I don't know of anyone who thinks natural language conveyance of mathematics is unimportant. — TonesInDeepFreeze

Fair. I was trying to convey the sense that there is this slightly annoying informal thing we have to do before we get on to doing math, properly, formally. And if you try to formalize that part ("We define a language L0, which contains the word 'Let', lower case letters, and the symbol '=', ..."), you'll find that you need in place some other formal system to legitimate that, and ― at some point we do have to just stop and figure out how to conceive of bootstrapping a formal system. And that bootstrapping will not be ex nihilo, but from the informal system ― if that's what it is ― that we are already immersed in, human culture, reasoning, language, blah blah blah.

I probably shouldn't have brought it up. It's another variation on the chicken-and-egg issue you pointed out.

Another way is to point to the coherency: There is credibility as both logic-to-math and math-to-logic are both intuitive and work in reverse nicely. — TonesInDeepFreeze

This is a nice point.

Circularity need not be vicious.

In particular, it's interesting to think of this whole complex of ideas as being "safe" because coherent ― you can jump on the merry-go-around anywhere at all, pick any starting point, and you will find that it works, and whatever you develop from the point where you began will serve, oddly, to secure the place where you started. And this will turn out to be true for multiple approaches to foundations for mathematics and logic.

Well that's just a somewhat flowery way of saying "bootstrapping" I guess.

Now I can't help but wonder if there's a way to theorize bootstrapping itself, but I am going to stop myself from immediately beginning to do that.

Thanks for very much for the conversation @TonesInDeepFreeze!

In particular, it's interesting to think of this whole complex of ideas as being "safe" because coherent ― you can jump on the merry-go-around anywhere at all, pick any starting point, and you will find that it works, and whatever you develop from the point where you began will serve, oddly, to secure the place where you started. And this will turn out to be true for multiple approaches to foundations for mathematics and logic. — Srap Tasmaner

Nice.

A working hypothesis: anything that can be said, can be said in a natural language. But not anything that can be said, can be said in a formal language.

What makes me hesitate to reduce logic to math has more to do with thinking about informal logic as still a part of logic, even though it doesn't behave in the same manner as formal logic — Moliere

If you wade through everything I've vomited here in the last day or so, I think you'll find me half backtracking on that ― although I still tend to think there's something like a "formal impulse" that you can scent underlying mathematics and logic, so perhaps even our informal reasoning. It's a very fog-enshrouded area.

It's already been mentioned a couple times in this thread that "follows from" is often taken as the core idea of logic, formal and informal. Logical consequence.

Another option is consistency, and it's the story that Peter Strawson tells (or told once, anyway) for the origins of logic: his idea was that if you can convince John that what he said is inconsistent, then he'll have to take it back, and no one wants to do that. So the core idea would be not whether one idea (or claim or whatever) follows from another, but whether two ideas (claims, etc) are consistent with each other. (I should dig out a quote. He tells it better than I do.)

Do you know about the ultimatum game? It's a standard experiment design in psychology, been done lots of times in all sorts of variations. You take pairs of subjects, and you offer one of them, say, $100, on this condition: they have to offer their partner a share; if the partner accepts the offer, they get the agreed upon amounts of money; if the partner refuses, they get nothing. ― Okay, I'm telling you that story (which you probably already know) because it's famous for completely undermining a standard assumption of rationality. Since the participants start with 0, the partner should be happy to get anything, to accept $1 out of $100, instead of walking away with nothing. But that's not what happens. The offers have to be fair, something close to 50-50. Not quite 50-50 is usually accepted, but lowball offers almost never are.

And the point is this: evidently, whether it's evolution or a cultural norm, we have a sense of fairness. And it can override what theory might say is rational. (The target here is Homo oeconomicus, the rational agent.)

Similarly, we might hunt for "logical consequence" or "consistency" as some sort of ur-concept upon which logic is built.

Said in natural language? Includes using parentheses to mark arbitrarily deep nested sub-sentences?

