Everything about Logic totally explained
Logic is the study of the principles of valid
inference and
demonstration. The word derives from
Greek λογική (
logike), fem. of
λογικός (
logikos), "possessed of reason, intellectual, dialectical, argumentative", from
λόγος logos, "word, thought, idea, argument, account, reason, or principle".
As a
formal science, logic investigates and classifies the structure of
statements and
arguments, both through the study of
formal systems of
inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of
validity,
fallacies and
paradoxes, to specialized analysis of reasoning using
probability and to arguments involving
causality. Logic is also commonly used today in
argumentation theory.
Traditionally, logic was considered a branch of
philosophy, a part of the classical
trivium of
grammar, logic, and
rhetoric. Since the mid-nineteenth century
formal logic has been studied in the context of
foundations of mathematics, where it was often called
symbolic logic. In 1903
Alfred North Whitehead and
Bertrand Russell attempted to establish logic formally as the cornerstone of mathematics with the publication of
Principia Mathematica. However, except for the elementary part, the system of Principia is no longer much used, having been largely superseded by
set theory. At the same time the developments in the field of Logic since
Frege,
Russell and
Wittgenstein had a profound influence on both the practice of philosophy and the ideas concerning the nature of philosophical problems especially in the English speaking world (see
Analytic philosophy). As the study of formal logic expanded, research no longer focused solely on foundational issues, and the study of several resulting areas of mathematics came to be called
mathematical logic. The development of formal logic and its implementation in computing machinery is fundamental to
computer science. Logic is now widely taught by university philosophy departments, more often than not as a compulsory discipline for their students, especially in the English speaking world.
Nature of logic
Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic language is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional
Aristotelian syllogistic logic, which deals solely with
categorical propositions.
- 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 are a good example of informal logic.
- Formal logic is the study of inference with purely formal content, where that content is made explicit. (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 isn't about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic. In many definitions of logic, logical inference and inference with purely formal content are the same. This doesn't render the notion of informal logic vacuous, because no formal language captures all of the nuance of natural language.)
- Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. 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.
"Formal logic" is often used as a synonym for symbolic logic, where informal logic is then understood to mean any logical investigation that doesn't involve symbolic abstraction; it's this sense of 'formal' that's parallel to the received usages coming from "
formal languages" or "
formal theory". In the broader sense, however, formal logic is old, dating back more than two millennia, while symbolic logic is comparatively new, only about a century old.
Consistency, soundness, and completeness
Among the valuable properties that
logical systems can have are:
» *
Consistency, which means that none of the theorems of the system contradict one another.
*
Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.
» *
Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.
Not all systems achieve all three virtues. The work of
Kurt Gödel has shown that no useful system of arithmetic can be both consistent and complete: see
Gödel's incompleteness theorems. The Chinese philosopher
Gongsun Long (ca.
325–
250 BC) proposed the paradox "One and one can't become two, since neither becomes two." In China, the tradition of scholarly investigation into logic, however, was repressed by the
Qin dynasty following the legalist philosophy of
Han Feizi.
The first sustained work on the subject of logic which has survived was that of
Aristotle. The formally sophisticated treatment of modern logic descends from the Greek tradition, the latter mainly being informed from the transmission of
Aristotelian logic.
Logic in Islamic philosophy also contributed to the development of modern logic, which included the development of "
Avicennian logic" as an alternative to Aristotelian logic.
Avicenna's system of logic was responsible for the introduction of
hypothetical syllogism,
temporal modal logic, and
inductive logic. The rise of the
Asharite school, however, limited original work on
logic in Islamic philosophy, though it did continue into the 15th century and had a significant influence on European logic during the
Renaissance.
In India, innovations in the scholastic school, called
Nyaya, continued from ancient times into the early
18th century, though it didn't survive long into the
colonial period. In the 20th century, western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the
Indian tradition of logic. According to
Hermann Weyl (1929):
Occidental mathematics has in past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry.
During the 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.
Topics in logic
Syllogistic 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 wasn't 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 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's some current interest in extending term logics), regarded as made obsolete by the advent of
sentential 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.
Predicate logic
Logic as it's studied today is a very different subject to that studied before, and the principal difference is the innovation of predicate logic. Whereas Aristotelian syllogistic logic specified the forms that the relevant part of the involved judgements took, predicate logic allows sentences to be analysed into subject and argument in several different ways, thus allowing predicate logic to solve the
problem of multiple generality that had perplexed medieval logicians. With predicate logic, for the first time, logicians were able to give an account of
quantifiers general enough to express all arguments occurring in natural language.
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 Theoretical Logic by
David Hilbert and
Wilhelm Ackermann in
1928. The analytical generality of the predicate logic allowed the formalisation of mathematics, and drove the investigation of
set theory, allowed the development of
Alfred Tarski's approach to
model theory; it's no exaggeration to say that it's the foundation of modern
mathematical logic.
Frege's original system of predicate logic wasn't first-, but second-order.
Second-order logic is most prominently defended (against the criticism of
Willard Van Orman Quine and others) by
George Boolos and
Stewart Shapiro.
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, who was concerned with the
alethic modalities of necessity and possibility, which he observed to be dual in the sense of
De Morgan duality. 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.
Deduction and reasoning
The motivation for the study of logic in ancient times was clear, as we've described: it's 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 earliest use of mathematics and
geometry in relation to logic and philosophy goes back to the ancient Greeks such as
Euclid,
Plato, and
Aristotle. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.
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've 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 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 (for example,
free logics,
tense logics) as well as various extensions of
classical logic (for example,
modal logics), and non-standard semantics for such logics (for example,
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 their 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 correctly formulate an argument.
Logic and computation
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;
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.
Argumentation theory
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.
Controversies in logic
Just as we've seen there's disagreement over what logic is about, so there's disagreement about what logical truths there are.
Bivalence and the law of the excluded middle
bivalent" or "two-valued"; that is, they're most naturally understood as dividing propositions into the true and the false propositions. Systems which reject bivalence are known as non-classical logics.
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. 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.
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.
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's a constructive logic, and is hence a logic of what computers can do.
Modal logic isn't 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. On the other hand, modal logic can be used to encode non-classical logics, such as intuitionistic logic.
Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.
Implication: strict or material?
It is obvious that the notion of implication formalised in classical logic doesn't 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 doesn't 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.
Tolerating the impossible
Closely related to questions arising from the paradoxes of implication comes the radical 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's 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.
Is logic empirical?
What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticising purported principles of logic? In an influential paper entitled "Is logic empirical?" 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.
Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity. 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's 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.
Further Information
Get more info on 'Logic'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://logic.totallyexplained.com">Logic Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
