Logic

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Logic (from the Greek λογική
logikē)[1]
is the study of arguments.[2]
Logic is used in most intellectual activities, but is studied primarily in the
disciplines of philosophy, mathematics, and computer science. Logic
examines general forms which arguments may take, which forms
are valid, and which are fallacies. It is one kind of critical thinking. In
philosophy, the study of logic figures in most major areas of focus:
epistemology, ethics, metaphysics. In mathematics, it is the study of valid inferences
within some formal language.[3]

Logic has origins in several ancient civilizations, including ancient India, China and Greece. Logic was
established as a discipline by Aristotle, who established its
fundamental place in philosophy. The study of logic was part of the classical trivium.

Averroes
defined logic as "the tool for distinguishing between the true and the
false;"[4]
Richard Whately, "the
Science, as well as the Art, of reasoning;" and Gottlob Frege, "the
science of the most general laws of truth." The article Definitions of
logic provides citations for these and other definitions.

Logic is often divided into two parts, inductive reasoning
and deductive reasoning.
The first is drawing general conclusions from specific examples, the second
drawing logical conclusions from definitions and axioms. A similar dichotomy,
used by Aristotle, is analysis and
synthesis. Here the first takes an object of study and examines its
component parts. The second considers how parts can be combined to form a
whole.

Logic is also studied in argumentation
theory.[5]

Contents

1 Nature

1.1 Logical
form

1.2 Deductive and inductive
reasoning

1.3 Consistency, validity,
soundness, and completeness

1.4 Rival conceptions of
logic

2 History

3 Topics in logic

3.1 Syllogistic
logic

3.2 Sentential
(propositional) logic

3.3 Predicate
logic

3.4 Modal
logic

3.5 Informal
reasoning

3.6 Mathematical
logic

3.7 Philosophical
logic

3.8 Logic and
computation

4 Controversies

4.1 Bivalence and the law
of the excluded middle

4.2 "Is logic
empirical?"

4.3 Implication: strict or
material?

4.4 Tolerating the
impossible

4.5 Rejection of logical
truth

5 See also

6 Notes

7 References

8 External links and further
readings

[edit]
Nature

The concept of logical form is central to
logic, it being held that the validity of an argument is determined by its
logical form, not by its content. Traditional Aristotelian syllogistic logic
and modern symbolic logic are examples of formal logics.

Informal logic is
the study of natural language arguments. The study
of fallacies is an especially
important branch of informal logic. The dialogues of Plato[6]
are good examples of informal logic.

Formal
logic is the study of inference with purely formal
content. An inference possesses a purely formal content if it can be
expressed as a particular application of a wholly abstract rule, that is, a
rule that is not about any particular thing or property. The works of Aristotle
contain the earliest known formal study of logic. Modern formal logic follows
and expands on Aristotle.[7]
In many definitions of logic, logical inference and inference with purely
formal content are the same. This does not render the notion of informal logic
vacuous, because no formal logic captures all of the nuance of natural
language.

Symbolic logic is the study of symbolic abstractions that capture
the formal features of logical inference.[8][9]
Symbolic logic is often divided into two branches: propositional
logic and predicate logic.

Mathematical
logic is an extension of symbolic logic into other areas, in
particular to the study of model theory, proof
theory, set theory, and recursion
theory.

[edit]
Logical form

Logic is generally accepted to be formal, in that it aims to analyze
and represent the form (or logical form) of any valid
argument type. The form of an argument is displayed by representing its
sentences in the formal grammar and symbolism of a logical language to make its
content usable in formal inference. If one considers the notion of form to be
too philosophically loaded, one could say that formalizing is nothing else than
translating English sentences in the language of logic.

This is known as showing the logical form of the argument. It is
necessary because indicative sentences of ordinary language show a considerable
variety of form and complexity that makes their use in inference impractical. It
requires, first, ignoring those grammatical features which are irrelevant to
logic (such as gender and declension if the argument is in Latin), replacing
conjunctions which are not relevant to logic (such as 'but') with logical
conjunctions like 'and' and replacing ambiguous or alternative logical
expressions ('any', 'every', etc.) with expressions of a standard type (such as
'all', or the universal quantifier ∀).

Second, certain parts of the sentence must be replaced with schematic
letters. Thus, for example, the expression 'all As are Bs' shows the logical
form which is common to the sentences 'all men are mortals', 'all cats are
carnivores', 'all Greeks are philosophers' and so on.

