logical notation symbols designed to achieve unambiguous formulation of principles and inferences in deductive logic. Such notations involve some regimentation of words, word order, etc., of natural language. Some schematization was attempted even in ancient times by Aristotle, the Megarians, the Stoics, Boethius, and the medievals. But Leibniz’s vision of a universal logical language began to be realized only in the past 150 years.
The notation is not yet standardized, but the following varieties of logical operators in propositional and predicate calculus may be noted. Given that ‘p’, ‘q’, ‘r’, etc., are propositional variables, or propositions, we find, in the contexts of their application, the following variety of operators (called truth-functional connectives). Negation: ‘-p’, ‘Ýp’, ‘p- ‘, ‘p’ ‘. Conjunction: ‘p • q’, ‘p & q’, ‘p 8 q’. Weak or inclusive disjunction: ‘p 7 q’. Strong or exclusive disjunction: ‘p V q’, ‘p ! q’, ‘p W q’. Material conditional (sometimes called material implication): ‘p / q’, ‘p P q’. Material biconditional (sometimes called material equivalence): ‘p S q’, ‘p Q q’. And, given that ‘x’, ‘y’, ‘z’, etc., are individual variables and ‘F’, ‘G’, ‘H’, etc., are predicate letters, we find in the predicate calculus two quantifiers, a universal and an existential quantifier: Universal quantification: ‘(x)Fx’, ‘(Ex)Fx’, ‘8xFx’. Existential quantification: ‘(Ex)Fx’, ‘(Dx)Fx’, ‘7xFx’.
The formation principle in all the schemata involving dyadic or binary operators (connectives) is that the logical operator is placed between the propositional variables (or propositional constants) connected by it. But there exists a notation, the so-called Polish notation, based on the formation rule stipulating that all operators, and not only negation and quantifiers, be placed in front of the schemata over which they are ranging. The following representations are the result of application of that rule: Negation: ‘Np’. Conjunction: ‘Kpq’. Weak or inclusive disjunction: ‘Apq’. Strong or exclusive disjunction: ‘Jpq’. Conditional: ‘Cpq’. Biconditional: ‘Epq’. Sheffer stroke: ‘Dpq’. Universal quantification: ‘PxFx’. Existential quantifications: ‘9xFx’. Remembering that ‘K’, ‘A’, ‘J’, ‘C’, ‘E’, and ‘D’ are dyadic functors, we expect them to be followed by two propositional signs, each of which may itself be simple or compound, but no parentheses are needed to prevent ambiguity. Moreover, this notation makes it very perspicuous as to what kind of proposition a given compound proposition is: all we need to do is to look at the leftmost operator. To illustrate, ‘p7 (q & r) is a disjunction of ‘p’ with the conjunction ‘Kqr’, i.e., ‘ApKqr’, while ‘(p 7 q) & r’ is a conjunction of a disjunction ‘Apq’ with ‘r’, i.e., ‘KApqr’. ‘- p P q’ is written as ‘CNpq’, i.e., ‘if Np, then q’, while negation of the whole conditional, ‘-(p P q)’, becomes ‘NCpq’. A logical thesis such as ‘((p & q) P r) P ((s P p) P (s & q) P r))’ is written concisely as ‘CCKpqrCCspCKsqr’. The general proposition ‘(Ex) (Fx P Gx)’ is written as ‘PxCFxGx’, while a truth-function of quantified propositions ‘(Ex)Fx P (Dy)Gy’ is written as ‘CPxFx9yGy’. An equivalence such as ‘(Ex) Fx Q – (Dx) – Fx’ becomes ‘EPxFxN9xNFx’, etc. Dot notation is way of using dots to construct well-formed formulas that is more thrifty with punctuation marks than the use of parentheses with their progressive strengths of scope. But dot notation is less thrifty than the parenthesis-free Polish notation, which secures well-formed expressions entirely on the basis of the order of logical operators relative to truth-functional compounds. Various dot notations have been devised. The convention most commonly adopted is that punctuation dots always operate away from the connective symbol that they flank. It is best to explain dot punctuation by examples: (1) ‘p 7 (q – r)’ becomes ‘p 7 .q P – r’; (2) ‘(p 7 q) P – r’ becomes ‘p 7 q. P – r’; (3) ‘(p P (q Q r)) 7 (p 7 r)’ becomes ‘p P. q Q r: 7. p 7r’; (4) ‘(- pQq)•(rPs)’ becomes ‘-p Q . r Q s’. Note that here the dot is used as conjunction dot and is not flanked by punctuation dots, although in some contexts additional punctuation dots may have to be added, e.g., ‘p.((. r) P s), which is rewritten as ‘p : q.r. P s’. The scope of a group of n dots extends to the group of n or more dots. (5) ‘- p Q (q.(r P s))’ becomes ‘- p. Q : q.r P s’; (6) ‘- p Q ((. r) P s)’ becomes ‘~p. Q : q.r. P s’; (7) ‘(- p Q (. r)) P s’ becomes ‘- p Q. q.r: P s’.
The notation for modal propositions made popular by C. I. Lewis consisted of the use of ‘B’ to express the idea of possibility, in terms of which other alethic modal notions were defined. Thus, starting with ‘B p’ for ‘It is possible that p’ we get ‘- B p’ for ‘It is not possible that p’ (i.e., ‘It is impossible that p’), ‘- B – p’ for ‘It is not possible that not p’ (i.e., ‘It is necessary that p’), and ‘B – p’ for ‘It is possible that not p’ (i.e., ‘It is contingent that p’ in the sense of ‘It is not necessary that p’, i.e., ‘It is possible that not p’). Given this primitive or undefined notion of possibility, Lewis proceeded to introduce the notion of strict implication, represented by ‘ ‘ and defined as follows: ‘p .% . – B (p.-q)’. More recent tradition finds it convenient to use ‘A’, either as a defined or as a primitive symbol of necessity. In the parenthesis-free Polish notation the letter ‘M’ is usually added as the sign of possibility and sometimes the letter ‘L’ is used as the sign of necessity. No inconvenience results from adopting these letters, as long as they do not coincide with any of the existing truthfunctional operators ‘N’, ‘K’, ‘A’, ‘J’, ‘C’, ‘E’, ‘D’. Thus we can express symbolically the sentences ‘If p is necessary, then p is possible’ as ‘CNMNpMp’ or as ‘CLpMp’; ‘It is necessary that whatever is F is G’ as ‘NMNPxCFxGx’ or as ‘LPxCFxGx’; and ‘Whatever is F is necessarily G’ as ‘PxCFxNMNGx’ or as PxCFxLGx; etc.
See also IMPLICATION, MODAL LOGIC, WELL -FORMED FORMULA , Appendix of Special Symbols. I.Bo.