Introduction to Logic

Logic, from Classical Greek λόγος logos (the word), is the study of the principles and criteria of valid inference and demonstration.

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 scope of logic is therefore large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory.

Traditionally, logic is studied as a branch of philosophy, one part of the classical trivium, which consisted of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied as the foundation of mathematics. In 1903 Bertrand Russell and Alfred North Whitehead established logic as the cornerstone of mathematics with the publication of Principia Mathematica. The development of formal logic and its implementation in computing machinery is the foundation of computer science.

Nature of logic
The crucial concept of form is central to discussions of the nature of logic, and it complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic language is actually a species of formal logic, and is distinguished from another class 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 major 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 is not about any particular thing or property. The first rules of formal logic that have come down to us were written by Aristotle.   We will see later that in many definitions of logic, logical inference and inference with purely formal content are the same thing.  This does not render the notion of informal logic vacuous, since one may wish to investigate logic without committing to a particular formal analysis.)


 * Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.

"Formal logic" is very often used with the alternate meaning of symbolic logic whereas informal logic is any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory".

Formal Logic
While formal logic is old, dating back more than two millennia, most of symbolic logic is comparatively new, and arises with the application of insights from mathematics to problems in logic. Generally, a symbolic logic is captured by a formal system, comprising a formal language including rules for creating expressions in the language, and a set of rules of derivation. The expressions will normally be intended to represent claims that we may be interested in, and likewise the rules of derivation represent inferences; such systems usually have an intended interpretation.

A formal system has axioms. An axiom is a sentence that is always true within the system. For example, many systems have as an axiom: "If P implies Q and P is the case, then Q is the case." To go along with the axioms the system will have a special rule of derivation, called the 'rule of substitution'. It says that you can derive from any axiom a sentence that is just like it, except that other sentences have been substituted for the 'P' and the 'Q'. For example, from the axiom above, we can conclude the following: "If R&S implies that T or U, and R&S is the case; then it is the case that T or U." (This assumes that "R&S" and "T or U" are expressions in the formal system.) Most formal systems have either a rich set of rules of derivation, but few or no axioms; or a rich set of axioms but only the derivation rule of substitution.

Sentences that are derived using the system's axioms and rules of derivation are called theorems.

Consistency, soundness, and completeness
There are three valuable properties that formal systems can have:


 * Consistency, which means that none of the theorems of the system contradict


 * Soundness, which means that the system's rules of derivation will never let you infer anything false, so long as you start with only true premises. So if a system is sound (and its axioms, if any, are true), then the theorems of a sound formal system are the truths.  All of the theorems of a system that has no axioms are its truths and sometimes the truths of such a system are called 'logical truths.' (Note that if a system is not consistent, it cannot be sound. This is because a contradiction is always false, so if two theorems contradict at least one is false.)


 * Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.

Not all systems achieve all three virtues. It has been proven by Kurt Gödel that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic cannot be both consistent and complete. These are called Gödel's incompleteness theorems.

Important families of formal systems
Formal logic encompasses a wide variety of logical systems. Various systems of logic discussed below include term logic, predicate logic, propositional logic, and modal logic, and formal systems are indispensable in all branches of mathematical logic. The table of logic symbols describes various widely used notations in symbolic logic.

Rival conceptions of logic
Logic arose (see below) from a concern with correctness of argumentation. The conception of logic as the study of argument is historically fundamental, and was how the founders of distinct traditions of logic, namely Plato, Aristotle, Mozi and Aksapada Gautama, conceived of logic. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference; so for example the Stanford Encyclopedia of Philosophy says of logic that it "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" (Hofweber 2004).

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

A third view of logic arises from the idea that logic is more fundamental than reason, and so that logic is the science of states of affairs (German: Sachverhalt) in general. Barry Smith locates Franz Brentano as the source for this idea, an idea he claims reaches its fullest development in the work of Adolf Reinach (Smith 1989). This view of logic appears radically distinct from the first: on this conception logic has no essential connection with argument, and the study of fallacies and paradoxes no longer appears essential to the discipline.

Occasionally one encounters a fourth view as to what logic is about: it is a purely formal manipulation of symbols according to some prescribed rules. This conception can be criticized on the grounds that the manipulation of just any formal system is usually not regarded as logic. Such accounts normally omit an explanation of what it is about certain formal systems that makes them systems of logic.

Relation to other sciences
Logic is related to rationality and the structure of concepts, and so has a degree of overlap with psychology. Logic is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive, so the overlap is not so marked. Gottlob Frege, however, was adamant about anti-psychologism: that logic should be understood in a manner independent of the idiosyncrasies of how particular people might reason.

Deductive and inductive reasoning
Originally, logic consisted only of deductive reasoning which concerns what follows necessarily from given premises. However, inductive reasoning&mdash;the process of deriving a reliable generalization from observations&mdash;has sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive validity. An inference is deductively valid if and only if there is no possible situation in which all the premises are true and the conclusion false. 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. deductive argument follows the pattern of a general premiss to a particular one,there is a very strong relationship between the premiss and the conlusion of the argument.

History of logic
While many cultures have employed intricate systems of reasoning and math, logic as an explicit analysis of the methods of reasoning received sustained development originally only in three places: India in the 6th century BC, China in the 5th century BC, and Greece between the 4th century BC and the 1st century BC.

The formally sophisticated treatment of modern logic descends from the Greek tradition, the latter mainly being informed from the transmission of Aristotelian logic. The traditions outside Europe did not survive into the modern era: in China, the tradition of scholarly investigation into logic was repressed by the Qin dynasty following the legalist philosophy of Han Feizi; in the Islamic world the rise of the Asharite school suppressed original work on logic.

However in India, innovations in the scholastic school, called Nyaya, continued into the early 18th century. It did not 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, after it was shown that Aristotle's ideas were largely compatible with faith, a greater emphasis was placed upon Aristotle's logic. During the later period of the medieval ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.

Topics in 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.

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 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 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 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 is 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 is no exaggeration to say that it is the foundation of modern mathematical logic.

Frege's original system of predicate logic was not 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 subparts 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 axiomatisations 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 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 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 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., 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.

Philosophy of language underwent a renaissance in the 20th century because of the work of Ludwig Wittgenstein.

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 have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.

Bivalence and the law of the excluded middle
The logics discussed above are all "bivalent" or "two-valued"; that is, they are 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.

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. On the other hand, modal logic can be used to encode non-classical logics, such as intuitionistic 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. 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 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.

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 is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the striking grounds that there are in fact, true contradictions.

Is logic empirical?
What is the epistemological status of the laws of logic? What sort of arguments 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 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.