Classifications for inconsistent theories.
John Grant · Notre Dame Journal of Formal Logic · 1978
In [2] N. C. A. da Costa surveys some interesting results about inconsistent formal systems.A formal system is said to be inconsistent if there is a formula φ such that both φ and~<ρ are theorems.The approach in [2] towards the study of inconsistent systems is basically syntactical.In this paper we investigate inconsistent theories from a model-theoretical point of view.However we do not analyze semanticaily the calculi presented in [2] as suggested on Page 508.Instead we define a notion of structure which allows for the possibility of built-in inconsistencies.These structures may then be models of inconsistent theories.We classify theories in 3 different ways.Intuitively, the higher a theory is in a classification, the more inconsistent it is.This way we obtain measures of inconsistency for theories. Terminology and ExamplesSince for the purposes of this paper it is convenient to deviate somewhat from the standard terminology, we explain our notations in this section.We deal with first-order languages of finite type with equality and without function symbols.A type μ = (n l9 . .., ft&) is always finite and nonempty.We use j, k, m, n for integers or possibly ω; α, β for infinite cardinals; φ, ψ for formulas (usually sentences); Γ for a set of sentences.The cardinality of a set A is denoted by \A\.We differentiate between equations and atomic formulas: an equation has the form t{ = tj while an atomic formula has the form St(t l9 . .., t n ) where S, is an Wt-ary relation symbol mentioned in μ, and the U are terms.We use the connectives ~, Λ, V, and the quantifiers 3, V.We give the following recursive definition of a formula:1) Every equation, negation of equation, atomic formula, and negation of atomic formula is a formula.2) If φ and ψ are formulas then so are φ *ψ, φyψ, ~φ(3x)φ y and (Vx)φ.3) An expression is a formula only if it follows from a finite number of applications of 1) and 2) that it is a formula.Sometimes we may write an expression where negation is applied to a formula which is neither an equation nor an atomic formula.Such an