ''modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive System
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Concepts A formal system has the following: * Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules). * Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal language. A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantics
Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction between sense and reference. Sense is given by the ideas and concepts associated with an expression while reference is the object to which an expression points. Semantics contrasts with syntax, which studies the rules that dictate how to create grammatically correct sentences, and pragmatics, which investigates how people use language in communication. Lexical semantics is the branch of semantics that studies word meaning. It examines whether words have one or several meanings and in what lexical relations they stand to one another. Phrasal semantics studies the meaning of sentences by exploring the phenomenon of compositionality or how new meanings can be created by arranging words. Formal semantics (natural language), Formal semantics relies o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphabet
An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from another in a given language. Not all writing systems represent language in this way: a syllabary assigns symbols to spoken syllables, while logographies assign symbols to words, morphemes, or other semantic units. The first letters were invented in Ancient Egypt to serve as an aid in writing Egyptian hieroglyphs; these are referred to as Egyptian uniliteral signs by lexicographers. This system was used until the 5th century AD, and fundamentally differed by adding pronunciation hints to existing hieroglyphs that had previously carried no pronunciation information. Later on, these phonemic symbols also became used to transcribe foreign words. The first fully phonemic script was the Proto-Sinaitic script, also descending from Egyptian hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String (computer Science)
In computer programming, a string is traditionally a sequence of character (computing), characters, either as a literal (computer programming), literal constant or as some kind of Variable (computer science), variable. The latter may allow its elements to be Immutable object, mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or word (computer architecture), words) that stores a sequence of elements, typically characters, using some character encoding. More general, ''string'' may also denote a sequence (or List (abstract data type), list) of data other than just characters. Depending on the programming language and precise data type used, a variable (programming), variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
