Section 1.1 - Propositional Connectives
Everything needs to start somewhere. Section 1.1 introduces the basic building blocks of propositional calculus. Much like all section 1.1's it starts out simple, thus lulling the reader into a sense that the whlie book will be straight forward and easy.
Several named truth functions are introduced in section 1.1, along with vocabulary that makes conversation smoother. I'll admit that my first impression was that special names for certain truth functions isn't necessary when functions are made variable, but as I read, and used the words, I began to see the merits. Having names like conjunction and antecedent make speaking about an expression easier. In fact, this section inspired me to name the switch statement. Further, ``principal connective'' provides a name for the outer function of a composition; something I've been missing.
On the whole, this section is simple, but lays the groundwork for the rest of the book...like you'd expect.
Concepts and Vocabulary
- Negation $(\neg)$: A unary truth function that returns the opposite truth value from its input variable $$\begin{array}{c|c|c} A & \neg A & (A)_\mathtt{0x1} \\ \hline 0 & 1 & 1 \\ 1 & 0 & 0 \end{array}$$
- Conjunction $(\land)$: A binary function that returns one only if both input variables are set to one. $$\begin{array}{c|c|c|c} A & B & A\land B & (A,B)_\mathtt{0x8} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{array}$$
- Conjuncts: Input variables to a conjunction.
- Disjunction $(\vee)$: A binary function that returns one so long as at least one input variable is set to one. $$\begin{array}{c|c|c|c} A & B & A\vee B & (A,B)_\mathtt{0xE} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}$$
- Disjuncts: Input variables to a disjunction
- Conditional $(\Rightarrow)$: A binary function that returns one for all condition values except when the first variable is one and the second variable is zero. $$\begin{array}{c|c|c|c} A & B & A\Rightarrow B & (A,B)_\mathtt{0xD} \\ \hline 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}$$
- Antecedent The first variable in a $\mathtt{0xD}$ truth function, or the second variable in a $\mathtt{0xB}$ truth function.
- Consequent The other variable in $\mathtt{0xD}$'s and $\mathtt{0xB}$'s.
- Biconditional $(\Leftrightarrow)$ A binary function that returns one only when both input variables have the same value. $$\begin{array}{c|c|c|c} A & B & A\Leftrightarrow B & (A,B)_\mathtt{0x9} \\ \hline 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{array}$$
- Propositional Connectives The symblis $\neg$, $\land$, $\vee$, $\Rightarrow$, and $\Leftrightarrow$.
- Statement Form An expression made of statement letters and connectives.
- Statement Letters Variable names
- Truth Function...Come on, do we really need to define this.
- Exclusive ``or'' $(\oplus)$ A binary function that returns one only if the input variables have different values. $$\begin{array}{c|c|c|c} A & B & A\oplus B & (A,B)_\mathtt{0x6} \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \end{array}$$
- Atomic Sentences An expression that is not composed of multiple truth functions.
- Principal Connectives Outer function in a composition.