That the concept of form is fundamental to logic was already recognized in
ancient times. Aristotle uses variable letters to represent valid inferences in
Prior Analytics,
leading Jan Łukasiewicz to
say that the introduction of variables was 'one of Aristotle's greatest
inventions'.[10]
According to the followers of Aristotle (such as Ammonius[disambiguation
needed]), only the logical principles stated in schematic terms
belong to logic, and not those given in concrete terms. The concrete terms
'man', 'mortal', etc., are analogous to the substitution values of the schematic
placeholders 'A', 'B', 'C', which were called the 'matter' (Greek 'hyle') of the
inference.

The fundamental difference between modern formal logic and traditional or
Aristotelian logic lies in their differing analysis of the logical form of the
sentences they treat.

In the traditional view, the form of the sentence consists of (1) a
subject (e.g. 'man') plus a sign of quantity ('all' or 'some' or 'no'); (2)
the copula which is
of the form 'is' or 'is not'; (3) a predicate (e.g. 'mortal'). Thus: all men
are mortal. The logical constants such as 'all', 'no' and so on, plus
sentential connectives such as 'and' and 'or' were called 'syncategorematic'
terms (from the Greek 'kategorei' – to predicate, and 'syn' – together with).
This is a fixed scheme, where each judgement has an identified quantity and
copula, determining the logical form of the sentence.

According to the modern view, the fundamental form of a simple sentence is
given by a recursive schema, involving logical
connectives, such as a quantifier with its bound variable, which are
joined to by juxtaposition to other sentences, which in turn may have logical
structure.

The modern view is more complex, since a single judgement of Aristotle's
system will involve two or more logical connectives. For example, the sentence
"All men are mortal" involves in term logic two non-logical terms "is a man"
(here M) and "is mortal" (here D): the sentence is given by the
judgement A(M,D). In predicate logic the
sentence involves the same two non-logical concepts, here analyzed as m(x) and d(x), and the sentence is given by ,
involving the logical connectives for universal
quantification and implication.

But equally, the modern view is more powerful: medieval logicians
recognized the problem of
multiple generality, where Aristotelean logic is unable to satisfactorily
render such sentences as "Some guys have all the luck", because both
quantities "all" and "some" may be relevant in an inference, but the fixed
scheme that Aristotle used allows only one to govern the inference. Just as
linguists recognize recursive structure in natural languages, it appears that
logic needs recursive structure.

[edit]
Deductive and
inductive reasoning

Deductive reasoning
concerns what follows necessarily from given premises (if a, then b). However,
inductive
reasoning—the process of deriving a reliable generalization from
observations—has sometimes been included in the study of logic. Correspondingly,
we must distinguish between deductive validity and inductive validity (called
"cogency"). An
inference is deductively valid if and only if there is
no possible situation in which all the premises are true but the conclusion
false. An inductive argument can be neither valid nor invalid; its premises give
only some degree of probability, but not certainty, to its conclusion.

The notion of deductive validity can be rigorously stated for systems of
formal logic in terms of the well-understood notions of semantics. Inductive validity
on the other hand requires us to define a reliable generalization of some set of
observations. The task of providing this definition may be approached in various
ways, some less formal than others; some of these definitions may use mathematical models
of probability. For the most part this discussion of logic deals only with
deductive logic.

[edit]
Consistency, validity, soundness, and completeness

Among the important properties that logical systems can
have:

Consistency,
which means that no theorem of the system contradicts another.[11]

Validity, which means
that the system's rules of proof will never allow a false inference from true
premises. A logical system has the property of soundness when the logical
system has the property of validity and only uses premises that prove true
(or, in the case of axioms, are true by definition).[11]

Completeness, which
means that if a theorem is true, it can be proven.

Soundness, which means
that the premises are true and the argument is valid.

Some logical systems do not have all four properties. As an example, Kurt
Gödel's incompleteness
theorems show that sufficiently complex formal systems of arithmetic cannot
be consistent and complete;[9]
however, first-order predicate logics not extended by specific axioms to be
arithmetic formal systems with equality can be complete and consistent.[12]

[edit]
Rival conceptions of
logic

Main article: Definitions of
logic

Logic arose (see below) from a concern with correctness of argumentation. Modern
logicians usually wish to ensure that logic studies just those arguments that
arise from appropriately general forms of inference. For example, Thomas
Hofweber writes in the Stanford
Encyclopedia of Philosophy that logic "does not, however, cover good
reasoning as a whole. That is the job of the theory of rationality. Rather it deals
with inferences whose validity can be traced back to the formal features of the
representations that are involved in that inference, be they linguistic, mental,
or other representations".[3]

By contrast, Immanuel Kant argued that
logic should be conceived as the science of judgment, an idea taken up in Gottlob
Frege's logical and philosophical work, where thought (German:
Gedanke) is substituted for judgment (German: Urteil). On this
conception, the valid inferences of logic follow from the structural features of
judgments or thoughts.

