Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus

The goal of the basic concepts introduced so far was to link the syntactical and the semantic levels of a formal language. The following figure demonstrates these two levels:


The advantage of the derivability property "$\vdash$" is that it is defined in a way capable for automation. Even a machine, (we have called the proving machine PM) would be able to derive one string from another formally (i.e. in a formal language $L$), according to its syntax and the defined rules of inference.

The disadvantage of the derivability property "$\vdash$" is that it only operates on strings without any meaning. But we want to construct a logical calculus to be able to derive true statements about a universe $U$, we call the domain of discourse. This is where interpretation $I(U,L)$ comes into play with its valuation function $[[]]_I$.

Ideally, our logical calculus should fulfill some desirable properties:

Now, we will introduce these desirable properties of a logical calculus as formal definitions.

  1. Definition: Soundness and Completeness of a Logical Calculus
  2. Definition: Consistency and Negation-Completeness of a Logical Calculus
  3. Definition: Negation of a String

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000