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Question.

What early published or citably attested examples (preferably in the mathematical literature) can you give of the following convention?

Let $\mathbb{S}$ denote some nonempty subset of some specified alphabet $\mathbb{A}$.

Then the convention is made that

  • in any formula, any letter from $\mathbb{S}$ must be interpreted to a strictly negative real number (i.e. $<0$), while any letter from $\mathbb{A}\setminus \mathbb{S}$ must be interpreted to a strictly positive real number (i.e. $>0$).

Remarks.

  • An example is $\mathbb{A}:=$Latin alphabet, $\mathbb{S}:=$consonants.

    Then, for example,

    $\mathrm{for}\ \mathrm{all}\ a,\mathrm{for}\ \mathrm{all}\ z,\qquad a\cdot z < 0\qquad (\mathrm{sen})$

    is a true sentence,

    while, needless to say, with the usual convention of indifference to the typographical shape of variable symbols, (sen) is false, since without the convention there are both $(a,z)$ with $a\cdot z<0$ and $(a,z)$ with $a\cdot z>0$.

  • This is a rare convention, yet it exists in the mathematical literature.

It can make formulas more concise.

To give a toy example (this is not the application in the published article which made me ask this question): this convention is useful to more concisely realize the idea of

  • 'necessarily false formula'
  • 'possibly yet not necessarily true formula'.

Needless to say, a formula (in the model-theoretic sense) is never true or false, only after interpretation via a specified satisfaction-relation.

Definition: a 'necessarily false formula'$:=$shorthand for the longer term 'necessarily false formula over a specified signature $\Sigma$, and w.r.t. a specified satisfaction relation $\vDash$, and w.r.t. a specified form-formula $\Phi$' (in the usual sense of model theory)$:=$shorthand for quadruple ($\varphi,\Phi,\Sigma,\vDash$) wherein $\Sigma$ is a signature, $\varphi$ and $\Phi$ are formulas over $\Sigma$, $\vDash$ is a satisfaction relation for formulas over $\Sigma$ (note that this implies that there is a specified structure $\mathfrak{R}$ in the background), such that $\varphi$ is obtained from $\Phi$ by substitution of variables (in the usual technical sense, where as usual substitution of variables is indifferent to sorts of variables; you may substitute any sort of variable for any other), and $\neg\ (\mathfrak{R}\vDash \varphi[\alpha])$ for every variable assignment $\mathfrak{\alpha}$ (variable assignment maps must respect sorts of variables).

For the example, if the above convention is in force, then

  • $a<z$ is a 'necessarily false formula' w.r.t. the form-formula $x<y$, the signature $<,\mathrm{negvar},\mathrm{posvar}$ and the (standard satisfaction relation $\vDash$ of) the structure $(\mathbb{R},<)$,
  • $z<a$ is a 'possibly yet not necessarily true' formula w.r.t. the form-formula $x<y$, the signature $<,\mathrm{negvar},\mathrm{posvar}$ and the (standard satisfaction relation $\vDash$ of) the structure $(\mathbb{R},<)$,

Now where's the difference to the usual situation in which variables are not sorted? In this situation, it is impossible to write a 'necessarily false formula' of the form '$x<y$' with $x,y$ arbitrary variables.

The whole question is related to what, in particular in categorical logic, are called sorted signatures. In a sense, here, the consonants have sort 'strictly negative real number' and the non-consonants have sort 'strictly positive real number'.

  • Example citations I will not give, since I am hoping for examples far older than the one I could give.
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    $\begingroup$ I have never seen this. There is a common convention of having letters from the middle of the alphabet represent integers and from the end represent arbitrary reals, or have Greek letters from the beginning of the alphabet represent (typically infinite) ordinals and those from the middle represent infinite cardinals (and $\omega $ having a specific meaning). $\endgroup$ – Andrés E. Caicedo Aug 11 '17 at 13:29
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    $\begingroup$ Your framework is so bizarre (I see no purpose for such a letter convention and have never heard of it before) that I do not find your reason to refuse to give even a single historical example of it compelling. Please show us at least one example. Then you can ask if anyone knows an earlier instance instead of hiding examples from us. I am almost tempted to suspect, since you give no example, that you are making this up and there are no examples. $\endgroup$ – KCd Aug 11 '17 at 15:45
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    $\begingroup$ Then I find your refusal to provide an example as defeating the presumed goal of encouraging others to take an interest in your question. If the instructor in a course introduces an abstract definition and then doesn't give any example of it even after the students ask for one, I don't think the instructor is helping the class care about the concept. $\endgroup$ – KCd Aug 11 '17 at 18:01
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    $\begingroup$ "A mathematical convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians". If you are reluctant to give specific sources could you at least point to an area or a group of mathematicians that employ this convention? It might help people find what you are looking for. $\endgroup$ – Conifold Aug 11 '17 at 19:40
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    $\begingroup$ I am voting to close this as its current formulstion is not serious. There are other sites for puzzles and games. $\endgroup$ – Andrés E. Caicedo Aug 12 '17 at 0:27
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It is not completely clear to me what you mean... Would this be an example?

In the FORTRAN programming language (1957)$^1$, variables beginning with letters I,J,K,L,M,N represent integers, and other variables represent floating point values.

$^1$ Backus, J. W.; H. Stern, I. Ziller, R. A. Hughes, R. Nutt, R. J. Beeber, S. Best, R. Goldberg, L. M. Haibt, H. L. Herrick, R. A. Nelson, D. Sayre, P. B. Sheridan (1957). "The FORTRAN Automatic Coding System". Western joint computer conference: Techniques for reliability. Los Angeles, California

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  • $\begingroup$ Dear @Gerald Edgar: thanks for pointing out. Well, obviously in the example you give the signs are not mentioned, and the reference you give does not critically use this convention (I suppose), in the sense of writing sentences which would be false without it. So this is not an example. $\endgroup$ – Peter Heinig Aug 11 '17 at 13:42

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