I think of mathematical logic sub-subject of formal logic.

I guess because logic and formal logic have under philosophy, and mathematical is a part of formal logic, we have mathematical logic under the wide umbrella too.

Mathematical logic is typically a course in Mathematics. But sometimes such things as set theory are taught as a Philosophy course.

Symbolic logic is often found as a Philosophy course. But it can be a warmup for mathematical logic, and it usually includes translation of natural language arguments; even if one goes on to use symbolic logic mainly for mathematics, it helps to first know how to translate natural language since so much of mathematical prose is in natural language.

And, of course, mathematical logic is extended in formal logics regarding all kinds of philosophical subjects - modal, epistemic, etc. And, of course, in philosophy of language. And, prominently, computing.

And, of course, philosophy of mathematics is steeped in considerations about mathematical logic, set theory and mathematics.

What does mathematics get out of pretending it's importing logic from elsewhere? — Srap Tasmaner

What pretending? Would you mention a specific writing?

you have to have sets (or an equivalent) to do much of anything in the rest of mathematics, but so what? — Srap Tasmaner

Set theory axiomatizes classical mathematics. And the language of set theory is used for much of non-classical mathematics. Those are two answers to "so what?"

Propositional logic deals in propositions. Your piece has the form of a modus ponens, but doesn't deal in propositions. That makes it interesting in several ways. But "not-a" is pretty well defined in propositional logic, in various equivalent ways. And by that I mean that the things we can do with negation in propositional logic are set. There are not different senses of "not-A" in propositional calculus. — Banno

OK thanks. It does seem to be a propositional statement in ordinary langauge.

1. If there is life (A) there is death (not-A)
2. There is life
3. Therefore there is death.

But as I've said before my undertsnding of formal logic leaves much to be desired.

Absolutely sure. — TonesInDeepFreeze

I'm okay with that.

The chicken and egg still bothers me, though, so one more point and one more question.

Another issue I have with treating logic as just "given" in toto, such that mathematics can put it to use, is that one of the central concepts of modern logic is nakedly mathematical in nature: quantifiers. If you rely on ∃ anywhere in constructing set theory (so that you can construct numbers), you're already relying on the concept of "at least one", which expresses both a magnitude and a comparison of magnitudes. Chicken and egg, indeed.

And if you need to identify the formula "∅ ⊂ ∅" as an instance of the schema "P → P", then you also have to have in place the apparatus of schemata and instances (those objects of Peter Smith's unforgiving gaze), which you presumably need both quantifiers and sets ― or at least classes of some kind ― to define rigorously. More chicken and egg.

And since we're wallowing in the muddy foundations , a quick question: somewhere I picked up the idea that all you need to add to, say, classical logic is one more primitive, namely ∈, in order to start building mathematics. I suppose you need the concepts (but no definitions!) of member and collection as what goes on the LHS and RHS respectively, but that's it. And there just is no way around ∈, no way to cobble it together from the other logical constants. Is that your understanding as well? Or is there a better way to pick out what logic lacks that keeps it from functioning as itself the foundations of mathematics?

What pretending? — TonesInDeepFreeze

Just a tendentious turn of phrase, not important.

Someplace to start writing without having to explain yourself. — fdrake

Kinda what I think. Also, at some point you'll have to say to the kiddies something like "group" or "collection" and just hope to God they know what you mean, because there is nothing anyone can say to explain it.

I think of mathematical logic sub-subject of formal logic. — TonesInDeepFreeze

Certainly. I almost posted the same observations about the dual existence of logic courses and research in academic departments (logic 101 in the philosophy department, advanced stuff in the math department, and so on).

― ― I suppose another way of putting the question about formal logic is whether we could get away with thinking of its use elsewhere, not only in the sciences, but in philosophy and the humanities, as, in essence, applied mathematics.