[edit]
History

Main article: History of logic

The earliest sustained work on the subject of logic is that of Aristotle.[13]
Aristotelian logic
became widely accepted in science and mathematics and remained in wide use in
the West until the early 19th century.[14]
Aristotle's system of logic was responsible for the introduction of hypothetical
syllogism,[15]
temporal modal
logic,[16][17]
and inductive logic.[18]
The Chinese logical
philosopher Gongsun Long (ca. 325–250
BC) proposed the paradox "One and one cannot become two, since neither becomes
two."[19]
In China, the tradition of scholarly investigation into logic, however, was
repressed by the Qin dynasty following the
legalist philosophy of Han Feizi.

Logic in Islamic philosophy, particularly Avicenna's
logic, was heavily influenced by Aristotelian logic.[20]
In Europe during
the later medieval period, major efforts were made to show that Aristotle's
ideas were compatible with Christian faith. During the
later period of the Middle Ages, logic became a main focus of philosophers, who
would engage in critical logical analyses of philosophical arguments.

In India, innovations in the scholastic school, called Nyaya, continued from ancient
times into the early 18th century with the Navya-Nyaya school. By the
16th century, it developed theories resembling modern logic, such as Gottlob
Frege's "distinction between sense and reference of proper names" and his
"definition of number," as well as the theory of "restrictive conditions for
universals" anticipating some of the developments in modern set
theory.[21]
Since 1824, Indian logic attracted the attention of many Western scholars, and
has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan,
and George Boole.[22]
In the 20th century, Western philosophers like Stanislaw Schayer and Klaus
Glashoff have explored Indian logic more extensively.

The syllogistic logic developed by
Aristotle predominated in the West until the mid-19th century, when interest in
the foundations of
mathematics stimulated the development of symbolic logic (now called mathematical logic).
In 1854, George Boole published An Investigation of the
Laws of Thought on Which are Founded the Mathematical Theories of Logic and
Probabilities, introducing symbolic logic and the principles of what is
now known as Boolean logic. In 1879,
Gottlob Frege published Begriffsschrift
which inaugurated modern logic with the invention of quantifier notation. From
1910 to 1913, Alfred North
Whitehead and Bertrand Russell
published Principia
Mathematica[8]
on the foundations of mathematics, attempting to derive mathematical truths from
axioms and inference rules in
symbolic logic. In 1931, Gödel raised serious problems
with the foundationalist program and logic ceased to focus on such issues.

The development of logic since Frege, Russell and Wittgenstein had a profound
influence on the practice of philosophy and the perceived nature of
philosophical problems (see Analytic
philosophy), and Philosophy of
mathematics. Logic, especially sentential logic, is implemented in computer
logic circuits and
is fundamental to computer science. Logic
is commonly taught by university philosophy departments often as a compulsory
discipline.

[edit]
Topics in logic

[edit]
Syllogistic 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.[23]
The parts of syllogistic logic, also known by the name term logic, are 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 consist of two propositions sharing a common term as premise, and a
conclusion which is 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.
However, it was not alone: the Stoics proposed a system of propositional logic
that was studied by medieval logicians. Also, the problem of
multiple generality was recognised in medieval times. Nonetheless, problems
with syllogistic logic were not seen as being in need of revolutionary
solutions.

Today, some academics claim that Aristotle's system is generally seen as
having little more than historical value (though there is some current interest
in extending term logics), regarded as made obsolete by the advent of
propositional logic and the predicate calculus.
Others use Aristotle in argumentation
theory to help develop and critically question argumentation schemes that
are used in artificial
intelligence and legal arguments.

[edit]
Sentential
(propositional) logic

Main article: Propositional
calculus

A propositional calculus or logic (also a sentential calculus) is a formal
system in which formulae representing propositions can be formed by combining
atomic propositions using logical connectives, and in which a system of formal
proof rules allows certain formulæ to be established as "theorems".

