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?


  • 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.


  • 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.


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?


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.

  • 5
    I would prefer rightarrow for implication and rightleftarrow for the biconditional because it accords better with model-theoretic practice
    – Chuck
    Commented Jun 9, 2011 at 21:02
  • Thanks for your input Chuck. I have very little understanding of model theory. I'm open to suggestions on all of these. I'm certainly not wedded to '⊃' & '≡'. Negation & conjunction are fair game obviously. Even '∀' can be discarded. The model operators '□' & '◇' can be used in other contexts too. Sheffer and 'therefore' are probably as safe as any of them. Anyone else got some suggestions? Do we even need to standardize these?
    – boehj
    Commented Jun 9, 2011 at 21:29
  • 1
    Agree with Chuck, my exposure to logical notation was with FOL which uses those symbols, and my (obviously bias) perspective thinks those make more sense.
    – dimo414
    Commented Jun 10, 2011 at 8:55
  • @dimo: No problems. That makes sense to me. I'm happy to switch around here. As I say, I'm not particularly attached to '⊃' & '≡'... or any of them for that matter. Thanks for your comments. Anyone else got an opinion on this?
    – boehj
    Commented Jun 10, 2011 at 9:00
  • By the way, in response to a couple of your objections, there's absolutely nothing wrong with being anal. Another word for that is "precise", and such implies both knowledge and caring, two things that are held in high esteem around here. Also, you mention that such a list might be "Potentially a lot of work for mods for minimal gain", but that's not true at all. If the list is assembled and maintained by the community, then it would be absolutely no work for the mods at all! And of course it would/should be maintained by the community. That's really who runs these sites in the first place. Commented Jun 10, 2011 at 11:36
  • 1
    For those that are unacquainted with Unix: unlike in Windows, in Unix the file /dev/sdb3 and the file /DEV/SDB3 are quite different files since Unix filesystem is usually case sensitive; so if you typed filenames into the terminal, using different case may produce distinctly different result, for example, cat * > /dev/sda1 is likely to trash your disk while cat * > /dev/sdA1 probably will just create some extra files.
    – Lie Ryan
    Commented Jun 10, 2011 at 13:10
  • @boehj: btw, I think you should make the glossary you're proposing into an answer so people can vote on the glossary separately from the question. We probably also need a "No, we should not standardize" answer.
    – Lie Ryan
    Commented Jun 10, 2011 at 13:19
  • @Lie: I'm actually just about to crash out. Been up far too long and the brain requires sleep. Now! Haha... If anyone else wants to make up such a glossary I'm fine with that. Otherwise I'll get around to it in about 14 hrs time. I'm dying here. :)
    – boehj
    Commented Jun 10, 2011 at 13:22
  • Some people are down with anal. Commented Jan 13 at 5:09

5 Answers 5


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

Binary operators

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.

  • Definitely +1 for laziness. I'm not unhappy with any of the changes here. Considering you're giving 'translations' such as 'and', 'or', etc. it may be worthwhile pointing out that 'or' is the inclusive 'or', not the exclusive 'or' we use in English. We're probably not going to see too much modal logic done here so it won't really hurt to lose (□) & (◇). I've never used them outside a modal context. They won't be missed, by me at least. Good work. Oh, it'd be nice to get some truth tables up here as well for use with the propositional calculus as well. What do others think?
    – boehj
    Commented Jun 10, 2011 at 12:49

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.


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.


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.

  • 1
    +1 - Joe you make some good points. I think we should steer clear of logic for logic's sake. Phil. of logic belongs here though; that's generally not all that involved as far as symbolism goes. But wherever phil. of logic goes, metaphysics follows. Theory of reference may ask symbolic logic to be done. This strikes a chord with me: "I think we should to provide quick, thoughtful interventions when possible." I'm still trying to figure out what the site should be about. I think if we let answers meander then the site will fail. A SE environment isn't set up for that. I think. Cheers Joe.
    – boehj
    Commented Jun 10, 2011 at 1:04
  • A glossary is a really good idea by the way, if we do start getting a significant number of logic-related problems (especially as we almost certainly won't be getting either syntax or Tex.)
    – Joseph Weissman Mod
    Commented Jun 10, 2011 at 1:17
  • I think you underestimate the amount of philosophy that makes use of and is concerned with serious technical logic. I would argue that philosophy.stackexchange needs to activate TeX so that we may more easily use formal symbolism.
    – JDH
    Commented Dec 7, 2013 at 22:11
  • @JDH I actually think that may well be the case at this point too, and I think we may want to start thinking about ways to articulate this need. It's been an active item of discussion since very early on...
    – Joseph Weissman Mod
    Commented Dec 7, 2013 at 22:23
  • If you are involved in any discussions with the SE people who can implement this, please tell them that I have had problems in several answers here, in which I had wanted to use TeX, but it was not available. I strongly advocate the availability of TeX. Of course, many philosophers will not make use of it, but for those involved with the more technical parts of the subject, it will be appreciated.
    – JDH
    Commented Dec 7, 2013 at 22:25
  • 1
    For example, this very question is about the use of certain symbols, and the quality of the symbols appearing in the posts on this page is very low in comparison with what would be possible with TeX.
    – JDH
    Commented Dec 7, 2013 at 22:27

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.

  • Thanks for your input @Lie. You make a decent point re: the conditionals we're likely to see here. I guess we'll see how it plays out. Personally, I'd want to see the quantifiers in there as (∀), (∃), etc. but that's partially because I've set up shortcut keys to deal with them. Still, if we have a resource we can copy/paste from then it doesn't really matter anyway. To be honest I don't think we'll see Sheffer strokes being used so I wouldn't worry too much about that. I reckon there will be some modal stuff - just not in symbolic form. Thanks again for your input.
    – boehj
    Commented Jun 10, 2011 at 13:15
  • I very much enjoy that your chosen symbols for the last three look very much like other symbols that we use in the English language. Commented Jun 10, 2011 at 13:26
  • Mosdef on tilde for negation, I would prefer => for therefore (or qed since it seems classy)
    – mfg
    Commented Jun 10, 2011 at 14:39
  • 1
    @mfg The Latin you're looking for is ergo, not QED.
    – user20
    Commented Jun 10, 2011 at 16:51
  • @Matthew ergo has an extra character (2 if you count auto-complete+space), I'm referring to qed. My choice is mostly based on taps on a mobile keyboard, without access to [ALT]+ combos.
    – mfg
    Commented Jun 10, 2011 at 20:03
  • @mfg QED doesn't mean "therefore".
    – user20
    Commented Jun 10, 2011 at 20:10
  • 1
    @mfg: Quod Erat Demonstrandum (Q.E.D.) literally means "which was what we want to demonstrate" (e.g. "therefore no composite numbers are also prime numbers. Q.E.D."). "ergo" means "therefore" (e.g. "Cogito ergo sum" means "I think, therefore I am").
    – Lie Ryan
    Commented Jun 10, 2011 at 20:25
  • 1
    i hang my head in shame at the conflation of two separate concepts; i have no rebuttal by ignorance considering i looked q.e.d. up and should have remembered better :)
    – mfg
    Commented Jun 11, 2011 at 16:33

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