By Trofimchuk E. P.
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Extra info for A. M. Samoilenkos method for the determination of the periodic solutions of quasilinear differential equations
1 Undeﬁned terms: ‘blub’, ‘glug’ and ‘to lie on’. Axioms: A1. Every blub lies on at least one glug. A2. For every glug, there are exactly two blubs which lie on it. A3. There are exactly ﬁve blubs. 1 gives an interpretation of the axiom system with blubs represented as points and glugs as lines, with the obvious interpretation of ‘to lie on’. Note that, in this interpretation, each of the axioms is a true proposition. A speciﬁc interpretation of the undeﬁned terms such that the axioms are true propositions is called a model of the axiom system.
For instance, using this predicate, we may form the proposition ‘A brick is heavier than a hamster’. The predicate ‘is heavier than’ is an example of a two-place predicate. If H denotes this predicate, then H (x, y) denotes the propositional function ‘x is heavier than y’. Two-place predicates can be quantiﬁed using the universal and existential quantiﬁers. However, two quantiﬁers are necessary to produce a proposition from a two-variable propositional function. The quantiﬁed expressions ∀x F(x, y) and ∃x F(x, y) are not propositions but propositional functions of the single variable y.
Bill has green eyes. ’ Expressing this in our propositional notation would given us an argument with premises p and q and a conclusion r . Our notation gives us no means of showing that different propositions are making statements about the same thing. Two propositions as similar as ‘Bill has green eyes’ and ‘Jeff has green eyes’ would have to be symbolized by p and q. We have as yet no means of expressing the fact that both propositions are about ‘green eyes’. Predicate Logic 29 A predicate describes a property of one or several objects or individuals.
A. M. Samoilenkos method for the determination of the periodic solutions of quasilinear differential equations by Trofimchuk E. P.