User:JMRyan/Sandbox

Logical form
We will not try to explain fully what logical form is. Both natural languages such as English and logical languages such as $$\mathcal{L_S}\,\!$$ 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 $$\mathcal{L_S}\,\!$$, 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 $$((\mathrm{P^0_0} \land \mathrm{Q^0_0}) \rightarrow \mathrm{R^0_0})\,\!$$ has the following three logical forms:


 * $$((\phi \land \psi) \rightarrow \chi)\,\!$$
 * $$(\phi \rightarrow \psi)\,\!$$
 * $$\phi\,\!$$

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 $$(\phi \rightarrow \psi)\,\!$$ has, among many others, the following instances.


 * $$(\mathrm{P^0_0} \rightarrow \mathrm{Q^0_0})\,\!$$
 * $$((\mathrm{P^0_0} \land \mathrm{Q^0_0}) \rightarrow \mathrm{R^0_0})\,\!$$
 * $$((\mathrm{P^0_0} \land \mathrm{Q^0_0}) \rightarrow (\mathrm{Q^0_0} \lor \mathrm{R^0_0}))\,\!$$

Formal semantics
The formal semantics for a formal language such as $$\mathcal{L_S}\,\!$$ 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.  $$\mathcal{L_S}\,\!$$ 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 $$\mathcal{L_S}\,\!$$ (or similar sentential language) based on a valuation. A valuation for sentence letters is extended by a set of rules to cover all formulae.