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Many times I find myself wondering exactly how to separate the conditions of a $\forall$ statement and its conclusions, for example:

$$\forall x,y\in S, x\circ y\not\in S, x=y.$$

This could mean: $$\forall x,y\in S, (x\circ y\not\in S, x=y).$$ Or $$\forall x,y\in S, x\circ y\not\in S, (x=y).$$ Or even something like $$\forall x,(y\in S, x\circ y\not\in S, x=y).$$ if $y$ had been defined elsewhere already.

I'm looking for advice on how to notate this kind of statement more clearly. What separates the conditions of a $\forall$? Is it proper to put parentheses around things how I have in these statements? Can/Should semicolons ever be used in these?

To be clear, the examples are logically nonsensical so that the notational ambiguity is more apparent.

Austin Weaver
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  • The fourth one seems wrong to me: I think that you are meaning "all $x$ and $y$ that belongs to $S$". – Mauro ALLEGRANZA Oct 19 '17 at 14:44
  • As you can see from your doubt, the colon is also ambiguous. Are you meaning: $\forall x \in S \ \forall y \in S \ (x \circ y \notin S \land x=y)$ ? – Mauro ALLEGRANZA Oct 19 '17 at 14:46
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    I don't actually have a meaning in mind for the statement (the random example I gave). The point is to find how each example could be stated with no ambiguity from notation. – Austin Weaver Oct 19 '17 at 14:49
  • See $\forall x,y\in S[\cdots]$ as an abbreviation of $\forall x\forall y[[x\in S\wedge y\in S]\implies\cdots]$ and don't use comma's within the brackets. you better use $\wedge$ there. – drhab Oct 19 '17 at 14:49
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    Conclusion: to avoid ambiguities, avoid too many short-cuts. Ink is quite cheap... – Mauro ALLEGRANZA Oct 19 '17 at 14:53
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    My advice would be to avoid commas completely. They're not used in any formalized language of logic I know, and the people who use them seem to be motivated by the notion that logical symbols are just an abbreviation for English words, and therefore they can use commas in the middle of a symbolic formula where English grammar want them. This is fundamentally misunderstood -- symbolic logic is a language of its own, with separate syntax and grammar, and it just creates confusion to try to mix natural-language punctuation into it. – hmakholm left over Monica Oct 19 '17 at 15:41
  • ... except perhaps as shorthand for quantifying over multiple variables at once, as in $(\forall x,y)\cdots$ -- but even there the savings are minimal. – hmakholm left over Monica Oct 19 '17 at 15:41

1 Answers1

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A caveat about the following: I am not a logician, so my experience has that limitation; also, my experience includes a PhD in computer science, which may have "polluted" my pure-math knowledge.

That said, in my experience a "pure" usage of the $\forall$ notation might look like this:

$$\forall x.\forall y.((x\in S \land y\in S)\implies (x\circ y\not\in S \land x=y)).$$

Note that I have gotten rid of all the commas, since I believe they are relatively informal and (in this case) ambiguous. Alternatively, maybe the original with the commas meant

$$\forall x.\forall y.((x\in S \land y\in S \land x\circ y\not\in S ) \implies (x=y)).$$

There's also a form in which we're allowed to indicate the domain as part of the $\forall$ notation itself (which I think the examples above show is not technically necessary):

$$\forall x:S.\forall y:S.((x\circ y\not\in S)\implies (x=y)).$$

At least, I think I've seen this with colons for the domain. Alternatively, using set notation, $$\forall x \in S.\forall y\in S.((x\circ y\not\in S)\implies (x=y)).$$

The dot after $\forall x\in S$ is (I think) a relatively common connector for such statements, but you might want to write it without any connector (but with a little bit of blank space) in that position: $$\forall x\;\forall y\;((x\in S \land y\in S)\implies (x\circ y\not\in S \land x=y))$$ or $$\forall x \in S\;\forall y\in S\;((x\circ y\not\in S)\implies (x=y))$$ or maybe even with extra parentheses like this: $$(\forall x \in S)(\forall y\in S)((x\circ y\not\in S)\implies (x=y)).$$

I think I've seen all of these at some point or another. They all seem reasonably unambiguous to me.

David K
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  • +1. My vague impression is that using dots to separate quantifier and body is done mostly in computer science, and may be influenced by the $\lambda$-calculus. Similarly, the colon in $\forall x:S$ has a bit of CS aroma to it; it looks influenced by type annotations. For a pure mathematics audience, one of the forms with parentheses will probably be most recognizable. – hmakholm left over Monica Oct 19 '17 at 15:46
  • @HenningMakholm Those things have the same "smell" to me too, which is why I tried to hint at possible CS crossovers in the first paragraph. What I don't know sometimes is whether something was imported into CS from math or whether it originated in CS. Thanks for calling out those notations in particular. – David K Oct 19 '17 at 16:53
  • x @David: My understanding is that the dots can be traced back to Peano who used them instead of nested brackets in general (see comments here) way before CS was invented -- but that they survive today almost exclusively in the vestigial form exhibited here. – hmakholm left over Monica Oct 19 '17 at 17:36