Map of Some Logical SystemsPeter Suber, Philosophy Department, Earlham College

D. Higher order logics

—not (generally) consistent

C. First order polyadic predicate logic

with or without identity

—consistent, semantically complete

—not decidable, syntactically complete, negation complete

B. First order monadic predicate logic

—consistent, semantically complete, decidable

—not syntactically complete, negation complete

A. Truth-functional propositional logic

—consistent, semantically complete, syntactically complete, decidable

—not negation complete

The purpose of this map is to show that these important systems of logic are related by subsumption, and that to "climb" to the next higher, more encompassing system, one must sacrifice some important metalogical properties.

In climbing from A to B, one gives up syntactic completeness and truth-functionality. In climbing from B to C, one gives up decidability; note that one must climb at least this high to have a "respectable" system of arithmetic. In climbing from C to D, one (generally) gives up consistency.

This file is an electronic hand-out for the course, Logical Systems.

Peter Suber,
Department of Philosophy,
Earlham College, Richmond, Indiana, 47374, U.S.A.

*peters@earlham.edu*. Copyright © 1997, Peter Suber.