positive logic[′päz·əd·iv ′läj·ik]
logic in which the only arguments considered acceptable are those not connected with refutations, that is, with proofs of the falseness of propositions. Since the expression “A is false” is merely a variant of the expression ”not-A,” positive logic rejects any methods of introducing negation that involve methods of indirect proof, including proof by contradiction. It also rejects explicit definitions of negation of the type ⌝ A = dfA ⊃ f, where ⌝ is the negation symbol, 3 indicates implication, and f is a propositional variable or some “admissible” absurd assertion. Thus, positive logic may be said to be a logic without negation.
The logical laws corresponding to correct arguments in positive logic (or the rules codifying the methods of such arguments) are described and cataloged in suitable logical calculi. The most important of these logical calculi are the positive implicative propositional calculus with the single logical operation of implication and the complete positive propositional calculus with conjunction, disjunction, implication, and equivalence.
The positive implicative propositional calculus is given by means of two axiom schemes:
(1) A ⊃ (B ⊃ A)
(2) (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C))
and the rules of modus ponens. The complete positive propositional calculus is given by adding to schemes (1) and (2) the following:
(3) (A & B) ⊃ A
(4) (A & B) ⊃ B
(5) A ⊃ (B ⊃ (A & B))
(6) (A ⊃ C) ⊃ ((8 ⊃ C) ⊃ ((A V B) ⊃ C))
(7) A ⊃ (A V B)
(8) B ⊃ (A V B)
An equivalence is defined to be an abbreviation for the expression (A ⊃ B) & (B ⊃ A). More powerful logical calculi are obtained from the calculi of positive logic by successive nonconservative extensions of their systems of axioms or rules of derivation. Thus, attaching the axiom scheme
(9) (A ⊃ B) ⊃ ((A ⊃ ⌝ B) ⌝ A)
or the corresponding rule of reductio ad absurdum to (1) and (2) gives Kolmogorov’s minimal logic (1925). The analogous addition to the complete positive propositional calculus gives Johansson’s minimal logic (1936). By appending to Johansson’s minimal logic the scheme
(10) ⌝ A ⊃ (A ⊃ B)
(a contradiction will entail an arbitrary assertion) and the scheme
(11) ⊃ A V A
(the law of the excluded middle), we obtain the intuitionist propositional logic and classical propositional logic, respectively.
Since all laws of positive logic are valid (provable) in intuitionist and classical logic—the converse naturally is untrue—positive calculi usually are considered subsystems; generally they are considered partial systems. It is significant, however, that positive calculi, taken “in and of themselves,” and “the same” calculi “within” a more powerful logic are calculi with a different semantics for the logical connectives (operations). The semantics for the former is determined only by the calculus’ own axioms or rules for the use of connectives, whereas the semantics for the latter is inherited from the more powerful logic.
REFERENCESChurch, A. Vvedenie ν matematicheskuiu logiku, vol. 1, subsec. 26. Moscow, 1960. (Translated from English.)
Rasiowa, H. and R. Sikorski. Matematika metamatematiki, ch. 11, subsecs. 1–6. Moscow, 1972. (Translated from English.)
M. M. NOVOSELOV