[edit]
Predicate logic

Main article: Predicate logic

Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic,
many-sorted logic, and
infinitary logic.

Predicate logic provides an account of quantifiers general enough
to express a wide set of arguments occurring in natural language. Aristotelian
syllogistic logic specifies a small number of forms that the relevant part of
the involved judgements may take. Predicate logic allows sentences to be
analysed into subject and argument in several additional ways, thus allowing
predicate logic to solve the problem of
multiple generality that had perplexed medieval logicians.

The development of predicate logic is usually attributed to Gottlob
Frege, who is also credited as one of the founders of analytical
philosophy, but the formulation of predicate logic most often used today is
the first-order logic presented in Principles
of Mathematical Logic by David Hilbert and Wilhelm Ackermann in
1928. The analytical generality of predicate logic allowed the formalisation of
mathematics, drove the investigation of set theory, and allowed the
development of Alfred Tarski's approach
to model theory. It provides
the foundation of modern mathematical
logic.

Frege's original system of predicate logic was second-order, rather than
first-order. Second-order logic is
most prominently defended (against the criticism of Willard Van Orman
Quine and others) by George Boolos and Stewart Shapiro.

[edit]
Modal logic

Main article: Modal logic

In languages, modality deals with
the phenomenon that sub-parts of a sentence may have their semantics modified by
special verbs or modal particles. For example, "We go to the games" can
be modified to give "We should go to the games", and "We can go to the
games"" and perhaps "We will go to the games". More abstractly, we
might say that modality affects the circumstances in which we take an assertion
to be satisfied.

The logical study of modality dates back to Aristotle,[24]
who was concerned with the alethic modalities of
necessity
and possibility, which
he observed to be dual in the sense of De Morgan duality.[citation
needed] While the study of necessity and possibility remained
important to philosophers, little logical innovation happened until the landmark
investigations of Clarence Irving
Lewis in 1918, who formulated a family of rival axiomatizations of the
alethic modalities. His work unleashed a torrent of new work on the topic,
expanding the kinds of modality treated to include deontic logic and epistemic logic. The
seminal work of Arthur Prior applied the
same formal language to treat temporal logic and paved
the way for the marriage of the two subjects. Saul Kripke discovered
(contemporaneously with rivals) his theory of frame semantics which
revolutionised the formal technology available to modal logicians and gave a new
graph-theoretic way of
looking at modality that has driven many applications in computational
linguistics and computer science, such
as dynamic
logic.

[edit]
Informal reasoning

Main article: Informal logic

The motivation for the study of logic in ancient times was clear: it is so
that one 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.
Half of the works of Aristotle's Organon treat inference as it
occurs in an informal setting, side by side with the development of the
syllogistic, and in the Aristotelian school, these informal works on logic were
seen as complementary to Aristotle's treatment of rhetoric.

This ancient 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.

Argumentation
theory is the study and research of informal logic, fallacies, and critical
questions as they relate to every day and practical situations. Specific types
of dialogue can be analyzed and questioned to reveal premises, conclusions, and
fallacies. Argumentation theory is now applied in artificial
intelligence and law.

[edit]
Mathematical logic

Main article: 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.[25]

The earliest use of mathematics and geometry in relation to logic
and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle.[26]
Many other ancient and medieval philosophers applied mathematical ideas and
methods to their philosophical claims.[27]

One of the boldest attempts 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.[8]
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.[28]
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.[citation
needed]

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.[29]
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.[30]

[edit]
Philosophical logic

Main article: 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 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., Kripke's technique of
supervaluations in the semantics of logic).

Logic and the philosophy of language are closely related. Philosophy of
language has to do with the study of how our language engages and interacts with
our thinking. Logic has an immediate impact on other areas of study. Studying
logic and the relationship between logic and ordinary speech can help a person
better structure his own arguments and critique the arguments of others. Many
popular arguments are filled with errors because so many people are untrained in
logic and unaware of how to formulate an argument correctly.

[edit]
Logic and
computation

Main article: Logic in computer
science

Logic cut to the heart of computer science as it emerged as a discipline: Alan
Turing's work on the Entscheidungsproblem
followed from Kurt Gödel's work on the
incompleteness
theorems, and the notion of general purpose computers that came from this
work was of fundamental importance to the designers of the computer machinery in
the 1940s.

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.

