It is possible to construct interpreted logical languages in which there are no basic individuals (known as ur-individuals) and thus no first-order quantifiers.A logical formula whose variables have been replaced by nonlogical concepts (meanings or referents) is called an interpreted proposition, or simply an interpretation.

Number game: Problems of logical inference Many challenging questions do not involve numerical or geometrical considerations but call for deductive inferences based chiefly on logical relationships.Logic in a narrow sense is equivalent to deductive logic.Until then, most logical inferences were expressed by means of natural language supplemented with a smattering of variables and, in some cases, by traditional mathematical concepts.

One must also consider their semantics, or the relations between sentences in the logical system and the structures (usually referred to as models) in which the sentences are true.An inference rule is said to be valid, or deductively valid, if it is necessarily truth-preserving.A logical system is essentially a way of mechanically listing all the logical truths of some part of logic by means of the application of recursive rulesi.