Set theory axiomatizes classical mathematics. And the language of set theory is used for much of non-classical mathematics That's one so what. — TonesInDeepFreeze

Sure sure, my point was to suggest that logic could live here too, and I'm really not sure why it doesn't. Set theory is needed for the rest of math and so is logic. There's your foundations, all in a box, instead of logic coming from outside mathematics ― that's what I was questioning, am questioning. (I suppose, as an alternative to reducing it to something acknowledged as being part of mathematics, which I admit doesn't seem doable.)

because there is nothing anyone can say to explain it. — Srap Tasmaner

You can give a few token examples of... collections... using blocks, hands of cards, sweeties, square of chocolate, desks in their class, and just hope that the kid can educe the idea of a collection through analogy. Eventually.

Though it is incredibly hard if someone struggles with analogies and abstractions.

There's your foundations, all in box, instead of logic coming from outside mathematics ― that's what I was questioning, am questioning. (I suppose, as an alternative to reducing it to something acknowledged as being part of mathematics, which I admit doesn't seem doable.) — Srap Tasmaner

Eventually the metalanguages terminate in predicates applied to undefined primitives and natural language statements. At some point that always happens. Even though such statements have clear conceptual content - though one might need to flesh that content out by derivation.

At the outset of talking about unions (U) and intersections (/\), we get an interesting consideration.

For any S, US = {x | there is a y in S such that x in y}

For any non-empty S, /\S = {x | for all y in S, x in y}.

Why non-empty?

U0 no problem. U0 = 0.

But why no /\0? Because:

Roughly put, the subset axiom is: For any set x and describable property P, there is the subset of x whose members are all and only those members of x that have property P.

Now, there is no set of which every set is a member. Why? Because:

Suppose there is a set V such that every set is a member of V. Then there would be the subset of V of all the sets that have the property of not being a member of itself. Then we have Russell's paradox.

So there is no such V.

Now, suppose there is /\0. But every set would be a member of /\0, as seen:

For all y, it is not the case that y in 0. So for all y, if y in 0 then x in y. So every x would be in /\0, so /\0 would be the universal set V, but there is no such set.

Note: I used the word 'because' in the sense of 'since' not causality.

natural language statements — fdrake

It's curious when you notice that mathematics textbooks have no alternative to saying things like "Let x = the number of oranges in the bag", and if you don't say things like that, you might as well not bother with the rest. (For similar reasons, doing it all in some APL-like symbolism would work, but no one would have any idea what the symbolism meant, if you didn't have "∈ means is a member of" somewhere.)

And if you have natural language, you have how humans live, human culture, evolution, and all the rest. There's your foundations.

Don't know what you're driving at. People use variables outside of math books too.

Don't know what you're driving at. — TonesInDeepFreeze

I imagined Srap and I were talking about how the formalism in mathematics doesn't start at its "grounds", in the axioms. I imagine Srap and I are reacting to an imagined enemy of a formalist who thinks that mathematics is somehow "just" symbol manipulation. Or alternatively just awed at how the root of the formalism is in as something as messy as natural language, despite how set in stone - settable in stone - the concepts of mathematics seem to be.

Just that there's at least here a dependence of mathematics on natural language, which gives the appearance of being purely pedagogical, or unimportant "set up" steps (still closely related to the thing about logical schemata, from above).

Algebra books set up problems this way, with a little bit of natural language, and then line after line of symbolism, of "actual" math.

If you get nervous about there being such a dependency, you might shunt it off to something you call "application".

I'm just wondering if the dependency is ever really overcome, especially considering the indefinability of "set" for example.

I keep throwing in more issues related to foundations, sorry about that.

...there is life... — Janus

Yep, nice. But is "There is life" then the negation of "There is death"? If we pars "There is life" as "there is something that is alive" then it's negation is "it is not the case that there is something that is alive", which is not the same as "there is something that is dead". There are things that are neither alive nor dead, so being dead is not the negation of being alive.

Or do we pars the whole first assumption as "everything that is alive will die" in which case we have an implicit temporality in "will" and need to include time. But then nothing is both alive and dead at the same time.

