Formal Logic/Sentential Logic/Formal Semantics

= Formal Semantics =

English syntax for 'Dogs bark' specifies that it consists of a plural noun followed by an intransitive verb. English semantics for 'Dogs bark' specify its meaning, namely that dogs bark.

In The Sentential Language, we gave an informal description of $$\mathcal{L_S}\,\!$$. We also gave a Formal Syntax. However, at this point our language is just a toy, a collection of symbols we can string together like beads on a necklace. We do have rules for how those symbols are to be ordered. But at this point those might as well be aesthetic rules. The difference between well-formed formulae and ill-formed expressions is not yet any more significant than the difference between pretty and ugly necklaces. In order for our language to have any meaning, to be usable in saying things, we need a formal semantics.

Any given formal language can be paired with any of a number of competing semantic rule sets. The semantics we define here is the usual one for modern logic. However, alternative semantic rule-sets have been proposed. Alternative semantic rule-sets of $$\mathcal{L_S}\,\!$$ have included (but are certainly not limited to) intuitionistic logics, relevance logics, non-monotonic logics, and multi-valued logics.

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 values 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.

Valuations
We can give a (partial) valuation $$\mathfrak{v}\,\!$$ as:


 * $$\mathrm{P_0}\ :\ \mbox{True}\,\!$$
 * $$\mathrm{P_1}\ :\ \mbox{False}\,\!$$
 * $$\mathrm{P_2}\ :\ \mbox{False}\,\!$$
 * $$\mathrm{P_3}\ :\ \mbox{False}\,\!$$

(Remember that we are abbreviating our sentence letters by omitting superscripts.)

Usually, we are only interested in the truth values of a few sentence letters. The truth values assigned to other sentence letters can be random.

Given this valuation, we say:


 * $$\mathfrak{v}[\mathrm{P_0}]\ =\ \mbox{True}\,\!$$
 * $$\mathfrak{v}[\mathrm{P_1}]\ =\ \mbox{False}\,\!$$
 * $$\mathfrak{v}[\mathrm{P_2}]\ =\ \mbox{False}\,\!$$
 * $$\mathfrak{v}[\mathrm{P_3}]\ =\ \mbox{False}\,\!$$

Indeed, we can define a valuation as a function taking sentence letters as its arguments and truth values as its values (hence the name 'truth value'). Note that $$\mathcal{L_S}\,\!$$ does not have a fixed interpretation or valuation for sentence letters. Rather, we specify interpretations for temporary use.

Extended valuations
An extended interpretation generates the truth values of longer sentences given an interpretation. For sentential logic, an interpretation is a valuation, so an extended interpretation is an extended valuation. We define an extension $$\overline{\mathfrak{v}}\,\!$$ of valuation $$\mathfrak{v}\,\!$$ as follows.

For all sentence letters $$\varphi$$ and $$\psi$$ from $$\mathcal{L_S}:$$


 * $$\mbox{i.} \quad \overline{\mathfrak{v}}[\varphi] = \mathfrak{v}[\varphi].\,\!$$


 * $$\mbox{ii.} \quad \overline{\mathfrak{v}}[\lnot\varphi]\ =\ \begin{cases}\mbox{True} & \mbox{if}\ \overline{\mathfrak{v}}[\varphi] = \mbox{False}; \\ \mbox{False} & \mbox{otherwise (i.e., if}\ \overline{\mathfrak{v}}[\varphi] = \mbox{True).}\end{cases}\,\!$$


 * $$\mbox{iii.} \quad \overline{\mathfrak{v}}[(\varphi \land \psi)]\ =\ \begin{cases}\mbox{True} & \mbox{if}\ \overline{\mathfrak{v}}[\varphi] = \overline{\mathfrak{v}}[\psi] = \mbox{True}; \\ \mbox{False} & \mbox{otherwise}.\end{cases}\,\!$$


 * $$\mbox{iv.} \quad \overline{\mathfrak{v}}[(\varphi \lor \psi)]\ =\ \begin{cases}\mbox{True} & \mbox{if}\ \overline{\mathfrak{v}}[\varphi] = \mbox{True}\ \mbox{or}\ \overline{\mathfrak{v}}[\psi] = \mbox{True}\ \mbox{(or both)}; \\ \mbox{False} & \mbox{otherwise}.\end{cases}\,\!$$


