How can I learn to read predicate logic sentences?

Question

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?

Answer 1

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.

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

Answer 2

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.

Answer 3

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.

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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 $v$?

Answer 4

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!

Answer 5

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.