That is, in setting the passage out as a series of proposition, the negation dissipates.

Yeah I think we're thinking about the same things.

But then nothing is both alive and dead at the same time. — Banno

Right. I think this is the nub. 'Not-A' should strictly be the negation of 'A'. We cannot say 'if something is alive, then it is dead' even if we can say 'if something is alive then it will be dead'.

Also, in ordinary language 'not-A' can alternatively be anything which is not A. As you point out death is not-A in the second sense but it not strictly the negation of life. The strict negation of life would be no life. Language is messy.

, Isn't formal language a part of natural language?

And Mathematics, also?

We understand each by what we do with it. Or rather, to understand a language is to be able to make use of it. Sets are not only defined by rules, but by our actually putting things into groups.

PI §201, yet again. There is a way of understanding a rule that is not found in setting it out but in following it.

Isn't formal language a part of natural language? — Banno

Not transparently so, to me? Consistent systems capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. But natural language does contain its own truth predicate and behaves... well it doesn't disintegrate. That's at the very least a type distinction between consistent formal systems and natural language - one can contain its own truth predicate without being crap, one cannot.

logic as just "given" in toto — Srap Tasmaner

It's more like the chicken and the egg (as you mention later in your post).

You can take the logic as given to base math on it.

And you can take a certain amount of math as given to base logic on it.

Choose your chicken or your egg.

Even formally: You need predicate logic as a basis for Z set theory. And, at least in usual formulations, you need at least some finitistic math to formalize the predicate calculus.

One way of thinking a way out of the bind is to take successive meta-theories, but that would be ad infinitum.

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.

Another way would be just to display the code for the formal theory without explanation or verification of any aspect of it, and then put in sequences of formulas and see which are ratified as proofs. That is, suppose you uploaded it to a highly intelligent life form, without explanation, and let those creatures discern that it works. Personally, though not necessarily philosophically, that tack doesn't appeal to me.

But, I don't see how to disagree that yes, ultimately, as humans, since we have finite time and can't in a lifetime escalate meta-theories infinitely, ultimately it will boil down to ostensive understanding, just as so much of thinking and use of language seems to do.

member and collection — Srap Tasmaner

Most writers seem to view 'member' and 'set' (or 'class' depending on the treatment) as the base notions. But formally we can do it with just 'member'.

there just is no way around ∈, no way to cobble it together from the other logical constants. — Srap Tasmaner

(1) 'e' is a non-logical constant.

(2) It is not precluded that we may define 'e' from primitives. von Neumann for example.

Just a tendentious turn of phrase, not important. — Srap Tasmaner

'tendentious' is definitely the Word of the Week. The runner-up is 'flows'.

whether we could get away with thinking of its use elsewhere, not only in the sciences, but in philosophy and the humanities, as, in essence, applied mathematics. — Srap Tasmaner

I must misunderstand you? Famously, formal logic is not just studied in philsophy but applied in philosophy.

Yep. Curious how rendering them into a more formal language detracts from their use.

:up:

I imagined Srap and I were talking about how the formalism in mathematics doesn't start at its "grounds", in the axioms. I imagine Srap and I are reacting to an imagined enemy of a formalist who thinks that mathematics is somehow "just" symbol manipulation. Or alternatively just awed at how the root of the formalism is in as something as messy as natural language, despite how set in stone - settable in stone - the concepts of mathematics seem to be. — fdrake

Just to be clear, one form of formalism is the extreme view that mathematics is mere symbol games. But formalism is not at all confined to such an extreme view.

/

Yes, even an extreme game formalist would have to admit that eventually we have to communicate in natural language. (Though he might try the "write the code and send it up in a spaceship to the advanced intelligence creatures" argument.)

And variables are used in both mathematics and everyday discourse. I can see the shape of an argument against the extreme game formalist based on the fact that variables (for example) probably originated naturally. So the argument is this?: Mathematics needs variables, and variables are ultimately understood naturally not formally.