Today, logic is extensively applied in the fields of artificial
intelligence, and computer science, and
these fields provide a rich source of problems in formal and informal logic. Argumentation
theory is one good example of how logic is being applied to artificial
intelligence. The ACM
Computing Classification System in particular regards:

Section F.3 on Logics
and meanings of programs and F. 4 on Mathematical
logic and formal languages as part of the theory of computer science: this
work covers formal
semantics of programming languages, as well as work of formal methods such as
Hoare
logic

Boolean logic as
fundamental to computer hardware: particularly, the system's section B.2 on Arithmetic
and logic structures, relating to operatives AND, NOT, and OR;

Many fundamental logical formalisms are essential to section I.2 on
artificial intelligence, for example modal logic and default logic in Knowledge
representation formalisms and methods, Horn clauses in logic
programming, and description
logic.

Furthermore, computers can be used as tools for logicians. For example, 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.

[edit]
Controversies

Just as there is disagreement over what logic is about, so there is
disagreement about what logical truths there are.

[edit]
Bivalence and the law of the excluded middle

Main article: Principle of
bivalence

The logics discussed above are all "bivalent" or
"two-valued"; that is, they are most naturally understood as dividing
propositions into true and false propositions. Non-classical logics
are those systems which reject bivalence.

Hegel developed his own dialectic logic that
extended Kant's transcendental logic but
also brought it back to ground by assuring us that "neither in heaven nor in
earth, neither in the world of mind nor of nature, is there anywhere such an
abstract 'either–or' as the understanding maintains. Whatever exists is
concrete, with difference and opposition in itself".[31]

In 1910 Nicolai A. Vasiliev
rejected the law of excluded middle and the law of contradiction and proposed
the law of excluded fourth and logic tolerant to contradiction.[citation
needed] In the early 20th century Jan Łukasiewicz
investigated the extension of the traditional true/false values to include a
third value, "possible", so inventing ternary logic, the first
multi-valued
logic.[citation
needed]

Logics such as fuzzy logic have since been
devised with an infinite number of "degrees of truth", represented by a real
number between 0 and 1.[32]

Intuitionistic
logic was proposed by L.E.J. Brouwer as the
correct logic for reasoning about mathematics, based upon his rejection of the
law of the
excluded middle as part of his intuitionism. Brouwer
rejected formalisation in mathematics, but his student Arend Heyting studied
intuitionistic logic formally, as did Gerhard Gentzen.
Intuitionistic logic has come to be of great interest to computer scientists, as
it is a constructive logic,
and is hence a logic of what computers can do.

Modal
logic is not truth conditional, and so it has often been proposed as a
non-classical logic. However, modal logic is normally formalised with the
principle of the excluded middle, and its relational
semantics is bivalent, so this inclusion is disputable.

[edit]
"Is logic
empirical?"

Main article: Is logic
empirical?

What is the epistemological status of
the laws of logic? What sort
of argument is appropriate for criticizing purported principles of logic? In an
influential paper entitled "Is logic empirical?"[33]
Hilary Putnam, building on
a suggestion of W.V. Quine, argued that in
general the facts of propositional logic have a similar epistemological status
as facts about the physical universe, for example as the laws of mechanics or
of general relativity,
and in particular that what physicists have learned about quantum mechanics
provides a compelling case for abandoning certain familiar principles of
classical logic: if we want to be realists about the
physical phenomena described by quantum theory, then we should abandon the principle of
distributivity, substituting for classical logic the quantum
logic proposed by Garrett Birkhoff and John von Neumann.[34]

Another paper by the same name by Sir Michael Dummett
argues that Putnam's desire for realism mandates the law of distributivity.[35]
Distributivity of logic is essential for the realist's understanding of how
propositions are true of the world in just the same way as he has argued the
principle of bivalence is. In this way, the question, "Is logic empirical?" can
be seen to lead naturally into the fundamental controversy in metaphysics on realism versus
anti-realism.

[edit]
Implication:
strict or material?

Main article: Paradox of
entailment

It is obvious that the notion of implication formalised in classical logic
does not comfortably translate into natural language by means of "if… then…",
due to a number of problems called the paradoxes of material
implication.

The first class of paradoxes involves counterfactuals, such as "If the moon
is made of green cheese, then 2+2=5", which are puzzling because natural
language does not support the principle of
explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of
strict implication,
which eventually led to more radically revisionist logics such as relevance logic.

