# How to efficiently read a predicate logic formula (best practices)

This no question about how to understand a predicate logic proposition in general, it's about fast understanding such a proposition.

E.g. as a simple example, the convergence definition for $(a_n)_{n\in \mathbb{N}}$ with $a_n\in M$ (for all $n\in \mathbb{N}$ ; $M$ ordered ; e.g. $M=\mathbb{R}$ ) is

$\exists a\in M \forall \epsilon \in M_{>0} \exists N\in \mathbb{N} \forall n\in \mathbb{N}\colon n\ge N \implies |a_n -a|<\varepsilon$

When you read something like that (unknown to you): What is your approach?

So far, mine is the following:

Part 1: The part, that does include the quantifier declations. (e.g. "$\exists a\in M \forall \epsilon \in M_{>0} \exists N\in \mathbb{N} \forall n\in \mathbb{N}$"

Part 2: The part, that does not include the quantifier declrations. (e.g. "$n\ge N \implies |a_n -a|<\varepsilon$")

a) Directly read the formula as it is:

1. Look very shortliy at part 1 to get a short impression of the quantifiers (mostly to see what variables are used).

3. Look closely to the part 1 again and read it carfully from right to left. While doing that, i look from time to time to part 2 again, to see how the read variable is used in context of part 2.

b) Transform the formula:

If the formula appears to be too "chaotic" for me, i transform it to an equivalent predicate logic propositions which composes of a declaration part (part 1) and a coresponding part (part 2). After that i do a).

My questions now are:

1. What is your approach/procedure to read a (new to you) formula efficiently?

2. Do you occasionally transform the formula in a better readable formula? And if yes: How do you transform (e.g. do you also separate the declaration part of the other part; do you write the discure universum conditions alltogether in the declaration part [e.g. short $\forall n \in \mathbb{N}\ge N$])?

• It seems to me that it's quite opinion-based and rather related to experience/being used to syntax rather than "procedures". – Boris E. Mar 31 '17 at 22:32
• @BorisEng I think the way of understanding a proof often results in a common way of understanding cognitive sequence. That's also why many people write prose when writing proofs (there is one train of thought). My question relates to what you do, if you just have a formula and no written prose. What heuristic leads, in your experience, often to that train of thought, which helps you understanding the formula? – SearchSpace Apr 11 '17 at 7:49

The best solution to the problem is not to create the problem. Write formulas in a readable way. You might have to rewrite it yourself.

1) Don't use the variable "a" to refer to 2 different things. Avoid using $n/N/\mathbb N$ to refer to different things. Sometimes a convention helps, something like upper case for sets (and maybe sequences), lower case for primitive values.

2) Don't try to shove everything into 1 formula. Give different names to different concepts.

$$\exists l \in M~ \forall \varepsilon \in M_{>0} ~\exists k \in \mathbb{N} ~\forall j \in \mathbb{N} ~\colon~ j \ge k \implies |a_j - l|<\varepsilon$$

Break it up:

Eventuality: $E(a, P) = \exists k \in \mathbb N ~ \forall j \in \mathbb N ~:~ j \ge k \implies P(a_j)$ : "Eventually every value in a sequence has property P"

Boundedness : $B(a, l) = \forall \varepsilon \in M_{>0} ~:~ E(a, a_j \mapsto |a_j - l| < \varepsilon)$ : "For every deviation $\varepsilon$, (eventually every value in a sequence has property) that it is within $\varepsilon$ of L".

Existance : $F(a) = \exists l ~:~ B(a, l)$ : "There exists a limit L such that (for every deviation $\varepsilon$, (eventually every value in a sequence has property) that it is within $\varepsilon$ of L)"

You probably will have to start with the first part of the formula that makes sense, and go from there. It could be inside out, or left to right, or right to left.

• This seems to be a good idea when writing proofs. Question to the example: $E(a,a_j \implies |a_j-l|<\varepsilon )$ here you mean $E(a,|a_j-l|<\varepsilon )$? This probably also reflects your procedure when reading proofs, roughly speaking, you go from the inside out? – SearchSpace Apr 11 '17 at 7:43
• @SearchSpace Oh I was just writing an anonymous function. Instead of $f(x) = x^2$ I wrote $f = (x \mapsto x^2)$. There's a lot of conventions. – DanielV Apr 11 '17 at 16:38