Definition:Alphabet of Propositional Calculus
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Definition
Propositional Symbols
The vocabulary of $\mathcal L_0$ can be any set $\mathcal P_0$ of arbitrary symbols.
It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts.
A typical set of letters would be, for example:
- $\mathcal P_0 = \left\{{p_1, p_2, p_3, \ldots, p_n, \ldots}\right\}$
The letters of $\mathcal L_0$, i.e. the elements of $\mathcal P_0$, are referred to as the propositional symbols of $\mathcal L_0$.
Some sources call the elements of $\mathcal P_0$ the propositional variables of $\mathcal L_0$.
Signs
The signs of the alphabet of $\mathcal L_0$ usually include (but may be a subset of) the following:
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \land\) | \(:\) | \(\displaystyle \)the conjunction sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \lor\) | \(:\) | \(\displaystyle \)the disjunction sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \implies\) | \(:\) | \(\displaystyle \)the conditional sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \iff\) | \(:\) | \(\displaystyle \)the biconditional sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \neg\) | \(:\) | \(\displaystyle \)the negation sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle (\) | \(:\) | \(\displaystyle \)the left parenthesis sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle )\) | \(:\) | \(\displaystyle \)the right parenthesis sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \top\) | \(:\) | \(\displaystyle \)the tautology sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \bullet\) | \(\displaystyle \) | \(\displaystyle \bot\) | \(:\) | \(\displaystyle \)the contradiction sign\(\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Connectives
- The signs $\land, \lor, \implies, \iff$ are called the binary connectives.
- The sign $\neg$ is called the unary connective.
Together they are known as the connectives of $\mathcal L_0$.
Length of Symbols
Whether subscripted or not, however they are defined, the elements of $\mathcal P_0$ are defined as having a length of $1$.
The signs of $\mathcal L_0$ are also defined to have length $1$.
Thus, each of the symbols of $\mathcal L_0$ is considered to have a length of $1$.
