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Logic, from Classical Greek λόγος (logos), originally meaning the word, but also referring to speech or reason is the science that evaluates reasoning within arguments.

Nature of logicEdit

Logic is generally understood and accepted as a set of rules that tell us when an argument's premises support their conclusion. An understanding of just what logic is, can be enhanced by delineating it from what it is not:

  • Logic is not the 'groundness of being' - that's metaphysics.
  • Logic is not a set of laws that governs the universe - that's physics.
  • Logic is not an immaterial "entity" that transcends reality - that's speculative theology.
  • Logic is not a set of laws that governs human behavior - that's psychology.
  • Logic is not a method for 'studying the world' - that's science.
  • Logic is not the method for assessing axioms - that's a matter of pure reason.
  • Logic is not a way of evaluating 'truth' - that's philosophy.

Arguments Edit

An argument is made up of groups of statements we call propositions - and every proposition contains a truth value. In the case of classic logic, this truth value is either "true" or false" - a rule we can call "bivalence". Formally: for any proposition P, P is either true or false. We use the term "proposition" instead of "sentence" because a proposition is more than just a sentence, it is a declarative sentence - asserting a specific and coherent claim, that is either true or false.

There are two types of propositions within an argument. The first, the premise, makes a commitment to truth, and is used as evidence to support the second type of proposition, the conclusion, which is the claim the argument is supposed to prove. An argument must at least imply one of each.

Here is an example:

  • Premise 1: Either that's a gun in your pocket, or you're glad to see me.
  • Premise 2: Uh... that's not a gun in your pocket.
  • Conclusion: So you must be glad to see me.

The study of logic, therefore, is the effort to determine the conditions under which one is justified in passing from the premises to the conclusion that logically must follow them. When an argument has this sort of justification, it is a valid argument. A valid argument with true premises is a sound argument, and guarantees the truth of its conclusion.

Informal, formal, and symbolic logicEdit

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. We shall start by giving definitions that we shall adhere to in the rest of this article:

  • 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. We will see later that on 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.)


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.

For example, consider a very simple formal system that has just the symbols "p", "q", and "and" in its language. Its only rules for creating expressions are (1) "'p and q' and 'q and p' are expressions" and (2) Any expression compounded with another by 'and" is also an expression". Its only rule of derivation is "from any expression of the form 'p and q', you may conclude 'p'". The intended interpretation of the "p" and "q" is that they stand for any sentence. The intended interpretation of 'and' is expressed by specifying when sentences that contain 'and' are true. Most systems would interpret 'and' like this: sentences containing 'and' are true only when the expressions on either side of it (both of them) are true.

A formal system can also have axioms. An axiom is a sentence that counts as always true within the system. For example, many systems have as an axiom the sentence "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 completenessEdit

There are three valuable properties that formal systems can have:

  • Consistency, which means that none of the theorems of the system contradict each other.
  • 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. This is called Gödel's Incompleteness Theorem.

Important families of formal systemsEdit

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 logicEdit

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).




Relation to other sciencesEdit


Deductive and inductive reasoningEdit


History of logicEdit


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 apparently descends from the Greek tradition, although it is suggested that the pioneers of Boolean logic were likely aware of Indian logic (Ganeri 2001). The Greek tradition itself comes mainly from the transmission of Aristotelian logic, which probably developed independently of Indian logic, and commentary upon it by Islamic philosophers to Medieval logicians. 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.

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 logicEdit


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 logicEdit





Predicate logicEdit


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 parts 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.



Modal logicEdit





Mathematical logicEdit







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 logicEdit




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

Logic and computationEdit




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 artifical intelligence. The ACM Computing Classification System in particular regards:


Argumentation TheoryEdit


Controversies in logicEdit


Bivalence and the law of the excluded middleEdit


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.





Implication: strict or material?Edit





Tolerating the impossibleEdit



Is logic empirical?Edit




  • Banda, Obdulio (1944), "Hacia el análisis ontológico-categorial de lo lógico", IIPCIAL, Fondo Editorial (2008).ISBN 978-9972-9982-2-5
  • 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.
  • Hodges, W. (2001), Logic. An introduction to Elementary Logic, Penguin Books.
  • Hughes, R.I.G. (ed., 1993), A Philosophical Companion to First-Order Logic, Hackett Publishing.
  • Priest, Graham (2004), "Dialetheism", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
  • Putnam, H. (1969), "Is Logic Empirical?", Boston Studies in the Philosophy of Science' 5.
  • Smith, B. (1989), "Logic and the Sachverhalt", The Monist 72(1), 52–69.

See alsoEdit

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External linksEdit


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Smallwikipedialogo.png This page uses content from the English-language version of Wikipedia. The original article was at Logic. The list of authors can be seen in the page history. As with Philosophy Wiki, the text of Wikipedia is available under the GNU Free Documentation License.
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