Logic

Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow one to distinguish good from bad arguments.

Nature of logic

Due to its fundamental role in philosophy, the nature of logic has been the object of intense disputation, and it is not possible to give a clear delineation of the bounds of logic in terms acceptable to all rival viewpoints. Nonetheless, the study of logic has, despite this fundamental controversy, been very coherent and technically grounded. Here we characterise logic, firstly by introducing the fundamental ideas about form, then outlining in broad terms some of the most influential rival conceptions of the subject, giving a brief overview of its history and then give an account of its relationship to other science, and then go on to provide an exposition of some essential concepts.

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Informal, formal and symbolic logic

The crucial concept of form is central to discussions of the nature of logic, and it complicates matters 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:

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• Informal logic is the study of natural language arguments and fallacies
• An inference possesses a purely formal structure 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 structure 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.
• Formal logic is the study of logical inference whose validity derives from its explicitly formal structure.
• Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.

The ambiguity is that "formal logic" is very often used with the alternate meaning of symbolic logic as we have defined it, with informal logic meaning 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".

Related Topics:
Formal language - Formal theory

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While on the above analysis, formal logic is old, dating back more than two millenia, symbolic logic is comparatively new, and arises with the application of insights from mathematical abstraction to problems in logic. Certain conventions have become prevalent in the symbolic analysis of logic: the logic is captured by a formal systems, comprising a formal language, which describes a set of formulas, a set of rules of derivation. The formulas 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.

Related Topics:
Formal system - Formal language - Intended interpretation

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Within this formal system, the rules of derivation and potential axioms then specify a set of theorems, which are formulas that are derivable using the rules of derivation. The most essential property of a logical formal system is soundness, which is the property that under interpretation, all of the rules of derivation are valid inferences. The theorems of a sound formal system are then truths. A minimal condition which a sound system should satisfy is consistency, meaning that no theorem contradicts another. Also of the essence completeness, meaning that everything true is also provable. However, when the language of logic reaches a certain degree of expressiveness (say, second-order logic), completeness becomes impossible to achieve in principle.

Related Topics:
Theorem - Soundness - Truth - Consistency

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In the case of formal logical systems, the theorems are often interpretable as expressing logical truths (tautologies), and in this way can such systems be said to capture at least a part of logical truth and inference.

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Formal logic encompasses a wide variety of logical systems. Various systems of logic we will discuss later can be captured in this framework, such as term logic, predicate logic and modal logic, and formal systems are indispensable in all branches of mathematical logic.

Related Topics:
Term logic - Predicate logic - Modal logic - Mathematical logic

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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 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).

Related Topics:
Aristotle - Mozi - Aksapada Gautama - Stanford Encyclopedia of Philosophy

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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 (Gedanke) is substituted for the judgement (Urteil). On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts.

Related Topics:
Immanuel Kant - Gottlob Frege

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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, 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.

Related Topics:
Franz Brentano - Adolf Reinach

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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 an account omits an explanation of what it is about certain formal systems that makes them systems of logic.

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History of logic

While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally only in three places: China in the 5th century BCE, and India and Greece between the 2nd century BCE and the 1st century BCE.

Related Topics:
China - 5th century BCE - India - Greece - 2nd century BCE - 1st century BCE

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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) but comes not wholly through Europe, but instead comes from the transmission of Aristotelian 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, and in India, though innovation in the scholastic school continued into the early 18th century, it did not survive long into the colonial period.

Related Topics:
Boolean logic - Aristotelian logic - Islamic philosophers - Medieval logic - Qin dynasty - Han Feizi - Asharite - 18th century - Colonial period

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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, that is, it describes how reasoning ought to take place, however, whereas psychology is descriptive, so the overlap is not so marked. Gottlob Frege, for example, was adamant about anti-psychologism: that logic should be understood in a manner independent of the idiosyncrasies of how particular people might reason.

Related Topics:
Psychology - Gottlob Frege - Anti-psychologism

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Deductive and inductive reasoning

Originally, logic consisted only of deductive reasoning which concerns what follows universally from given premises. However, it is important to note that inductive reasoning—the study 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. 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 our discussion of logic deals only with deductive logic.

Related Topics:
Deductive reasoning - Inductive reasoning - Formal logic - Semantics - Mathematical model

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