I'm studying axiomatic set theory and even though I know some predicate logic I still struggle to understand some symbolizations, such as:

Union axiom: $(\forall x)(\exists y)(\forall u)(u \in y \iff (\exists v)[v\in x \land u\in v])$

This axiom is no SO hard but I still take a little to understand what it says, when symbols should facilitate the understanding of definitions etc. I guess.

How can I learn to read (and most importantly understand) long complex symbolizations?


5 Answers 5


How did you learn to read?

You were told the basics, then you learned by reading hundreds upon hundreds of words, each repetition being a little easier, until it became second nature.

How did you learn to ride a bike?

You were shown the basics, then you learned by riding for ages, each meter being a little easier, until it became second nature.

How did you learn to multiply?

You were told the basics, then you learned by multiplying hundreds of pairs of numbers, each pair being a little easier to multiply, until it became trivially easy.

How do you think you will learn to read predicate logic?

  • $\begingroup$ Yeah but since nobody tells me the basics, because there are no books that I know of, that explain you how to symbolize/read and they always explain the same things I dont know what to do $\endgroup$
    – Abcd
    Jan 20, 2020 at 13:45
  • $\begingroup$ @Abcd but you do know the basics, you listed many symbols in your OP. $\endgroup$
    – AlvinL
    Jan 20, 2020 at 14:39

Welcome to our little flock! I can understand your frustration, but you just need practice and it will become easier. I'm no master, but I find that it helps to translate the symbols to simple words or phrases:

  • $\forall$ = 'for all'
  • $\exists$ = 'there is/exists'

and so on. Thus your axiom becomes 'For all x there is (at least one) y, so that for all u ...'. That's the easy part - the hard part is learning the maths, so to speak; you need to get used to a mindset - a way of thinking. Like the odd things that a set isn't defined simply as 'A set is ...', but rather pops into existence as something outof the blue, which has certain properties.

The other thing that helped me with axiomatic set theory was the fact that I had already learned 'naive' set theory: the one where these things are defined rather more intuitively with things like 'A set is a collection of stuff'. Axiomatic set theory is simply an attempt at fixing some of the issues in naive set theory, so you avoid Russel's Paradox and similar.


Couple of pointers:

First, as many have already pointed out, it just takes practice

Second, a simple but effective training regime is to start with simple sentences, and gradually work your way to more complex sentences

Third, do not see this as a one-way street from logic symbols to English. Rather, you should learn the two-way street between them. That is, you should learn how to translate English into logic at the same time as you learn to translate logic back into English; it is only yhen that your brain really learns to see the connections.

Fourth, keep a close eye to the grammatical structure of the sentences: see where the parentheses are, so you see the scope of the quantifiers, and the order of the operators. The translation obviously depends on exactly those things.

Fifth, many predicate logic sentences that you will encounter follow the basic structure of the four Aristotelean sentences, exactly because so many English sentences are of this format:

'All $P$'s are $Q$'s': $\forall x (P(x) \to Q(x))$

'Some $P$'s are $Q$'s': $\exists x (P(x) \land Q(x))$

'No $P$'s are $Q$'s': $\forall x(P(x) \to \neg Q(x))$ (which is the same as 'All $P$'s are not $Q$'s')

'Some $P$'s are not $Q$'s: $\exists x (P(x) \land \neg Q(x))$

Commit these four patterns to memory, and you'll find they pop up all the time in predicate logic sentences.

Good luck!


Akin to learning a language. You likely know some vocabulary: $\in , \exists, \forall, \neg, \land, \lor, \Rightarrow, \Leftrightarrow$. The good thing about mathematics is that it's a relatively simple language to learn, compared to, say, spoken languages such as English or Russian or what have you.

I can tell you I didn't speak or write English after 'reading' the dictionary for a few days. Neither was that the case with (predicate) logic. It takes practice, please be patient.

Specifically regarding axioms of set theory. Read the formula piece by piece or identify the roles of each variable. What is regarded as the 'union' in that array of symbols? What is $x$ or $v$?


It always helps if you try to write down the expressions in English: so

$$\forall y, \exists x :$$ becomes "for all $y$, there exists an $x$ such that".

Also when you're doing the writing, try to place commas and colons between the expressions ($\forall y, \exists x : \forall u\in x$...). It makes reading the symbols easier and also creates parallels with a language you're already familiar with.

You can also read them from the end to the beggining, or by parts, if it helps.

Another exercise that can be useful is to do the inverse process: pick up logical chains, like syllogisms, and try to write them in mathematical form - define each variable/set of objects and then try to make out how you'd write the relations between them.

If all you need is a dictionary of sorts, here's a pretty useful one. It has the symbols, their translation and their LaTeX codes, which you will need A LOT on this site.

  • $\begingroup$ I strongly disagree with the suggestion to sprinkle commas and colons among formal sentences: they have a tendency to give rise to frustrating ambiguity or misinterpretation. $\endgroup$
    – ryang
    Feb 25 at 19:17
  • $\begingroup$ It has been my experience that students facing logic sentences for the first time find them less intimidating and more readable when punctuation is present. It helps bring the mathematical sentencing closer to something they're already familiar with - so teaching them to use the commas where you would already place them in a sentence, and ":" in place of "such that", helps create that connection and makes learning easier. $\endgroup$
    – Rye
    Feb 27 at 10:44
  • $\begingroup$ I did, and I understood your point. But as you stated yourself in that link, if you read the colon as "such that", and try to translate logic sentences into English, you can see the grammatical errors arise when they are misused. Hence, the importance of doing exercises focused on building that correspondence when you are learning. It really helps spot mistakes and making sure you are writing correctly and unambiguously. $\endgroup$
    – Rye
    Feb 27 at 10:57
  • $\begingroup$ Actually, my previous link (now deleted, to declutter) was not an optimal example, because, as you rightly suggest, the colon after the universal quantifier doesn't even correspond to a natural-language reading. Here's an explicit example of what I mean by the punctuation not merely being superfluous but actually introducing ambiguity: ambiguity of . , : as a quantifier suffix. $\endgroup$
    – ryang
    Feb 27 at 11:28
  • $\begingroup$ To be clear: you may be writing using the "correct" punctuation according to natural language rules, however the ambiguity is when the reader requires a separate explainer from you to decode whether your : , . is merely abbreviation/punctuation imported from natural language, or whether it functions as a delimiter, which these symbols do commonly signify in formalisations. $\endgroup$
    – ryang
    Mar 2 at 5:59

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