Notation to assign a statement or predicate to a symbol

Question

If I want the statement "$4>2$" be represented by the letter $p$, how should it be written? $$p:4>2\\ p= 4>2\\ p \equiv “4>2”$$

On the other hand, if I have two predicates $A(x)$ and $B(x)$, and I want to represent the conjunction of both predicates using the symbol $C(x),$ how should it be written? $$C(x) = A(x) \wedge B(x)\\ C(x) \equiv A(x) \wedge B(x)$$

Answer 1

All these are unambiguous:

• Let $P$ represent the formula $4>2.$
• We use $C(x)$ to denote the formula $A(x) \wedge B(x).$
• $P\overset {\;\mathsf {def}}{\iff} 4>2$
• $C(x)\overset {\;\mathsf {def}}{\iff} A(x) \wedge B(x)$

Although = normally connects logic terms rather than logic formulae, these too are unambiguous:

• $P:=4>2$
• $C(x):= A(x) \wedge B(x)$

Don't write this, as it suggests that the LHS and RHS are logically equivalent by derivation instead of by definition:

$$C(x) \equiv A(x) \wedge B(x).$$

Answer 2

Hint: It might be convenient to consider the usage of Iverson brackets \begin{align*} [P]= \begin{cases} 1&\qquad\text{if $P$ is true}\\ 0&\qquad\text{if $P$ is false} \end{cases} \end{align*} We can this way define \begin{align*} \color{blue}{\mathrm{Let}\ S:=[4>2]} \end{align*} which has the advantage that since $S\in\{0,1\}$ it can be used in calculations.