Informal logic in the context of "J. Anthony Blair"

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as ${\displaystyle \exists x(M(x)\land B(x))}$. The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.

Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic.

Ralph Henry Johnson (born 1940) is a Canadian-American philosopher, born in Detroit, Michigan. Johnson has been credited as one of the founding members of the informal logic movement in North America, along with J. Anthony Blair who co-published one of the movement's most influential texts, Logical Self-Defense, with Johnson. Alongside its founder, Blair, Johnson co-directed the Centre for Research in Reasoning, Argumentation, and Rhetoric at the University of Windsor. As Johnson and Blair write in the preface to the newest edition of Logical Self-Defense on the influential nature of the text:

A co-premise is a premise in reasoning and informal logic which is not the main supporting reason for a contention or a lemma, but is logically necessary to ensure the validity of an argument. One premise by itself, or a group of co-premises can form a reason.

The Centre for Research in Reasoning, Argumentation, and Rhetoric (CRRAR) is an interdisciplinary research group within the University of Windsor, Canada, which supports research in the fields of argumentation, informal logic, and rhetoric. Notable members include the widely published argumentation theorist Douglas N. Walton, and early founders of the informal logic field Ralph Johnson and J. Anthony Blair.

Members of the centre collaborate with local and international projects related to argumentation and informal logic, including the Ontario Society for the Study of Argumentation (OSSA), the International Society for the Study of Argumentation (ISSA), and the Association for Informal Logic and Critical Thinking (AILACT).