I'm reading a Mathematical Logic book (A course in mathematical logic, Bell.M ) and the author is saying that the symbols of a formal language don't have a well-defined shape, he's claiming that they are abstract entities.
I think he is saying that even though symbols are usually defined by its shape, the symbols of a formal language have 'abstract' shapes.
He goes on explaining that we couldn't possible be able to define an exact shape in a formal language because that wouldn't be reproduceble in all metalanguages studying the symbols of that objective language ( the formal language ).
He proposes them, that when we are studying a formal language ( objective language ) by means of a meta-language, and we want to reference the objects of the formal language ( its symbols ), we use as name, metalinguistic symbols.
By doing this, we don't have to worry with the "shape" of the symbols in the objective languaeg when we are changing the meta-language that is studying the formal language, because the shape is not well-defined, they are allowed to vary with the meta-language.

All in all, i think he is claiming that the symbols, lexicon or alphabet of a formal language, and hence the syntax is independent of its visual representation.

I'm curious if this approach is worth to make and also if people are worried to make such distinction ( or is it only him ) ?

Also, i would like to be corrected if i misunderstood the point.


2 Answers 2


If this is referring to A Course in Mathematical Logic by John Bell and Moshé Machover (round about p.8), then you are perhaps making heavy weather of a fairly simple point.

Suppose, for example, we are interested in the propositional calculus. It evidently doesn't matter exactly what style of arrow we use in the object language as its conditional (or whether we use a horseshoe rather than an arrow), etc. That's agreed on all sides. The question is how should we handle the point when dealing with propositional logic?

One option is to start with a fixed object language, do the meta-theory for the logic of arguments as presented in this fixed object language, then say at the end "and obviously, the same applies to any variant object-language which shares such-and-such features."

Or we can take Bell and Machover's neater(?) line and build in generality from the start. So we allow for lots of variations in our object languages. And when talking about propositional arguments in one language or another we augment our English metalanguage with a symbol '$\to$' (for example) to denote the conditional symbol in the object language. But we go on to say with Bell and Machover that "What the latter symbol actually looks like is of no importance, and the reader may give free range to his imagination." If you like, this is to give a rather more abstract characterization of the object language.

This second approach is very common. At some point we need to get into the story the fact that shape of symbols (choice of fonts, italics vs bold, etc) are irrelevant to logic. To repeat: roughly speaking, we either take a fixed object language, do the meta-theory for that, then generalize -- or we do what Bell and Machover do, which is to take the meta-theory to be general from the start.

You pays your money and you makes your choice.

  • 1
    $\begingroup$ I'd hesitate to claim that the OP is "making heavy weather". They're simply requesting confirmation for their interpretation of this remark, and some further explanation about it. This is IMO actually a commendable thing to do; too many people gloss over details and have to come back for another pass because they didn't really get it. $\endgroup$
    – Lord_Farin
    Jun 29, 2013 at 19:39
  • $\begingroup$ That's exactly the mindset i adopted some months ago about learning, that i only move further when my current topic is at a lot clear to me.The problem is that, as opposed to what Peter said, it was my first encounter with such idea and the point was still not that 'simple' to me, that's why i looked for clarification here, to see if i understood the idea.Anyways, thanks a lot for the answer Peter, it helped to clarify even more the idea $\endgroup$
    – nerdy
    Jun 29, 2013 at 20:24
  • $\begingroup$ @PeterSmith My question is related, so I decided to ask as an comment to possibly supplement the original question. When defining arbitrary sum in categorical language, one uses notions of X-index family of objects and maps. So, I wonder, if naive set-theory is being used here as a meta-language to express categorical notions. The usual binary sum can be easily expressed by using precise natural language or logic. I was quite confused by such a usage because I have never encountered such an expression. Any further clarification will be much appreciated. Thank you! $\endgroup$
    – Jaspreet
    Mar 10, 2018 at 4:29

I agree with Peter Smith's answer, but I would go further. There is no reason to imagine shapes at all for the symbols of the object language. They can be just as abstract as, say, numbers. I don't imagine the number 3 as having a shape (in contrast to symbolic representations of it, like "$3$" or "iii" or "three" or "drei"). Similarly, I don't imagine conjunction as having a shape (in contrast to symbolic representations like "$\land$" or "$\cdot$" or "&" or "and" or "und").

  • $\begingroup$ Indeed! (A side remark: One of the things that can puzzle readers of Gödel's original 1931 incompleteness paper is that he takes this abstract line and treats formulae of the object language as sequences of natural numbers [not number-signs].) $\endgroup$ Jun 29, 2013 at 19:45

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