The second class of paradoxes involves redundant premises, falsely suggesting
that we know the succedent because of the antecedent: thus "if that man gets
elected, granny will die" is materially true if granny happens to be in the last
stages of a terminal illness, regardless of the man's election prospects. Such
sentences violate the Gricean maxim of
relevance, and can be modelled by logics that reject the principle of monotonicity of
entailment, such as relevance logic.

[edit]
Tolerating the
impossible

Main article: Paraconsistent
logic

Hegel was deeply critical of any
simplified notion of the Law of
Non-Contradiction. It was based on Leibniz's idea that this law of
logic also requires a sufficient ground to specify from what point of view (or
time) one says that something cannot contradict itself. A building, for example,
both moves and does not move; the ground for the first is our solar system for
the second the earth. In Hegelian dialectic, the law of non-contradiction, of
identity, itself relies upon difference and so is not independently
assertable.

Closely related to questions arising from the paradoxes of implication comes
the suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent
logic are the most important approaches here, though the concerns are
different: a key consequence of classical logic and some
of its rivals, such as intuitionistic
logic, is that they respect the principle of
explosion, which means that the logic collapses if it is capable of deriving
a contradiction. Graham Priest, the main
proponent of dialetheism, has argued for
paraconsistency on the grounds that there are in fact, true contradictions.[36]

[edit]
Rejection of logical
truth

The philosophical vein of various kinds of skepticism contains many kinds of
doubt and rejection of the various bases upon which logic rests, such as the
idea of logical form, correct inference, or meaning, typically leading to the
conclusion that there are no logical truths. Observe that this is opposite to
the usual views in philosophical
skepticism, where logic directs skeptical enquiry to doubt received wisdoms,
as in the work of Sextus Empiricus.

Friedrich Nietzsche
provides a strong example of the rejection of the usual basis of logic: his
radical rejection of idealisation led him to reject truth as a "mobile army of
metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn
out and without sensuous power; coins which have lost their pictures and now
matter only as metal, no longer as coins".[37]
His rejection of truth did not lead him to reject the idea of either inference
or logic completely, but rather suggested that "logic [came] into existence in
man's head [out] of illogic, whose realm originally must have been immense.
Innumerable beings who made inferences in a way different from ours
perished".[38]
Thus there is the idea that logical inference has a use as a tool for human
survival, but that its existence does not support the existence of truth, nor
does it have a reality beyond the instrumental: "Logic, too, also rests on
assumptions that do not correspond to anything in the real world".[39]

[edit]
See also

Digital
electronics (also known as digital logic
or logic gates)

Fallacies

Logic puzzle

Logic symbols

Mathematics

List
of mathematics articles

Outline of
mathematics

Metalogic

Outline of
logic

Book:Logic

Books are
collections of articles that can be downloaded or ordered in
print.

Logic
portal

Philosophy

List of
philosophy topics

Outline of
philosophy

Reason

Straight
and Crooked Thinking (book)

Table of
logic symbols

Truth

[edit]
Notes

^ "possessed of
reason, intellectual, dialectical, argumentative", also related to λόγος
(logos), "word, thought, idea, argument, account, reason, or principle"
(Liddell & Scott 1999; Online Etymology Dictionary 2001).

^ Gensler, Harry (2002). Introduction to Logic. 1 (1st ed.). Routledge.
ISBN 9780415226745. http://books.google.com/?id=sAe6GppFPDAC&lpg=PR1&dq=introduction%20to%20logic&pg=PA1#v=onepage&q&f=false.

^ a
b
Hofweber, T. (2004). "Logic
and Ontology". In Zalta, Edward N.
Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/logic-ontology.

^ Averroes,
In Arist. Physicam I, textus 35, ed. Juntina, IV, fol. 11vb.

^ Cox, J. Robert; Willard, Charles
Arthur, eds (1983). Advances in Argumentation Theory and Research.
Southern
Illinois University Press. ISBN 978-0809310500.

^ Plato (1976). Buchanan, Scott.
ed. The Portable Plato. Penguin. ISBN 0-14-015040-4.

^ Aristotle (2001). "Posterior
Analytics". In Mckeon, Richard. The Basic Works. Modern Library. ISBN 0-375-75799-6.

^ a
b
c
Whitehead, Alfred
North; Russell, Bertrand
(1967). Principia Mathematica to *56. Cambridge
University Press. ISBN 0-521-62606-4.

^ a
b
For a more modern treatment, see Hamilton, A. G.
(1980). Logic for Mathematicians. Cambridge University Press. ISBN 0-521-29291-3.

