Logical particles & notation: Is an effort to standardize worth the trouble?

Question

As the site grows we will probably be seeing more and more symbolic logic around the place. I wonder if it's worthwhile trying to standardize this from an early stage. Or is such a program simply too proscriptive and downright anal?

PROS:

• With an associated glossary, everyone will be on the same page.
• It's arguably easier for beginners to learn just one system of symbols initially.
• Aside from writing your own text-book, it's a rare (impossible?) thing to be able to standardize a system. If we can do this from the outset, then why shouldn't we?
• The basic core of symbols is comprised of relatively few members.

CONS:

• If you've already learned, and are familiar with, one specific system, why should you be forced into using another set of symbols?
• Those consulting the web or text-books will come across other notation systems anyway.
• Not all world keyboards make it easy to type the common particles for any given system.
• There are just too many symbols to cover.
• This would potentially be a lot of work for the moderators, and for minimal gain.
• The phil.SE may turn out to have a much greater natural language vs. formal language bent.

COMPROMISE:

Perhaps a reasonable middle ground would be to expect that users composing questions or answers try their best to be consistent within that particular slab of text. If there are spectacularly egregious fluctuations in technique within one body of text, then it may be appropriate for a mod to step in and clean things up a bit for the sake of clarity.

Is there any precedence on other SEs for strict adherence to defined sets of symbols? On the Unix.SE, for example, everyone knows what /dev/sdb3 means; it would reflect poorly on the user if she were to write /DEV/SDB3 instead (although people reading would still understand). On the Math.SE I imagine, although I'm not sure, that the use of symbols is quite strictly adhered to, lest readers may not understand what is being discussed.

Where does philosophy sit on this issue?

THE BEGINNINGS OF A PROPOSED GLOSSARY:

FWIW, I use the following basic symbols:

• negation (¬)
• conjunction (∧)
• disjunction (∨)
• material implication (⊃)
• biconditional (≡)
• Sheffer stroke (↑)
• universal quantification (∀)
• existential quantification (∃)
• 'it is necessary that' (□)
• 'it is possible that' (◇)
• therefore (∴)

Alternatively, we could take the already well-stocked Wikipedia page describing logical connectives. The disadvantage to this is that multiple forms are given for each symbol.

Answer 1

I think that this is a great idea.

Not so much because I'm particularly eager to see this become a site about formal logic (although to be fair, formal logic undoubtedly does belong on this site, given that the philosophy of logic is indeed a subset of philosophy) but because such notation is very likely to be used in a number of answers, and standardizing the notation in use is always a good idea. This way, we can make sure that everyone is on the same page.

Additionally, assembling such a "glossary" is important to providing a canonical resource for newcomers to the site who might not be at all familiar with the notation or its meaning. We can simply point them to this resource (or make it easy enough for them to find it themselves), rather than re-explaining the same issue repeatedly in comments.

Plus, it makes it much easier for me to find the symbols to copy and paste them into posts. +1 for laziness.

And here on Meta is a great place to start making this list. Despite not being at all an expert on formal logic (and despite having earned a grade of "C" in the only formal logic class I've ever taken!), I propose the following modifications to your system. These are the symbols with which I am personally most familiar, and the ones I've modified seem at least somewhat more intuitive (but perhaps that's my personal bias talking?).

Unary operators

• negation (not)  ¬
• universal quantification (true for all)  ∀
• existential quantification (true for some; there exists)  ∃

Binary operators

• conjunction (and)  ∧
• disjunction (or) (∨)
• material implication (if...then conditional)  →
• biconditional (if and only if)  ↔
• Sheffer stroke (not both)  ↑
• logical NOR (neither...nor)  ↓
• inference (implies; is derived from)  ⊢

Other signs and symbols

• therefore  ∴
• 'it is necessary that' (□)
• 'it is possible that' (◇)

---

I actually am not convinced that we need the bottom two. Their meanings are far less universally defined than the others. For example, □ can mean anything from "it is necessary that" to "it is provable that" to "it is obligatory that" to "it is believed that". And ◇ is even more potentially ambiguous, signifying "it is possible that", "it is not necessarily not", and even "it is not provable not". Their meanings can almost always be expressed using other symbolic expressions, and much more clearly so at that. But I suppose they're worth including for completeness nevertheless.

Answer 2

I would argue no. I think trying to define a specific set of symbols that ought to be used on this site would be too restrictive. After all, there is very far from this universal agreement among logicians themselves.

