First-order logic without sets?

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Discussion Overview

The discussion revolves around the challenges of studying first-order logic without relying on set theory. Participants express concerns about the circularity of definitions when set theory concepts are used to explain first-order logic, and seek resources that focus solely on first-order logic as a foundation for later studying set theory.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant notes that many books on first-order logic use set theory concepts early on, which they find problematic for establishing a clear logical path from first-order logic to set theory.
  • Another suggests starting with references for "symbolic logic," which may de-emphasize set theory, and introduces the distinction between sets and classes, proposing that class theory can be developed alongside first-order logic without circularity.
  • A participant emphasizes the importance of formal proofs in first-order logic and mentions a specific book that they find useful for its practical applications and theoretical discussions.
  • Concerns are raised about definitions in first-order logic that incorporate set-theoretic concepts, which some participants argue leads to circular reasoning.
  • One participant challenges the necessity of numbering axioms in mathematical theories, arguing that the concept of propositions suffices without the need for numerical labels.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of set theory in defining first-order logic, with some agreeing that circular definitions are problematic while others believe that the concepts can coexist without contradiction. The discussion remains unresolved regarding the best approach to studying first-order logic independently of set theory.

Contextual Notes

Participants highlight limitations in existing literature, particularly the reliance on set theory in foundational discussions of first-order logic, which may not align with their preferred learning path.

Who May Find This Useful

This discussion may be of interest to students and educators in mathematical logic, philosophy of mathematics, and those exploring foundational aspects of logic and set theory.

  • #31
diotimajsh said:
Hmmm. How much set theory do you want to rule out, jordi? If a system of logic uses the concepts of membership and quantification but doesn't start out with anything else set-theoretical, would that be too much?

No, not at all: it would be exactly what I am looking for.
WVO Quine had a 1937 paper, "New Foundations for Mathematical Logic", where he built up some other key notions for set theory from just the above two primitive notions plus the Scheffer stroke (logical NAND). It's kind of a revised (and very readable) version of something Russell and Whitehead did in Principia Mathematica. While Russell and Whitehead attempted to show how all of mathematics (as known at the time) could be derived from logical principles, Quine's paper shows how to get to the Principia's starting point from a somewhat more minimal set of assumptions. (Consequently, it's waaaaay shorter and much more to the point; and since it was written later than the Principia, it uses rather less obnoxious notation.)

"New Foundations" is on JSTOR, but I could email you a copy if you don't have access.
I do not have access to JSTOR. Would it be too much abusing if I sent to you my email address via PM?


For something more modern though, what kind of logic textbooks have you been looking into? If you've been looking primarily at mathematical logic books, maybe you should try a philosophy-oriented logic book. Those tend to focus on intuitive logical principles first and then work toward set theory later.

(Unfortunately, I don't have any really great recommendations. I used Hurley's A Concise Introduction to Logic as an undergrad, I think, but I didn't really like it, and I can't remember whether it resorts to set theory for its definitions.)

I tend to be quite picky with the books I use, and the philosophy-oriented logic books (I have browsed some of them) are too verbose for me.

What I would like is a modern book, with the title "everything you need before starting with set theory" (ie, logic and language). But probably the Quine's paper would be enough.
 
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  • #32
jordi:

"X is a (blank)" is another way of saying "X belongs to the set of all (blank)".

Being something is a set-theoretic notion.
 
  • #33
csprof2000 said:
jordi:

"X is a (blank)" is another way of saying "X belongs to the set of all (blank)".

Being something is a set-theoretic notion.

False. Proof: let (blank) := set

"X is a set" is meaningful. "X belongs to the set of all sets" is not meaningful, because the set of all sets does not exist.
 
  • #34
A second argument, more a philosophical one, is that "is a" is a basic concept. Instead, sets are a "secondary" concept, based on basic concepts.
 
  • #35
jordi said:
No, not at all: it would be exactly what I am looking for.

I do not have access to JSTOR. Would it be too much abusing if I sent to you my email address via PM?
Certainly not, go ahead.

So, glancing through this paper again, it turns out it does use somewhat more outdated notational conventions than I'd thought. Turns out the version I'm familiar with was re-released in a compilation of essays (Quine's From a Logical Point of View), and they must have updated the notation then.

I dunno, see what you make of it. Hopefully this older version is still useful to you. I find it a tad confusing to follow though.
 
  • #36
jordi said:
False. Proof: let (blank) := set

"X is a set" is meaningful. "X belongs to the set of all sets" is not meaningful, because the set of all sets does not exist.
More detail please. In this thread, we have used the word "set" in at least three different senses:
(1) A set[/color] in some model of ZFC[/color]
(2) A set[/color] in some model of ZFC[/color]
(3) A synonym for "logical predicate"

Furthermore, the corresponding logical predicate would be
(blank) is a predicate​
which, incidentally, is also meaningless.


(Unless, of course, you meant it as a metalogical predicate, or as a predicate[/color] about strings[/color] -- but then that wouldn't have anything to do with the "logic = set theory" correspondence)



And, for the record, there are forms of set theory that do have a set of all sets.
 

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