^ Łukasiewicz, Jan
(1957). Aristotle's syllogistic from the standpoint of modern formal
logic (2nd ed.). Oxford University Press. p. 7. ISBN 978-0198241447.

^ a
b
Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book fifth edition.
New York, NY: McGraw-Hill, 2009.

^ Mendelson, Elliott (1964). "Quantification Theory:
Completeness Theorems". Introduction to Mathematical Logic. Van
Nostrand. ISBN 0412808307.

^ E.g., Kline
(1972, p.53) wrote "A major achievement of Aristotle was the founding of the
science of logic".

^ "Aristotle", MTU Department of Chemistry.

^ Jonathan Lear
(1986). "Aristotle and Logical Theory". Cambridge
University Press. p.34. ISBN
0521311780

^ Simo Knuuttila
(1981). "Reforging the great chain of being: studies of the history of
modal theories". Springer Science & Business. p.71. ISBN
9027711259

^ Michael
Fisher, Dov M. Gabbay, Lluís Vila (2005). "Handbook of temporal reasoning in artificial
intelligence". Elsevier. p.119. ISBN
0444514937

^ Harold Joseph
Berman (1983). "Law and revolution: the formation of the Western legal
tradition". Harvard
University Press. p.133. ISBN
0674517768

^ The four
Catuṣkoṭi logical divisions are formally very close to the four opposed
propositions of the Greek tetralemma, which in turn
are analogous to the four truth values of modern relevance logic Cf.
Belnap (1977); Jayatilleke, K. N., (1967, The logic of four alternatives, in
Philosophy East and West, University of Hawaii Press).

^ Goodman, Lenn Evan (1992). Avicenna. Routledge.
p. 184. ISBN 978-0415019293.

^
Kisor Kumar Chakrabarti (June 1976). "Some Comparisons Between Frege's Logic and Navya-Nyaya
Logic". Philosophy and Phenomenological Research (International
Phenomenological Society) 36 (4): 554–563. doi:10.2307/2106873. http://www.jstor.org/stable/2106873. Retrieved 2010-03-24. "This paper consists
of three parts. The first part deals with Frege's distinction between sense
and reference of proper names and a similar distinction in Navya-Nyaya logic.
In the second part we have compared Frege's definition of number to the
Navya-Nyaya definition of number. In the third part we have shown how the
study of the so-called 'restrictive conditions for universals' in Navya-Nyaya
logic anticipated some of the developments of modern set theory.".

^ Jonardon Ganeri (2001). Indian logic: a reader.
Routledge. pp. vii, 5,
7. ISBN 0700713069.

^ "Aristotle". Encyclopædia
Britannica. http://www.britannica.com/EBchecked/topic/346217/history-of-logic/65920/Aristotle.

^ Peter Thomas
Geach (1980). "Logic Matters". University
of California Press. p.316. ISBN
0520038479

^ Stolyar, Abram A. (1983). Introduction to Elementary
Mathematical Logic. Dover Publications. p. 3. ISBN 0-486-64561-4.

^ Barnes, Jonathan (1995). The Cambridge Companion to
Aristotle. Cambridge University Press. p. 27. ISBN 0-521-42294-9.

^ Aristotle (1989). Prior
Analytics. Hackett Publishing Co.. p. 115. ISBN 978-0872200647.

^ Mendelson, Elliott (1964). "Formal Number Theory:
Gödel's Incompleteness Theorem". Introduction to Mathematical Logic.
Monterey, Calif.: Wadsworth & Brooks/Cole Advanced Books & Software.
OCLC 13580200.

^ Brookshear, J. Glenn (1989). "Computability: Foundations
of Recursive Function Theory". Theory of computation: formal languages,
automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub.
Co.. ISBN 0805301437.

^ Brookshear, J. Glenn (1989). "Complexity". Theory of
computation: formal languages, automata, and complexity. Redwood City,
Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.

^ Hegel, G. W.
F (1971) [1817]. Philosophy of
Mind. Encyclopedia of the Philosophical Sciences. trans. William
Wallace. Oxford: Clarendon Press. p. 174. ISBN 0198750145.

^ Hájek, Petr (2006). "Fuzzy Logic". In Zalta, Edward N..
Stanford
Encyclopedia of Philosophy. http://plato.stanford.edu/entries/logic-fuzzy/.

^ Putnam, H. (1969). "Is
Logic Empirical?". Boston Studies in the Philosophy of Science
5.

