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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? $$C(x) = A(x) \wedge B(x)\\ C(x) \equiv A(x) \wedge B(x)$$

ryang
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Mohammad muazzam ali
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  • For the first, I’d say that since we’re defining ‘p’, we should say $p \overset {\mathsf d \mathsf e \mathsf f}{=} 4>2$. For the second, the best thing would be just to assume $C(x) \iff (A(x) \land B(x))$. – PW_246 Feb 11 '23 at 13:40
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    I would personally just write "Let $p$ be the statement $4>2$". Alternatively, you could put it in a LaTeX equation environment and it would automatically get a number. You can then use the number (with "\ref" and "\label" commands). For predicates I would indeed use notation instead of text. – student91 Feb 11 '23 at 13:48
  • @student91 which notation would you use for predicates? – Mohammad muazzam ali Feb 12 '23 at 03:39
  • @Mohammadmuazzamali you can read the answers of the others for that. Both Pdubya and ryang have excellent suggestions – student91 Feb 12 '23 at 19:14

2 Answers2

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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:

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

ryang
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  • I think $P:=2>4$ is nice, when you can do it with numbers, why shouldn't you be able to do it with booleans? Maybe a note should be added when the notation is used for the first time... – fweth Feb 11 '23 at 15:01
  • I'm a little confused by which notation should be used for writing that the LHS and RHS are the exact same thing. Do you mean that we should use '$=$' or the double arrow with 'def' on top of it or '$\equiv$'? – Mohammad muazzam ali Feb 12 '23 at 03:39
  • @Mohammadmuazzamali Ah, okay, does this answer your question? After reading that answer: my final sentence above means that defining predicate $Cx$ as $Ax∧Bx$ is a more fundamental action than discovering that the former is logically equivalent (≡) to the latter. – ryang Feb 12 '23 at 05:35
  • Try $p \iff (4\gt 2)$, and $C(x)\iff (A(x) \land B(x))$ for proposition $p$ and unary predicates $A, B$ and $C$. – Dan Christensen Feb 12 '23 at 18:06
  • @ryang from that answer, if i'm understanding it correctly, i believe you're suggesting using either the symbols $\overset{\text{def}}{\iff}$ or $:\overset{\text{def}}=:$ for assigning a predicate to another predicate(s)? – Mohammad muazzam ali Feb 13 '23 at 07:00
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    @Mohammadmuazzamali I don't mind either, but opinions vary. – ryang Feb 14 '23 at 12:58
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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.

Markus Scheuer
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