Logic





Roughly speaking, logic is the study of prescriptive systems of reasoning,
for example, 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. See also:
list of topics in logic.

Aristotelian logic

Aristotelian logic was pioneered by Aristotle. Although it is possible that
Aristotle was taught by someone else, the earliest study of reasoning can be
attributed to Aristotle. Aristotle and his followers held that two of the
most important principles of logic are the law of non-contradiction and the
law of excluded middle. This kind of logic is now given various names to
distinguish it from more recent systems of logic, e.g., Aristotelian logic
or classical two-valued logic.

The law of non-contradiction 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.

Formal logic

Formal logic, also called symbolic logic, is concerned primarily with the
structure of reasoning. Formal logic deals with the relationships between
concepts and provides a way to compose proofs of statements. In formal
logic, concepts are rigorously defined, and sentences are translated into a
precise, compact, and unambiguous symbolic notation.

Some examples of symbolic notation are:

Lowercase letter p, q and r with italic font are conventionally used to
denote propositions:

     p: 1 + 2 = 3

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,

     p: 1 + 1 = 2 and "logic is the study of reasoning."

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:

     p: n is an odd integer.

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,

     for all n in D, P(n).

This can be written also as:

     [\forall n\in D, P(n)]

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.

[edit]

Mathematical logic

Mathematical logic is the use of formal logic to study mathematical
reasoning. At the beginning of the twentieth century, philosophical
logicians including(Frege, Russell) attempted to prove that mathematics
could be entirely reduced to logic. They held that in discovering the
"logical form" of a sentence, you were somehow revealing the "right" way to
say it, or uncovering some previously hidden essence. The reduction failed,
but in the process, logic took on much of the notation and methodology of
mathematics, and nowadays logic is accepted as an accurate way to describe
mathematical reasoning.

Philosophical logic

Philosophical logic is essentially a continuation of the traditional
discipline that was called "Logic" before it was supplanted by the invention
of Mathematical logic. It is concerned with the elucidation of ideas such as
reference, predication, identity, truth, quantification, existence, and
others. Philosophical logic has a much greater concern with the connection
between natural language and logic. See Philosophical logic.

Predicate logic

Gottlob Frege, in his Begriffsschrift, discovered a way to rearrange many
sentences to make their logical form clear, to show how sentences relate to
one another in certain respects. Prior to Frege, formal logic had not been
successful beyond the level of sentential logic: it could represent the
structure of sentences composed of other sentences using such words as
"and", "or", and "not," but it could not break sentences down into smaller
parts. It could not show how "Cows are animals" entails "Parts of cows are
parts of animals."

Sentential logic explains the workings of words such as "and", "but", "or",
"not", "if-then", "if and only if", and "neither-nor". Frege expanded logic
to include words such as "all", "some", and "none". He showed how we can
introduce variables and "quantifiers" to rearrange sentences.

   * "All humans are mortal" becomes "All things x are such that, if x is a
     human then x is mortal." which may be written symbolically

          [\forall x (H(x)\Rightarrow M(x))]

   * "Some humans are vegetarian" becomes "There exists some (at least one)
     thing x such that x is human and x is vegetarian" which may be written
     symbolically

          [\exists x (H(x)\wedge V(x))].

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.no,

Frege adds to sentential logic (1) the vocabulary of quantifiers
(upside-down 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.

Multi-valued Logic

Systems which go beyond these two distinctions are known as non-Aristotelian
logics, or multi-valued logics.

In the early 20th century Jan ?ukasiewicz 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.

Logic and computers

Logic is extensively used in the fields of artificial intelligence, and
computer science.

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
computer-assisted. 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