^ Birkhoff, G.; von Neumann, J.
(1936). "The Logic of Quantum Mechanics". Annals of
Mathematics (Annals of Mathematics) 37 (4): 823–843. doi:10.2307/1968621. http://jstor.org/stable/1968621.

^ Dummett, M. (1978).
"Is Logic Empirical?". Truth and Other Enigmas. ISBN 0-674-91076-1.

^ Priest, Graham (2008).
"Dialetheism". In Zalta, Edward N..
Stanford
Encyclopedia of Philosophy. http://plato.stanford.edu/entries/dialetheism.

^ Nietzsche,
1873, On
Truth and Lies in a Nonmoral Sense.

^ Nietzsche,
1882, The Gay
Science.

^ Nietzsche,
1878, Human, All Too
Human

[edit]
References

Nuel Belnap, (1977). A
useful four-valued logic. In Dunn & Eppstein, Modern uses of
multiple-valued logic. Reidel: Boston.

Brookshear, J. Glenn (1989). Theory of
computation: formal languages, automata, and complexity. Redwood City,
Calif.: Benjamin/Cummings Pub. Co.. ISBN 0805301437.

Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological
Studies in Contemporary Physics. Boston Studies in the Philosophy of
Science. D. Reidel Publishing Company: Dordrecht, Netherlands. ISBN
90-277-0377-9.

Finkelstein, D. (1969). "Matter, Space, and Logic". in R.S. Cohen and M.W.
Wartofsky (eds. 1974).

Gabbay, D.M., and
Guenthner, F. (eds., 2001–2005). Handbook of Philosophical Logic. 13
vols., 2nd edition. Kluwer Publishers: Dordrecht.

Hilbert, D., and Ackermann, W,
(1928). Grundzüge der theoretischen Logik (Principles
of Mathematical Logic). Springer-Verlag. OCLC 2085765

Susan Haack, (1996).
Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago
Press.

Hodges, W., (2001).
Logic. An introduction to Elementary Logic, Penguin Books.

Hofweber, T., (2004), Logic
and Ontology. Stanford
Encyclopedia of Philosophy. Edward N. Zalta
(ed.).

Hughes, R.I.G., (1993, ed.). A Philosophical Companion to First-Order
Logic. Hackett Publishing.

Kline, Morris (1972). Mathematical Thought
From Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.

Kneale,
William, and Kneale, Martha, (1962). The Development of Logic.
Oxford University Press, London, UK.

Liddell, Henry George;
Scott,
Robert. "Logikos". A Greek-English
Lexicon. Perseus Project. http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2363716. Retrieved 8 May 2009.

Mendelson, Elliott, (1964). Introduction to Mathematical Logic.
Wadsworth & Brooks/Cole Advanced Books & Software: Monterey, Calif. OCLC 13580200

Harper, Robert (2001). "Logic". Online
Etymology Dictionary. http://www.etymonline.com/index.php?term=logic. Retrieved 8 May 2009.

Smith, B.,
(1989). "Logic and the Sachverhalt". The Monist 72(1):52–69.

Whitehead, Alfred
North and Bertrand Russell,
(1910). Principia
Mathematica. Cambridge University Press: Cambridge, England. OCLC 1041146

[edit]
External links
and further readings

Find more about
Logic on Wikipedia's sister
projects:

Definitions
from Wiktionary

Images and
media from Commons

Learning
resources from Wikiversity

News stories
from Wikinews

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Textbooks
from Wikibooks

LogicWiki, an external wiki

Introductions and tutorials

An Introduction to Philosophical Logic, by Paul Newall,
aimed at beginners.

forall x: an introduction to formal logic, by P.D. Magnus,
covers sentential and quantified logic.

Logic Self-Taught: A Workbook (originally prepared for
on-line logic instruction).

Nicolas Rescher. (1964).
Introduction to Logic, St. Martin's Press.

Essays

"Symbolic Logic" and "The Game of
Logic", Lewis Carroll,
1896.

Math & Logic: The history of formal mathematical, logical,
linguistic and methodological ideas. In The Dictionary of the History
of Ideas.

Reference material

Translation Tips, by Peter Suber, for translating from
English into logical notation.

Ontology and
History of Logic. An Introduction with an annotated
bibliography.

Reading lists

The London Philosophy
Study Guide offers many suggestions on what to read, depending on the
student's familiarity with the subject:

Logic
& Metaphysics

Set
Theory and Further Logic

Mathematical Logic

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