In natural language, how would you say?:

∀x ∃y ∀z ((P(x) ∧ ∃u (Q(y) ∨ (R(u) ∧ ∀v (S(v) → T(z, v))))) → ¬(∀w (U(w) ∧ ∃t (V(x, t) → W(t, w))) ∧ ∃p(X(p) ∧ ∀q (Y(q) → Z(p, q)))) ∨ (A(x, y, z) ∧ ∀b ∃c (D(b, c) → (E(x, b, c) ∧ ∃d (F(d) ∧ G(d, x, y)))))

And throw in some math and modal operators too.

∀x ∈ ℝ ∃y ∈ ℝ ∀z ∈ ℕ ((x² + y² = z² ∧ ∃u ∈ ℤ (sin(u) + cos(y) ≥ 1 ∨ (∫₀ᵘ eᵗ dt = eᵘ - 1 ∧ ∀v ∈ ℚ (v > 0 → d/dv (v²) = 2v)))) → ¬(∀w ∈ ℂ (|w| ≤ 1 ∧ ∃t ∈ ℝ⁺ (log(t) ≤ 0 → t ≤ 1)) ∧ ∃p ∈ ℕ (p! = ∏ₖ₌₁ᵖ k ∧ ∀q ∈ ℝ (q ≠ 0 → 1/q ≠ 0))) ∨ (A(x, y, z) ∧ ∀b ∈ ℝ ∃c ∈ ℝ (D(b, c) → (E(x, b, c) ∧ ∃d ∈ ℕ (F(d) ∧ G(d, x, y)))))) ∧ ∀r ∈ ℝ (◻H(r) → ◇I(r))

(I hope that's well formed and displays correctly - it was made by a bot.)

∀x ∃y ∀z ((P(x) ∧ ∃u (Q(y) ∨ (R(u) ∧ ∀v (S(v) → T(z, v))))) → ¬(∀w (U(w) ∧ ∃t (V(x, t) → W(t, w))) ∧ ∃p(X(p) ∧ ∀q (Y(q) → Z(p, q)))) ∨ (A(x, y, z) ∧ ∀b ∃c (D(b, c) → (E(x, b, c) ∧ ∃d (F(d) ∧ G(d, x, y))))) — TonesInDeepFreeze

For all x there exists a y for all z such that if P is a property of x and there exists a u such that -- Q is a property of y or u is a property of R and allv's such that if v is S then the orderd pair z,v is T then it is not the case for all w such that w is U and there exists a t such that if t,x is V then t, w is W AND there exists p such that p is X and All q such that if q is Y then the ordered pair p q is Z OR x, y, z is A and for all b there exists a c such that (if the ordered pair c, b is D then x, b,c is E and there exists a d such that d is F and d, x, y is G.



Obviously.

I guess that' similar to the prisoner's dilemma.

So the core idea would be not whether one idea (or claim or whatever) follows from another, but whether two ideas (claims, etc) are consistent with each other. — Srap Tasmaner

Okay, but consistency is defined in terms of derivability (which, in first order, is equivalent with entailment).

Of course, you can use special formatting to do it, with a convention as to what it signifies. But is that natural? That is, try to do it spoken.

Heh I wouldn't be surprised if I made a few mistakes.

I did write it out while following the symbols though :D -- but I take your point that it's not something I'd ever say outside of logic.

Impressive that you did it without a bot. I just let the bot give me the English translation, but it too used specially formatting - bullet points and indentations.

But don't let the gratuitous complexity distract from the point. We could find examples in actual mathematics that might not be so complex but still tough. And then in the primitive language.

yeah it seems hard to emphasize the parenthetical points if we only typed it out in lowercase without punctuation.

For writing, I would accept ordinary punctuation, but don't know about formatting or special characters given an ad hoc role.

Also, I recognize that the burden is not just to show that it would be difficult to use only English but that it would not be possible.

By the way, I greatly enjoyed the video linked in the 'Logical Nihilism' thread. I have a lot of thoughts about it, and a lot of reading to do about it, but just not the time to put it together as a good post now. — TonesInDeepFreeze

Cheers. I've been thinking and reading for three years. Still reading and thinking.

