Formal Semantics and Truth Conditions

The formal structures representing the behavior of an automaton -or several automata- as such are only objects. If one wants to talk about these objects, stating e.g. that such an object has or has not a certain property then one needs a language S whose expressions E have some interpretations I mapping these expressions to certain aspects of a denoted object. In the 'light' of an interpretation $ I$ one can associate some properties of the denoted object as (denotational) meaning of the interpreted expression. This is illustrated in the figure 2.6.

Figure 2.6: Logic and Language
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In the figure one can see an expression 'The sun doesn't shine at 6.36 pm in Schoeneck Oct-27, 2009.' This is an expression of a language $ S$, in this case some variant of English. The expression consist of subexpressions which finally are composed of alphabetical signs. The overall synthesis of all these sub parts into one big expression is guided by syntactial rules representing the syntax or grammar G of the language $ S$. Purely snytactical rules are completely insensitive for information beyond the syntax. In natural languages this is not the case. Here we can find several kinds of interactions of the syntactical rules with information beyond pure syntax.

For the sake of simplicity we continue with the assumption, that we have here only some kind of a pure syntax. From natural language we know that expressions of a language are usually associated with some meaning, i.e. natural language expressions have usually some kind of an interpretation relating these expressions with some object beyond the expression. In simple cases we will have identifiable objects which function as denotational objects or as denotata. These denotational objects constitute that was usually is called the meaning of these expressions. The disciplin which is dealing with the meaning of language expressions is traditionally called semantics2.1.

The meaning of natural language expressions is usually part of our empirical world experience. Grounded in concrete experiences of the world through our body and brain we can asociate representations of the experiences with the associated expressions as a kind of representational meaning. The representational meaning then can be checked against the available empirical evidence whether the actual experience has some correspondence or similarity with the induced representational meaning. If there appears to be a 'sufficient similarity' we usually are inclined to say that the representational meaning is satisfied by the actual exprience; the representational meaning is then also judged to be true.

For scientific purposes it is often necessary to make the denoted empirical meaning of certain kinds of expressions more explicit. In this case one constructs a formal structure serving as an artificial object or as an artificial model. If the model shall be an aid to represent certain parts of the real world then the model must be sufficient similar to those properties of the real experience which are understood as necessary parts of the subject. If one calls that part of reality which shall be modeled the intended empirical domain then the formal model is called a formal domain model.

If such a formal (domain) model $ \cal{M}$ as well as the purely syntactical rules specifying a formal language $ \cal{S}$ are available then one can construct an interpretation $ \cal{I}$ mapping $ \cal{M}$ onto $ \cal{S}$ and vice versa ($ \cal{I}$ : $ \cal{M}$ $ \longleftrightarrow$ $ \cal{S}$). Thus having an interpretation $ \cal{I}$ and a certain kind of formal expression $ f \in \cal{S}$ is given then one can construct some kind of formal meaning $ I(f) = m$ with $ m \in \cal{M}$. If the constructed formal meaning $ m$ is 'indeed' an element of $ \cal{M}$ then $ M$ satisfies $ m$ which usually is described as ' $ f$ is true in $ \cal{M}$ under the interpretation $ \cal{I}$'. The constructed formal meaning $ m$ itself is a kind of a condition which has to be satisfied by some model $ \cal{M}$ that an expression $ f$ can become true. Therefore the constructed formal meaning $ m$ is a kind of truth condition which has to be met by a model $ \cal{M}$.

Gerd Doeben-Henisch 2010-03-03