Consistent systems capable of first order arithmetic can't contain their own truth predicate — fdrake

I'm not so sure of this, since Kripke's theory of truth contains it's own truth predicate, and there is considerable work around its relation to arithmetic, I don't think we can yet rule out a Kripke-style first order arithmetic. I might be mistaken.

That is, the creativity of logicians is such that it might be better not to specify such a demarcation between formal ind informal languages, lest they invent a counter instance.

Just that there's at least here a dependence of mathematics on natural language, which gives the appearance of being purely pedagogical, or unimportant "set up" steps (still closely related to the thing about logical schemata, from above). — Srap Tasmaner

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

the indefinability of "set" — Srap Tasmaner

'set' can be defined from 'element of'.

Keep 'em coming.

I like to be corrected -- it helps me learn. Alot of this is fuzzy in my head so I'm all ears -- formal training was an eternity ago, light, and now I just read logic books on my own in my free time for fun like a nerd.

Learning is one of the nice things about TPF.

Isn't formal language a part of natural language? — Banno

I agree with on truth, or at least that's basically been an intuition that my other argument in the other logic thread relies upon, and I'm suspicious of substitution with respect to natural language -- it has more boundaries to it than we'd formally expect. That's why I conceded the point to @TonesInDeepFreeze about ironic statements, in natural language, don't fit the form of the OP.

@Srap Tasmaner -- My introduction to propositional logic and set theory came from a math class, so I do think there's some overlap between math and logic. 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 -- at least by my consideration. I can understand a reductio without a formalization of it, and it always seems to me that that underlying, vague intuition of reasoning is basically what we check our formalisms against, in particular circumstances.

Isn't formal language a part of natural language? — Banno

In one view, we have a formal object-language, and an informal or formal meta-language that includes the formal object-language.

I don't have the background to think through this unfortunately.

Consistent languages capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. But natural language does contain its own truth predicate and behaves... well it doesn't disintegrate. That's at the very least a type distinction between consistent formal languages and natural language - one can contain its own truth predicate without being crap, one cannot. — fdrake

I'm not sure that I recall Tarski correctly (perhaps he does mention the notion of a 'consistent or inconsistent language'?) But usually languages are not consistent of inconsistent; sets of formulas are consistent or inconsistent. In this part, he's talking about an interpreted formal language. We have that there would be a contradiction (in the meta-theory, whether formal or informal) if such an interpreted formal language had its own truth predicate.

Yes, aside from paraconsistency, we would not comfortably bear contradiction, while we can bear paradox in natural language. But, we need to keep in mind that wide-open natural language doesn't provide the desiderata of formal languages.

In one view, we have a formal object-language, and an informal or formal meta-language that includes the formal object-language. — TonesInDeepFreeze

Yes, and I rather like that. But as I understand it, Kripke's theory of truth involves one language, avoiding separating a meta language from an object language. It does this by only assigning a truth value to certain formulae, not to all.

The point is not that Kripke's theory of truth can be used as a basis for arithmetic (That seems to be a topic of some discussion amongst the academicians). So while fDrake is quite right,

Consistent systems capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. — fdrake

Kripke's system plays with consistency, creating a formal language that contains it's own truth predicate.

Hence, it might be premature to use "not containing it's own truth predicate" as a way to demarcate between formal and natural languages.

And a further point, whatever such demarcation might be offered, some clever logician might find a way to undermine it.

All speculative.

Interesting. Very much bears looking into.

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.

Yep - well beyond my level, but I wish it wasn't.

So rounding back to your chat with , I'm reticent to place any firm boundary between formal and natural languages. Of course we could specify such a boundary, arbitrarily. That's cheating.

From what I've understood, Kripke also avoids being strictly paraconsistent becasue he does not use a third truth value, but just does not assign a truth value at all to oddities like the Liar.

Very clever.