...how would you say? — TonesInDeepFreeze

With difficulty... "For anything, there is something..." and beyond that it gets cumbersome and potentially ambiguous, but can we be sure it could not be put into a page of explanation in not-so-plain English? Is a software licence in a natural language or in a formal language? Should we ask @Hanover? He does lawyering, apparently. Or feed it into ChatGPT...

In simpler terms, the statement is saying: For every choice of x, we can find a y so that, for all z, if certain conditions about x, y, and other variables are true, then one of two broad cases must occur: Either a set of conditions that mostly focus on a specific chain of relations among various entities don’t hold together. Or, a different set of conditions, involving relationships among several other variables, do hold. — ChatGPT

:wink:

The reason we have formal languages is that they make such things easier and clearer; a mediaeval logician would be in great difficulty.

But is that natural? — TonesInDeepFreeze

Yep. We might stipulate a definition of natural language... is French a different natural language to English, or are both just dialects of one natural language? What about Lao?

I'm not offering an answer here, just pointing out that the difference between formal and informal languages is more intractable than it might appear.

I'm not offering an answer here, just pointing out that the difference between formal and informal languages is more intractable than it might appear. — Banno

Hey, that's my job! :D

...replaced by AI...

Well, it's not the first time, it won't be the second time...

Similarly, we might hunt for "logical consequence" or "consistency" as some sort of ur-concept upon which logic is built. — Srap Tasmaner

Whether we can specify a form for "logical consequence" that will apply universally is the bone of contention in Logical Nihilism

If you get stuck I will pay you to weed my veggies.

Only if it pays in both carrots and tomatoes and turnips.

Parsnips, I do. Not too keen on turnips.

Oh, either/or. I'm not picky. It's the Epicure in me.

Cool. I will throw in the occasional beetroot.

Ooooo... makes good wine!

the difference between formal and informal — Banno

We do have a definition of 'formal language': the set of well formed formulas is a recursive set*; and perhaps add unique readability. [EDIT: That might be only a terse synopsis of a definition that might need refinement and other clauses. And I have in mind mainly the kind of languages used in mathematical logic.]

* More exactly, the set of Godel numbers of well formed formulas is a recursive set.

/

Might be interesting to adduce a formal sentence and demonstrate somehow that it can't be said in English alone (not just that all known attempts failed).

Oh yeah.

I guess that' similar to the prisoner's dilemma. — TonesInDeepFreeze

It's related, yes.

consistency is defined in terms of consequence — TonesInDeepFreeze

Suppose I hold beliefs A and B. And suppose also that A → C, and B → ~C. That's grounds for claiming that A and B are inconsistent, but only because C and ~C are inconsistent. How would we define the bare inconsistency of C and ~C in terms of consequence?

Or did you have something else in mind?

Now it could be that the LNC, so beloved on this forum, functions as a minimal inconsistency guard, and from that you get the rest. ― This is a fairly common strategy with programming languages these days, to define a small subset of the language that's enough to compile the full language's interpreter or VM or whatever.

It could also be that the "starter versions" of consequence or consistency look a little different. I've been reading about some interesting work with gorillas, which suggests they grasp some "proto-logical" concepts. Negation, for example, is pretty abstract, but they seem to recognize and reason about rough opposites ― here/there, easy/hard, that sort of thing. Researchers have worked up a pretty impressive repertoire of "nearly logical" thinking among gorillas, though obviously their results are open to interpretation.

Anyway, suggests another type of bootstrapping.

( Might be worth mentioning that it looks like we're in the presence of one of Austin's trouser words, since the goal in Strawson's story is avoiding inconsistency, and that's what naturally came to mind above. )

Is a software licence in a natural language or in a formal language? — Banno

I speak from the Anglo legal perspective, particularly American. Ambiguity in contracts feeds an industry, and even should there be clarity, ambiguity will be argued because the value of one's claim or obligation will be greatly affected by what the word means.

But in the law, we have a whole system to decide what things mean. And they mean what the person or people authorized to say it means.

But Americans like risk, so we keep things vague and subject to argument. Trials and hearings have the element of surprise, so compromises become of great value. We over pay sometimes just for certainty.

