# construction of well-formed formulas

## Primary tabs

Type of Math Object:
Feature
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### Proposition : Sign or Object?

There's a generic issue that comes up here, so I thought I'd open it up for discussion.

Most of the time we can get away with using the word "proposition" in an ambiguous way -- to mean either a syntactic expression or a mathematical object, but there are times when we have to observe the distinction between a string in a formal language and one or another formal objects that we probably have in mind denoting with that string.

The simplest tactic I know for dealing with that is the one I learned in programming, which is to use words like "function" or "proposition" to describe the objects denoted and to use phrases like "function name" or "propositional expression" when we want to be clear that we are talking about a syntactic thingy.

The same issue arises with respect to a term like "connective", which is properly speaking a syntactic thing, and that is not itself a mathematical operation, but only used, when interpreted, to denote one.

### Re: Proposition : Sign or Object?

I agree that "proposition" isn't the optimal choice in this context. It has been used to denote so many diverse concepts that it inevitably brings with it considerable historical baggage. In addition, the consensus holds propositions to be intensional entities and, therefore, outside the realm of set-theoretic mathematics.

### Re: Proposition : Sign or Object?

Maybe I should just use "well-formed formulas" or "terms" to denote such strings, or, like you said, "propositional expressions"... and connectives should really be "logical symbols"? What is your preference?

### Re: Proposition : Sign or Object?

Actually my preference is the same as yours -- it's just that there are expository situations that force us to draw the distinction between the syntax denoting and the object denoted a little more clearly, especially in pedagogy or in translating between different conventions of usage.

Any extra bit phrasing like "expression", "formula", or "sentence" works well enough when you need to emphasize that you are talking about a string in a specific calculus or language. That was the computer science strategy, and it worked well enough with the least bit of fuss. We don't really want to get all model-theoretic in writing an intro to rudiments of logic.

Probably best to avoid "term" as many writers in logic use that for noun phrases like "x" and "f(x)" as opposed to clauses and sentences that have truth values, and some sources even use "term" for the denoted entity(!)

### Re: Proposition : Sign or Object?

It's a handy enough word, just so long as we make it clear what tradition of usage we are following in what context.

I don't think there's any chance we'll reform the usage of any tradition that has an established following -- about all we can do is navigate this or that passage between the various conventions.

I must have missed the vote on that intensional entity proposition -- maybe my absentee ballot got lost in the post?

### Re: Proposition : Sign or Object?

Jon Awbrey wrote:

"I must have missed the vote on that intensional entity proposition -- maybe my absentee ballot got lost in the post?"

I had in mind the vast literature associated with "propositional attitudes", etc.: http://plato.stanford.edu/entries/propositions/ .

### Re: Proposition : Sign or Object?

Sure, I was just trying to avoid wandering too far afield from the expository issue at hand.

It's not really all that necessary to discuss here, not just yet, but Peircean pragmatists take a different attitude toward propositional attitudes than what we get from the Frege-Russell school of thought. Pragmatists formalize interpretive attitudes in the medium of of triadic sign relations, and the more rudimentary properties of these can be handled moderately well in set theory or even in a more categorical style of relation theory.