Paradigms of Logic

Paradigms of logic

Throughout history, there has been interest in distinguishing good 
from bad arguments, and so logic has been studied in some more or 
less familiar form. Aristotelian logic has principally been concerned 
with teaching good argument, and is still taught with that end today, 
while in mathematical logic and analytical philosophy much greater 
emphasis is placed on logic as an object of study in its own right, 
and so logic is studied at a more abstract level.

Consideration of the different types of logic explains that logic is 
not studied in a vacuum. While logic often seems to provide its own 
motivations, the subject develops most healthily when the reason for 
our interest is made clear.
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Aristotelian logic

Main article: Aristotelian logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics 
constituting the first explicit work in formal logic, introducing the 
syllogistic. The parts of syllogistic, also known by the name term logic, 
were the analysis of the judgements into propositions consisting of two 
terms that are related by one of a fixed number of relations, and the 
expression of inferences by means of syllogisms that consisted of two 
propositions sharing a common term as premise, and a conclusion which 
was a proposition involving the two unrelated terms from the premises.

Aristotle's work was regarded in classical times and from medieval times 
in Europe and the Middle East as the very picture of a fully worked out 
system. It was not alone: the Stoics proposed a system of propositional 
logic that was studied by medieval logicians; nor was the perfection of 
Aristotle's system undisputed; for example the problem of multiple 
generality was recognised in medieval times. Nonetheless, problems with 
syllogistic were not seen as being in need of revolutionary solutions.

Today, Aristotle's system is mostly seen as of historical value (though 
there is some current interest in extending term logics), regarded as 
made obsolete by the advent of the predicate calculus.

Predicate logic
First-order predicate calculus or first-order logic (FOL) is a theory in 
symbolic logic that permits the formulation of quantified statements such 
as "there is at least one X such that..." or "for any X, it is the case 
that...", where X is an element of a set called the domain of discourse. 
A first-order theory is a theory that can be axiomatised as an extension 
of first-order logic by adding a recursive set of first-order sentences 
as axioms.

First-order logic is distinguished from higher-order logic in that it does 
not allow statements such as "for every property, it is the case that..." 
or "there exists a set of objects such that..."

Nevertheless, first-order logic is strong enough to formalize all of set 
theory and thereby virtually all of mathematics. Its restriction to 
quantification over individuals makes it difficult to use for the purposes 
of topology, but it is the classical logical theory underlying mathematics. 
It is a stronger theory than sentential logic, but a weaker theory than 
arithmetic, set theory, or second-order logic.

Modal logic

Modal logic, or (less commonly) intensional logic is the branch of logic 
that deals with sentences that are qualified by modalities such as can, 
could, might, may, must, possibly, and necessarily, and others. Any logical 
system making use of modal operators, such as possibly, or necessarily is 
thus also called a modal logic. Modal logics are characterized by semantic 
intensionality: non-modal logics all have the feature that the truth value 
of a complex sentence is determined by the truth values of its 
sub-sentences. They are thus extensional. In modal logics, by contrast, 
this does not hold: both "George W. Bush is President of the United States" 
and "2+2=4" are true, yet "Necessarily, George W. Bush is President of the 
United States" is false, while "Necessarily, 2+2=4" is true. Necessity and 
possibility are the most widely discussed modalities in work on modal logic,
 and most work on necessity and possibility focuses on the so-called alethic 
 modalities, but there are other senses of necessity and possibility, and 
 other modalities as well.

A formal modal logic represents modalities using modal sentential operators. 
The basic set of modal operators are usually given to be \Box and \Diamond. 
In alethic modal logic the \Box represents necessity and the \Diamond 
possibility. A sentence is said to be

    * possible if it might be true (whether it is actually true or actually 
    false);
    * necessary if it could not possibly be false;
    * contingent if it is not necessarily true, i.e., is possibly true, and 
    possibly false. A contingent truth is one which is actually true, but 
    which could have been otherwise.

Dialectical logic

The motivation for the study of logic in ancient times was clear, as we 
have described: it is so that we may learn to distinguish good from bad 
arguments, and so become more effective in argument and oratory, and 
perhaps also, to become a better person.

This motivation is still alive, although it no longer takes centre stage 
in the picture of logic; typically dialectical logic will form the heart 
of a course in critical thinking, a compulsory course at many universities, 
especially those that follow the American model.

Mathematical logic

Mathematical logic really refers to two distinct areas of research: the 
first is the application of the techniques of formal logic to mathematics 
and mathematical reasoning, and the second, in the other direction, the 
application of mathematical techniques to the representation and analysis 
of formal logic.

The boldest attempt to apply logic to mathematics was undoubtedly the 
logicism pioneered by philosopher-logicians such as Gottlob Frege and 
Bertrand Russell: the idea was that mathematical theories were logical 
tautologies, and the programme was to show this by means to a reduction 
of mathematics to logic. The various attempts to carry this out met with 
a series of failures, from the crippling of Frege's project in his 
Grundgesetze by Russell's Paradox, to the defeat of Hilbert's Program by 
Gödel's incompleteness theorems.

Both the statement of Hilbert's Program and its refutation by Gödel depended 
upon their work establishing the second area of mathematical logic, the 
application of mathematics to logic in the form of proof theory. Despite 
the negative nature of the incompleteness theorems, Gödel's completeness 
theorem, a result in model theory and another application of mathematics 
to logic, can be understood as showing how close logicism came to being true: 
every rigorously defined mathematical theory can be exactly captured by a 
first-order logical theory; Frege's proof calculus is enough to describe the 
whole of mathematics, though not equivalent to it. Thus we see how 
complementary the two areas of mathematical logic have been.

If proof theory and model theory have been the foundation of mathematical 
logic, they have been but two of the four pillars of the subject. Set theory 
originated in the study of the infinite by Georg Cantor, and it has been the 
source of many of the most challenging and important issues in mathematical 
logic, from Cantor's theorem, through the status of the Axiom of Choice and 
the question of the independence of the continuum hypothesis, to the modern 
debate on large cardinal axioms.

Recursion theory captures the idea of computation in logical and arithmetic 
terms; its most classical achievements are the undecidability of the 
Entscheidungsproblem by Alan Turing, and his presentation of the 
Church-Turing thesis. Today recursion theory is mostly concerned with the 
more refined problem of complexity classes -- when is a problem efficiently 
solvable? -- and the classification of degrees of unsolvability.

Philosophical logic

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. Philosophical logic is essentially a continuation of the traditional 
discipline that was called "Logic" before it was supplanted by the invention of 
Mathematical logic. Philosophical logic has a much greater concern with the 
connection between natural language and logic. As a result, philosophical 
logicians have contributed a great deal to the development of non-standard 
logics (e.g., free logics, tense logics) as well as various extensions of 
classical logic (e.g., modal logics), and non-standard semantics for such 
logics (e.g., supervaluation semantics).

Logic and computation

Logic is extensively applied in the fields of artificial intelligence, and 
computer science, and these fields provide a rich source of problems in formal 
logic.

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. 
In logic programming, a program consists of a set of axioms and rules. Logic 
programming systems such as Prolog compute the consequences of the axioms and 
rules in order to answer a query.

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.

There are also various systems for reasoning about computer programs. Hoare 
logic is one of the earliest of such systems. Other systems are CSP, CCS, 
pi-calculus for reasoning about concurrent processes or mobile processes. 
There is interest in the idea of finding a logical calculus that naturally 
captures computability; the computability logic of Japaridze is an example 
of a recently embarked research programme in this direction.