In most parts of mathematics I've seen logical symbols like $\implies, \exists, \forall$, etc.

But I haven't seen $\land, \lor, \lnot$. Like " $\forall x\in\mathbb{R} \land y>0$ " instead of " $\forall x\in\mathbb{R}$ and $y>0$ ".

Is there any reason for this (my professor just said not to use them)? Are they used outside pure logic context?

  • $\begingroup$ Most people prefer natural language over being too formal, that's all. Note, however, that your examples do not make sense in either case. $\endgroup$ – Git Gud Sep 14 '13 at 22:26
  • $\begingroup$ It's nice to say things in words, when it is not particularly inconvenient to do so. $\endgroup$ – PVAL-inactive Sep 14 '13 at 22:26
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    $\begingroup$ They are used outside logic. Ordinary language is ordinarily better. Indeed $\forall x$, $\exists x$, are sometimes overused by students who mistakenly think that makes what is written "more mathematical." $\endgroup$ – André Nicolas Sep 14 '13 at 22:29
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    $\begingroup$ I can't really answer this within the scope of math.stackexchange as my answer involves my opinions about how to write mathematics. To write mathematics well, you need to know how to explain mathematical concepts and arguments (1) in words and (2) in symbols. You then need to choose a good combination of (1) and (2) for each statement you have to make. The best authors often manage to say the same thing both in words and in symbols, but in a natural way, so the reader doesn't object to the repetition. The apparent unwillingness of professors like yours to engage fully in (2) is a sad thing. $\endgroup$ – Rob Arthan Sep 14 '13 at 23:14
  • $\begingroup$ @Rob: In so far as it can be characterized in black and white, it’s a good thing. (1) Fully symbolic mathematical exposition is in general extremely hard to read. (2) Students have a tendency to overuse formal symbolism at the expense of clarity. (3) A great many students need to learn that a proof is just a special kind of expository prose, not a parallel list of statements and reasons: it is composed of (paragraphs of) sentences, and using words for the connective tissue that gives a road map for the logic of the argument generally makes the prose more readable. $\endgroup$ – Brian M. Scott Sep 14 '13 at 23:29

You certainly can, but it's rarely done. Sometimes it's better to use usual language for the sake of clarity, rather than leaving everything in logical notation.


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