# User:JMRyan/Sandbox

## Example truth table

 $\mathrm{P}\,\!$ $\lnot \mathrm{P}\,\!$ T F F T

## Example derivation

 1 $\mathrm{P} \land \mathrm{Q} \,\!$ Premise 2 $\mathrm{P} \lor \mathrm{R} \rightarrow \mathrm{S}\,\!$ Premise 3 $\mathrm{S} \land \mathrm{Q} \rightarrow \mathrm{T}\,\!$ Premise 4 $\mathrm{P}\,\!$ 1 KE 5 $\mathrm{Q}\,\!$ 1 KE 6 $\mathrm{P} \lor \mathrm{R}\,\!$ 4 DI 7 $\mathrm{S}\,\!$ 2, 6 CE 8 $\mathrm{S} \land \mathrm{Q}\,\!$ 5, 7 KI 9 $\mathrm{T}\,\!$ 3, 8 CE

## Example subderivation

 1. $(\mathrm{P} \rightarrow \mathrm{Q}) \rightarrow\mathrm{R}\,\!$ Premise 2. $\mathrm{S} \land \mathrm{Q}\,\!$ Premise
 3. $\mathrm{P} \rightarrow \mathrm{Q}\,\!$ Assumption 4. $\mathrm{Q}\,\!$ 2 KE
 5 $\mathrm{P} \rightarrow \mathrm{Q}\,\!$ 3-4 CI 6 $\mathrm{R}\,\!$ 1, 5 CE

## 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.