Yet it is agreeable that the logical notation used should be common enough. I would suggest that as long as the symbols used are in common use among logicians, mathematicians, or philosophers (this list on Wikipedia is a decent list, mainly because it shows the common alternatives such as right arrow instead of horseshoe for material implication, etc), and that the author is consistent with his use of symbols (e.g., distinguishes between identity and strict equivalence), then it is fine.

My primary doubt is about older notation which shouldn't really be considered obsolete just because it is not seen much any more. It is ultimately nothing more than a change in fashion that logicians now days prefer infix notation with parentheses rather than polish notation or the dot notation of the Principia. Yet choosing a notation that even people who are familiar with introductory logic would have probably never encountered would put too high a burden on even diligent readers who are interested in the subject matter.

In any case, it should be a guideline rather than a rule about notation, that suggests high communicability should be the goal of the post. We should always welcome members to rewrite a post using an unfamiliar notation, such as polish notation, to a more communicable notation at least for the current time period, unless there is good reason for adopting the older notation, such as if you are discussing the polish logicians.

Answer 3

I don't see the need for having a uniform glossary, or even any advantage to having this. People will disagree, and have all kinds of perfectly acceptable notation, much of which will have historical origins in different parts of the subject. For example, your favored symbol ⊃ for material implication is very seldom used in mathematical logic circles, and philosophers with a stronger mathematical training in logic will likely find → to be more familiar. And so what if people use different notation?

My standard logic notation differs in several respects from yours. Are you saying that people should edit my posts here to change my notation to align with yours? That seems misguided to me, as well as unnecessary, and furthermore seems likely sometimes to result in error.

So let's skip the idea of having a house standard for notation.

For comparison, I will note that MathOverflow does not have any policy of house-standard notation for mathematical terms, and they get along just fine without it.

Answer 4

It's a tricky question. There are definitely philosophical issues associated with pure logic, and with mathematical stuff more generally. But I think this raises a more general issue.

Philosophy questions get placed on this stack, not math or logic problems. After all, this isn't the modal logic stack, right? Given this stack is ideally for philosophy experts, you may very well get a better answer at math.se (or SO for that matter) on a pure logic problem.

However, there are plenty of philosophical consequences and implications of various maths and technologies. So where do we stand? Should we allow all the stacks to flow their philosophical questions about their domains to us? Presumably they'll have some thoughts on this as well.

I think we should aim to provide quick, thoughtful interventions when possible. That is to say, I'm not sure we should be expected to help you finish your logic homework. I really still think that belongs technically to the other stacks. Again, that being said, I think we are due at least some of the interesting philosophical questions that will usually end up closed as off-topic on the other stacks.

If these philosophical questions actually do intricately involve real logical, mathematical or technological problems, and are completely inaccessible without the attendant proofs or source code, we will probably want to have support for them. In that spirit I say we try to make the case for LaTex/source code highlighting, when and if it seems like they're actually demanded. At this point anyway this is all a little proscriptive.

Answer 5

I sometimes used symbolic expressions when explaining something that can be easier explained in logical symbols as it avoids the ambiguity in natural language. However, I'm also sloppy, in that I'm too lazy to lookup the symbol table or do weird keyboard dances, so I prefer a set that can be typed in a regular keyboard. I've often used this set of symbols:

• conjunction: ^ or and
• disjunction: v or or
• negation: ~ or not
• strict implication: ->
• biconditional: <->
• identity: = (logically the same as biconditional, but used when emphasizing identity rather than condition)
• for all: for all
• for each: for each
• sheffer stroke: ~(a ^ b) (I don't use it often, so no special symbol)
• logical nor: ~(a v b) (I don't use it often, so no special symbol)
• therefore: therefore
• 'it is necessary that': necessarily
• 'it is possible that': possibly

Note how material conditional is missing from the table, since it is quite rare in discussions that you really wanted material conditional; on the other hand, there is the strict conditional which is usually more useful.

The pros of this set is practicality, it is much easier to type so people will be more inclined to adhere to it. The cons is that they are not proper logic symbols.

I don't think P.SE is going to get LaTeX, but if we ended up agreeing on using proper symbols (i.e. if the votes does not go to my set) then it'd be good if we get a set of copy-pasteable/clickable symbols below the text editor (like in Wikipedia edit page) so we don't have to hunt around for those symbols. Either way, I'd follow suit with the consensus.