We will not try to explain fully what logical form is. Both natural languages such as English and logical languages such as exhibit logical form. A primary purpose of a logical language is to make the logical form explicit by having it correspond directly with the grammar. In the context of , a logical form is indicated by a metalogical expression containing no sentence letters but containing metalogical variables (in our convention, Greek letters). A single formula can have several logical forms of varying degrees of explicitness or granularity. For example, the formula has the following three logical forms:
Obviously, the first of these is the most explicit or fine-grained.
We say that a formula is an instance of a logical form. For example, the formula has, among many others, the following instances.
The formal semantics for a formal language such as goes in two parts.
Rules for specifying an interpretation. An interpretation assigns semantic values to the non-logical symbols of a formal syntax. The semantics for a formal language will specify what range of vaules can be assigned to which class of non-logical symbols. has only one class of non-logical symbols, so the rule here is particularly simple. An interpretation for a sentential language is a valuation, namely an assignment of truth values to sentence letters. In predicate logic, we will encounter interpretations that include other elements in addition to a valuation.
Rules for assigning semantic values to larger expressions of the language. For sentential logic, these rules assign a truth value to larger formulae based on truth values assigned to smaller formulae.For more complex syntaxes (such as for predicate logic), values are assigned in a more complex fashion.
An extended valuation assigns truth values to the molecular formulae of (or similar sentential language) based on a valuation. A valuation for sentence letters is extended by a set of rules to cover all formulae.
