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We shall start by giving definitions that we shall adhere to in the rest of this article: 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".
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.
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, Aksapada Gautama, Farabi, and Avicenna, conceived of logic.
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. 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'.
Sentences that are derived using the system's axioms and rules of derivation are called theorems.
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.
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.