What this has to do with logic is that any argument goes so long as it's colorable. And this sparks creativity if you enjoy such chaos.

So how do you know what's what? You rely upon past decisions, and the art of the analogy and the ability to distinguish comes into play. Such is the significance of precedent. That we've been wrong for 100 years might hold more sway than a rigorous reevaluation. If you can't have clarity from the past, you'd have it nowhere.

Persuasion is the skill of the lawyer. Sometimes that has to with other than being strictly right. But what is "right" anyway?

A man posts a vague and somewhat mysterious advertisement for a job opening. Three applicants show up for interviews: a mathematician, an engineer, and a lawyer.

The mathematician is called in first. "I can't tell you much about the position before hiring you, I'm afraid. But I'll know if you're the right man for the job by your answer to one question: what is 2 + 2?" The mathematician nods his head vigorously, muttering "2 + 2, yes, hmm." He leans back and stares at the ceiling for a while, then abruptly stands and paces around a while staring at the floor. Eventually he stops, feels around in his pockets, finds a pencil and an envelope, and begins scribbling fiercely. He sits, unfolds the envelope so he can write on the other side and scribbles some more. Eventually he stops and stares at the paper for a while, then at last, he says, "I can't tell you its value, but I can show that it exists, and it's unique."

"Alright, that's fine. Thank you for your time. Would you please send in the next applicant on your way out." The engineer comes in, gets the same speech and the same question, what is 2 + 2? He nods vigorously, looking the man right in the eye, saying, "Yeah, tough one, good, okay." He pulls a laptop out of his bag. "This'll take a few minutes," he says, and begins typing. And indeed after just a few minutes, he says, "Okay, with only the information you've given me, I'll admit I'm hesitant to say. But the different ways I've tried to approximate this, including some really nifty Monte Carlo methods, are giving me results like 3.99982, 3.99991, 4.00038, and so on, everything clustered right around 4. It's gotta be 4."

"Interesting, well, good. Thank you for your time. I believe there's one last applicant, if you would kindly send him in." The lawyer gets the same speech, and the question, what is 2 + 2? He looks at the man for a moment before smiling broadly, leans over to take a cigar from the box on the man's desk. He lights it, and after a few puffs gestures his approval. He leans back in his chair, putting in his feet up on the man's desk as he blows smoke rings, then at last he looks at the man and says, "What do you want it to be?" — Srap Tasmaner

Suppose I hold beliefs A and B. And suppose also that A → C, and B → ~C. That's grounds for claiming that A and B are inconsistent, but only because C and ~C are inconsistent. — Srap Tasmaner

Okay.

How would we define the bare inconsistency of C and ~C in terms of consequence? — Srap Tasmaner

What sense of "consequence"? Entailment?

Do you mean how to define 'inconsistent'?

(First order in this post and generally in posts unless said otherwise.)

Or how to show that {C, ~C} is inconsistent? It's trivial. {C ~C} |- C & ~C. But yes, that uses conjunction intro, which is deduction. And since we have {C ~C} |- C & ~C, we have {C ~C} |= C & ~C. (soundness)

Or semantically, it's trivial to show that {C, ~C} is unsatisfiable. And we have that any unsatisfiable set is inconsistent. (completeness)

Or, we could define 'inconsistent' as "proves a formula C and proves ~C". Then, even more trivially, {C ~C} |- C and {C ~C} |- ~C. But even that uses a deduction rule (whatever you call it - inferring a sentence by virtue of it being in the set of premises.) And since we have {C ~C} |- C and {C ~C} |- ~C, we have {C ~C} |= C and {C ~C} |= ~C.


/

Df. a set of sentences is inconsistent if and only if it proves a contradiction.

Th. a set of sentences is inconsistent if and only if it entails a contradiction.

Df. a set of sentences is satisfiable if and only if there is an interpretation in which all the sentences are true.

Th. a set of sentences is inconsistent if and only if it is not satisfiable.

Now it could be that the LNC, so beloved on this forum, functions as a minimal inconsistency guard, and from that you get the rest. — Srap Tasmaner

I don't know what you mean by "minimal inconsistency guard".