I don't know of anyone who thinks natural language conveyance of mathematics is unimportant. — TonesInDeepFreeze
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
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
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
∀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
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
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
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......how would you say? — TonesInDeepFreeze
:wink: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
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?But is that natural? — TonesInDeepFreeze
Whether we can specify a form for "logical consequence" that will apply universally is the bone of contention in Logical NihilismSimilarly, we might hunt for "logical consequence" or "consistency" as some sort of ur-concept upon which logic is built. — Srap Tasmaner
the difference between formal and informal — Banno
I guess that' similar to the prisoner's dilemma. — TonesInDeepFreeze
consistency is defined in terms of consequence — TonesInDeepFreeze
Is a software licence in a natural language or in a formal language? — Banno
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
How would we define the bare inconsistency of C and ~C in terms of consequence? — Srap Tasmaner
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
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