# Logic

**Logic** (Greek *λογίζω* – I think, I reason; from *λόγος* – reason) refers the patterns in reasoning behind arguments. In philosophy, logic is a sub-branch of epistemology that deals with and attempts to guide the faculty of human reason. It is often studied alongside mathematics.

*Encyclopedia Britannica* declares:

- "Laws of thought, traditionally, the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. That is, (1) for all propositions p, it is impossible for both p and not p to be true, or symbolically ∼(p · ∼p), in which ∼ means “not” and · means “and”; (2) either p or ∼p must be true, there being no third or middle true proposition between them, or symbolically p ∨ ∼p, in which ∨ means “or”; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x."
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Aristotle was the first to formalize the practice of reasoning. In particular, Aristotle developed a formalization of the categorical syllogism, the three laws of logic, and the Square of Opposition. While modus ponens and its complement modus tollens were known to the medieval philosophers, it is not clear when these central laws of deduction were first formalized.

The next easily identifiable major development in logic comes from Boole in 1847 with the creation of Boolean algebra and Boolean logic. In the 1870s, Peirce introduces a logic of quantifiers, followed by Gottlob Frege's 1879 *Begriffsschrift*, which contains the first complete formalization of the propositional calculus. Frege's work was developed in service of his logicist project, which was shown to be inconsistent by Bertrand Russell in 1903 with what is famously called Russell's paradox. In 1913 Russell and Whitehead published *Principia Mathematica*, a landmark work that sought to define mathematics with first-order logic.

By the 1950s, with the work of many logicians including Hilbert, Emile Post, Alfred Tarski and Kurt Godel, most of the major results in first-order logic had been proved, and in the 1960s Saul Kripke added a completeness proof for modal logic.

Most logical systems are bivalent; that is, they admit only two truth-values, true or false. However, there is a fair amount of work done on trivalent systems of logic, specifically paraconsistent logic and relevance logic (a subset of paraconsistent logic). These logics were developed in response to the paradoxes that classical logic can create, namely, that anything can follow from a falsehood and the principle of explosion.

Formal logic requires, since Frege, a distinction between the object language and the metalanguage. The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with an alphabet limited to variable meanings and a symbolic system for describing the relationships between the variables.

## Contents

## Informal Logic

The art of rhetoric is sometimes called "informal logic", and the classic fallacies are often described as "logical fallacies", though, strictly speaking, most of them are simply emotionally effective ways to build an invalid argument. Generally speaking, "informal logic" consists of "common sense" and a collection of other rather loose rules that people employ while making most decisions and even in debate. It is unstructured, and depends largely on one's view of "the reasonable". The thresholds of what is "reasonable" and what is not are inexact and subject to change with the receipt of sufficient contrary evidence – and again, what constitutes "sufficiency" in this context might vary from person to person.

## Deduction and Induction

Deductive logic is characterized by certainty: in a valid argument, when the premises are true, the conclusion **must** be true. Take a classic argument of deductive logic:

**P1:**All men are mortal**P2:**Socrates is a man.**C:**Socrates is mortal.

The conclusion is "contained" in the premises. In this sense the conclusion can be said to "follow" from the premises. The conclusion, "Socrates is mortal", is also less informative than the premises, which imply not just that Socrates is mortal but that all men are also mortal. As a result, deductive logic is not thought to add to knowledge, merely to clarify it.

Inductive logic is ampliative, but is famously less certain. In a strong inductive argument, even when the premises are true, it is still possible for the conclusion to be false. An example of an inductive argument is:

**P1:**The first student is wearing red**P2:**The second student is wearing red**P(n):**The*n*th student is wearing red**Conclusion:***All*students are wearing red

Given the amount of evidence, it is reasonable to inductively conclude that all students are wearing red; however, one's senses could fail or a student could be hiding who is wearing blue. Given this, the conclusion is never certainly true and can only be highly probably true. The event of moving from "*n* examples are this way" to a universal statement that "*all* examples are this way" is commonly known as the problem of induction.

## Uses of logic in other disciplines

Logic is a necessary discipline in philosophy, because it deals with how we study and interact with propositions. Logic and mathematics are also closely connected, and much of mathematics can be reduced to first-order logic, though Gödel's incompleteness theorem shows that not all of mathematics can be so reduced.

- In computer science, logic dictates how a machine will follow a set of instructions, including how to test its "world," evaluate it, and act according to that evaluation, except that instead of establishing propositions, a computer following a program is usually choosing between and among different commands to execute.
- Logic is also used extensively in theology, and especially the study of the Bible
- Logic, and especially formal logic, informs the discipline of critical thinking – which, by no coincidence, takes its name from the Greek word for a judge. Indeed, judges and juries in courts of law must apply logic to evaluate the arguments and evidence and arrive at their decisions.

## Propositional logic

Propositional logic is the study of logical propositions. It is sometimes called *zeroth*-order logic because it does not deal with subjects or predicates but is specifically the study of the truth values of logical propositions and the logical connectives used to make compound propositions. Lewis Carroll wrote a book on it for young students.

## Recommended reading

- Aristotle, "Organon"
- Gottlob Frege, "Begriffsschrift" and "The Foundations of Arithmetic"
- Bertrand Russell and Albert N Whitehead, "Principia Mathematica"
- Bas van Fraassen and J C Beall, "Possibility and Paradox"
- Geoffrey Hunter, "Metalogic"
- Patrick Hurley, "A Concise Introduction to Logic, 12th Edition"