 * $$\mbox{v.} \quad \overline{\mathfrak{v}}[(\varphi \rightarrow \psi)]\ =\ \begin{cases}\mbox{True} & \mbox{if}\ \overline{\mathfrak{v}}[\varphi] = \mbox{False}\ \ \mbox{or}\ \ \overline{\mathfrak{v}}[\psi] = \mbox{True}\ \ \mbox{(or both)}; \\ \mbox{False} & \mbox{otherwise.}\end{cases}\,\!$$


 * $$\mbox{vi.} \quad \overline{\mathfrak{v}}[(\varphi \leftrightarrow \psi)]\ =\ \begin{cases} \mbox{True} & \mbox{if}\ \overline{\mathfrak{v}}[\varphi] = \overline{\mathfrak{v}}[\psi]; \\ \mbox{False} & \mbox{otherwise}.\end{cases}\,\!$$

Example
We will determine the truth value of this example sentence given two valuations.
 * $$(1) \quad \mathrm{P} \land \mathrm{Q} \rightarrow \lnot( \mathrm{Q} \lor \mathrm{R})\,\!$$

First, consider the following valuation:
 * $$\mathrm{P}\ :\ \mbox{True}\,\!$$
 * $$\mathrm{Q}\ :\ \mbox{True}\,\!$$
 * $$\mathrm{R}\ :\ \mbox{False}\,\!$$

(2) By clause (i):
 * $$\overline{\mathfrak{v}}[\mathrm{P}]\ =\ \mbox{True}\,\!$$
 * $$\overline{\mathfrak{v}}[\mathrm{Q}]\ =\ \mbox{True}\,\!$$
 * $$\overline{\mathfrak{v}}[\mathrm{R}]\ =\ \mbox{False}\,\!$$

(3) By (1) and clause (iii),
 * $$\overline{\mathfrak{v}}[\mathrm{P} \land \mathrm{Q}]\ =\ \mbox{True}.\,\!$$

(4) By (1) and clause (iv),
 * $$\overline{\mathfrak{v}}[\mathrm{Q} \lor \mathrm{R}]\ =\ \mbox{True}.\,\!$$

(5) By (4) and clause (v),
 * $$\overline{\mathfrak{v}}[\lnot( \mathrm{Q} \lor \mathrm{R})]\ =\ \mbox{False}.\,\!$$

(6) By (3), (5) and clause (v),
 * $$\overline{\mathfrak{v}}[\mathrm{P} \land \mathrm{Q} \rightarrow \lnot( \mathrm{Q} \lor \mathrm{R})]\ =\ \mbox{False}.\,\!$$

Thus (1) is false in our interpretation.

Next, try the valuation:
 * $$\mathrm{P}\ :\ \mbox{True}\,\!$$
 * $$\mathrm{Q}\ :\ \mbox{False}\,\!$$
 * $$\mathrm{R}\ :\ \mbox{True}\,\!$$

(7) By clause (i):
 * $$\overline{\mathfrak{v}}[\mathrm{P}]\ =\ \mbox{True}\,\!$$
 * $$\overline{\mathfrak{v}}[\mathrm{Q}]\ =\ \mbox{False}\,\!$$
 * $$\overline{\mathfrak{v}}[\mathrm{R}]\ =\ \mbox{True}\,\!$$

(8) By (7) and clause (iii),
 * $$\overline{\mathfrak{v}}[\mathrm{P} \land \mathrm{Q}]\ =\ \mbox{False}.\,\!$$

(9) By (7) and clause (iv),
 * $$\overline{\mathfrak{v}}[\mathrm{Q} \lor \mathrm{R}]\ =\ \mbox{True}.\,\!$$

(10) By (9) and clause (v),
 * $$\overline{\mathfrak{v}}[\lnot( \mathrm{Q} \lor \mathrm{R})]\ =\ \mbox{False}.\,\!$$

(11) By (8), (10) and clause (v),
 * $$\overline{\mathfrak{v}}[\mathrm{P} \land \mathrm{Q} \rightarrow \lnot( \mathrm{Q} \lor \mathrm{R})]\ =\ \mbox{True}.\,\!$$

Thus (1) is true in this second interpretation. Note that we did a bit more work this time than necessary. By clause (v), (8) is sufficient for the truth of (1).