Sentential Logic

$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $

• A sentence is the basic unit of sentential logic.
• A sentence is a mathematical object that can be assigned a truth value: either T or F

The symbols of the language of sentential logic are taken from the following sets:

1. The set of sentence symbols: Upper case english letters, possibly with subscripts
   1. A, Z, \(A_{11}\)
2. The set of logical constant symbols:
   1. And: \(\land\)
   2. Or: \(\lor\)
   3. Not: \(\lnot\)
   4. Iff: \(\iff\)
   5. If then: \(\to\)

A sentence is a string \(\alpha\) of symbols of the language, astisfying one of the following conditions (recursive definition)

1. Atomic Sentences (Single letter sentences)
2. Compound sentences