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Roughly speaking, logic is the study of prescriptive systems of reasoning, i.e. systems that people ought to use to reason deductively and inductively. How people actually reason is usually studied under other headings, including cognitive psychology. Logic is a branch of mathematics, and a branch of philosophy.
As a science, logic defines the structure of statement and argument and devises formulae by which these are codified. Implicit in a study of logic is the understanding of what makes a good argument and what arguments are fallacious. Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic.
Following are more specific discussions of some systems of logic.

The law of noncontradiction states that no proposition is both true and false and law of excluded middle states that a proposition must either be true or false. In combination, these laws require two truth values that are mutually exclusive. A proposition can be either true or false, but cannot be both at the same time.
Some examples of symbolic notation are:
Lowercase letter p, q and r with italic font are conventionally used to denote propositions:
This statement defines p is 1 + 2 = 3 and that is true.
Two propositions can be combined using conjunction, disjunction or conditional. They are called binary logical operators. Such combined propositions are called compound propositions. For example,
In this case, and is a conjunction. The two propositions can differ totally from each other.
In mathematics and computer science, one may want to state a proposition depending on some variables:
This proposition can be either true or false according to the variable n.
A proposition with free variables is called propositional function with domain of discourse D. To form an actual proposition, one uses quantifiers. For every n, or for some n, can be specified by quantifiers: either the universal quantifier or the existential quantifier. For example,
This can be written also as:
When there are several free variables free, the standard situation in mathematical analysis since Weierstrass, the quantifications for all ... there exists or there exists ... such that for all (and more complex analogues) can be expressed.
Aristotelian logic was capable of dealing with objects and not just sentences, but it was known to have a number of fallacies, and no way had been found to remove these systematically. Frege's insight was that by discovering "quantifiers" in sentences, we can reduce many more of them into structures of termandpredicate which we can operate on logically without running into the fallacies of Aristotelian logic.
Sentential logic explains the workings of words such as "and", "but", "or", "not", "ifthen", "if and only if", and "neithernor". Frege expanded logic to include words such as "all", "some", and "none". He showed how we can introduce variables and "quantifiers" to rearrange sentences.
Frege treats simple sentences without subject nouns as predicates and applies them to "dummy objects" (x). The logical structure in discourse about objects can then be operated on according to the rules of sentential logic, with some additional details for adding and removing quantifiers. Frege's work started contemporary formal logic; Aristotelian logic as a substantive research project effectively perished.
Frege adds to sentential logic (1) the vocabulary of quantifiers (upsidedown A, backward E) and variables, (2) a semantics that explains that the variables denote individual objects and the quantifiers have something like the force of "all" "some" in relation to those objects, and (3) methods for using these in language. To introduce an "All" quantifier, you assume an arbitrary variable, prove something that must hold true of it, and then prove that it didn't matter which variable you chose, that would have held true. An "All" quantifier can be removed by applying the sentence to any particular object at all. A "Some" (exists) quantifier can be added to a sentence true of any object at all; it can be removed in favor of a term about which you are not already presupposing any information.
In the early 20th century Jan Lukasiewicz[?] investigated the extension of the traditional true/false values to include a third value, "possible".
Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1. Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.
In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human reasoning. Logic programming is an attempt to make computers do logical reasoning and Prolog programming language is commonly used for it.
In symbolic logic and mathematical logic, proofs by humans can be computerassisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.
In computer science, Boolean algebra is the basis of hardware design, as well as much software design.
Quote Logic, logic, logic. Logic is the beginning of wisdom, Valeris, not the end. From Star Trek VI: The Undiscovered Country
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