# User:JMRyan/Sandbox

## Example truth table

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

## Example derivation

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

## Example subderivation

 1. ${\displaystyle (\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow \mathrm {R} \,\!}$ Premise 2. ${\displaystyle \mathrm {S} \land \mathrm {Q} \,\!}$ Premise
 3. ${\displaystyle \mathrm {P} \rightarrow \mathrm {Q} \,\!}$ Assumption 4. ${\displaystyle \mathrm {Q} \,\!}$ 2 KE
 5 ${\displaystyle \mathrm {P} \rightarrow \mathrm {Q} \,\!}$ 3-4 CI 6 ${\displaystyle \mathrm {R} \,\!}$ 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 ${\displaystyle {\mathcal {L_{S}}}\,\!}$ 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 ${\displaystyle {\mathcal {L_{S}}}\,\!}$, 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 ${\displaystyle ((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\rightarrow \mathrm {R_{0}^{0}} )\,\!}$ has the following three logical forms:

${\displaystyle ((\phi \land \psi )\rightarrow \chi )\,\!}$
${\displaystyle (\phi \rightarrow \psi )\,\!}$
${\displaystyle \phi \,\!}$

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

${\displaystyle (\mathrm {P_{0}^{0}} \rightarrow \mathrm {Q_{0}^{0}} )\,\!}$
${\displaystyle ((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\rightarrow \mathrm {R_{0}^{0}} )\,\!}$
${\displaystyle ((\mathrm {P_{0}^{0}} \land \mathrm {Q_{0}^{0}} )\rightarrow (\mathrm {Q_{0}^{0}} \lor \mathrm {R_{0}^{0}} ))\,\!}$

## Formal semantics

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