On Assertion and Negations in Logics and Mathematics
Two features of logic are discussed: 1) the details of asserting a proposition, that is, assigning to it the value ‘true ‘ or ‘untrue’; 2) the extension of negation to cases when a sub-proposition is negated. Various consequences follow for mathematics as well as for logic: we examine the formulation of several paradoxes, and the use of indirect proof-methods when the theorem is asserted.```
In this article the word ‘logic’ or phrase ‘a logic’ is confined to some systematic account of methods of forming propositions, making deductions with and from them, and working with their truth-values. When this logic is axiomatised, uses notations, admits paradoxes, and/or links closely to some branches of mathematics, it is often called a ‘formal’ or symbolic’ logic. We treat almost exclusively two-valued (or ‘bivalent’) formal logic, called ‘L2’, where the truth-values are ‘true’ and ‘untrue’. It divides into the propositional calculus, in which a proposition R is an indivisible “atom”, related to other propositions by logical connectives; and the functional calculus, where a proposition is split to reveal its propositional functions of one or of several variables (of individuals) x, y, …. Existential and universal quantification (‘there exists’, and ‘for all’ or ‘for every’) can apply to propositions, functions and individuals.
The first Part of this article deals with two deviations from normal practice. One concerns the assertion of a proposition, that is, the assignment to it of one of those two truth-values; while commonly done, the details of assertion are not usually pursued. The other concerns negation: normally TUL uses only ‘external’ negation, where an entire proposition is negated; but we take in also an important kind of ‘internal’ negation when a sub-proposition is negated.
The second Part of this article treats three contexts important in logic and mathematics where assertion and negations play significant roles: the propositional paradoxes, such as the liar paradox; indirect proof-methods, especially that by contradiction; and the distinction between implication and inference. Several features of our study contribute to an attempt in a sequel paper to characterise logical knowledge, and to distinguish it especially from mathematical knowledge.
All propositions are in indicative mood, and in either active or passive voice; we do not handle questions or commands, which have their own logical features. We do not treat more informal uses of a logic, such as logistics or the logic of the situation; or theories that have been called ‘inductive logics’, which belong to the philosophy of science and probability theory. However, they are all parts of ‘logical knowledge’, which refers to the totality of logics in general.