User:JMRyan/Sandbox

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Example truth table[edit]

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


Example derivation[edit]

 
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[edit]

 
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[edit]

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[edit]

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.