A chaotic (inconsistent) world in which the definiendum no longer entails the definiens and vice versa? — Pippen

Perhaps the first thing to point out is that it's not at all clear what this could possibly mean: if a definiendum no longer entails its definiens, in what sense could the one count as a definiendum, and the other, its definiens? What could it mean to say that tomorrow, bachelors might no longer be unmarried men? Does it mean that they will all be married (by forced decree, perhaps?): but this would be an 'empirical' change. So it can't be that. But if not that, then what? But that's what Hume was concerned with: experience. But if not experience, then - it's not clear what it could even mean to extend the problem of induction to logic and math.

Sometimes I think things can disappear

There exists disagreement as to the extent to which Hume considered the problem of induction to be epistemological versus metaphysical. Considered epistemologically, the problem of induction is simply the problem of predicting the physical consequences of an experiment, given that there are invariably rival explanatory hypotheses that are equally agreeable with respect to general principles of logical deduction or causation.

However when considered ontologically, logical deduction is also called into question. The underlying issue is that there doesn't exist agreement regarding the relationship of deduction to induction. For example, deduction might be considered to be special case of induction in which there is believed to exist perfect certainty for a conclusion with respect to a given premise.

Some philosophers identify deduction with the semantic notion of synonymy, yet for most people for most of the time, deduction is identified with the consequences of physical calculation as demonstrated by our reliance on computers. Hence deduction in practice is treated as a special case of (fallible) induction.

There is no problem of induction.

That is the way the world is. There is no certainty. Certainty about the real world is an illusion or perhaps a delusion.

Hume's problem of induction is not just about certainty. It's about plausibility as well - any kind of empirical inference. And it's a problem, if Hume's or similar arguments have force, because we clearly don't believe their conclusion. So you can oppose the argument (that we have no warrant for empirical inferences) or you can bite the bullet and accept its destructive conclusion, but you can't deny that there is a problem here.

A problem is only a problem if you think it's a problem.

What are these destructive conclusions of which you speak?

What prevents us from imagining that we all wake up tomorrow and a circle is no longer round, because we find ourselves in a chaotic (inconsistent) world in which the definiendum no longer entails the definiens and vice versa? — Pippen

Knowing that we'll not call squares by any other name...

Hello.
If I understand correctly, you are asking if, if it is possible for some constant to change, then can the laws of logic change too?

The answer is no, because asking if a thing is "possible" is to ask if it is "logically possible". Thus the statement "it is logically possible for logic to change" is nonsense. The laws of logic is the reference point around which everything can possibly change. The reference point cannot change around itself.

I really do not get it, why Hume judged at his time, it was possible at any time, the sun could not rise tomorrow (experience as an unreliable source of knowledge) — Pippen

Nowadays, Hume's intuition about the sun is considered to be quite right:

The Solar System will remain roughly as we know it today until the hydrogen in the core of the Sun has been entirely converted to helium, which will occur roughly 5 billion years from now. This will mark the end of the Sun's main-sequence life. At this time, the core of the Sun will contract with hydrogen fusion occurring along a shell surrounding the inert helium, and the energy output will be much greater than at present. The outer layers of the Sun will expand to roughly 260 times its current diameter, and the Sun will become a red giant. Because of its vastly increased surface area, the surface of the Sun will be considerably cooler (2,600 K at its coolest) than it is on the main sequence.[51] The expanding Sun is expected to vaporize Mercury and render Earth uninhabitable.

that from p and p -> q from tomorrow no more q follows (logic as an unreliable source of knowledge) — Pippen

As long as P does not change, then Q will keep necessarily following. If P is a Platonic abstraction, then there are no reasons for it to change.

For example, Pythagoras' theorem will forever remain provable from Euclid's classical axioms. You would have to modify Euclid's Platonic abstractions, i.e. the basic beliefs (axioms) that construct the abstract world of classical geometry, to effect a change that would render Pythagoras' theorem unsound.

You could also try to modify the axioms of first-order logic to make the inferences, in the proof for Pythagoras' theorem, invalid.

The difference between Hume's physical world and the abstract, Platonic worlds on which logic operates, is that there are no changes possible outside our control in abstract, Platonic worlds.

What prevents us from imagining that we all wake up tomorrow and apply other logical rules? — Pippen

You actually can. Hilbert calculi are an exercise on doing exactly that:

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege[1] and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.[1] Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference.

It does not seem to be possible to create more powerful systems of logic by adding axioms:

Because Hilbert-style systems have very few deduction rules, it is common to prove metatheorems that show that additional deduction rules add no deductive power, in the sense that a deduction using the new deduction rules can be converted into a deduction using only the original deduction rules.

In fact, this is a well-known phenomenon in mathematics. Beyond the basic construction, adding axioms usually does not make a system stronger. The system rarely knows more through these new axioms. It will just trust more.

For example, arithmetic theory can "see" all theorems and proofs of all other systems, including the "more powerful" ones, such as set theory. So, it actually knows these theorems, but it does not trust them. More axioms often means that your system does not become more knowledgeable or more powerful, but only more gullible.

What are these destructive conclusions of which you speak? — A Seagull

I just said, didn't I? (Assuming you are replying to me.)

Said what?

A problem is only a problem if you think it's a problem. — A Seagull

No, that's not how problems usually work.

What are these destructive conclusions of which you speak? — A Seagull

The undermining of knowledge. Biting the bullet is admitting that the ancient skeptics were right.

But if not experience, then - it's not clear what it could even mean to extend the problem of induction to logic and math. — StreetlightX

Which gives logic and math a kind of atemporal, aspatial quality. Which is odd, given that we inhabit temporal, spatial universe of change.

For example, deduction might be considered to be special case of induction in which there is believed to exist perfect certainty for a conclusion with respect to a given premise. — sime

This makes sense only if you ignore the difference between probability and certainty. Any conclusion with a probability less than a 100% is from an inductive argument and anything else (100% probabilty/certainty) is from a deductive argument.

Also induction has fewer forms among which arguments from analogy and statistical arguments are the only ones I remember.

Deduction, on the other hand, probably has an infinite number of valid forms. If not they at least outnumber the forms available in induction.

If I recall correctly the problem of induction, although exposing a limitation of inductive logic, is more about a specific class of induction viz. science where statistics plays a huge role as conclusions about how nature works are drawn from a finite sample space of observations.

1. The present will resemble the past.

Why?

Because

2. The present has resembled the past

Why?

Because

3. The present has resembled the past

As you can see this is a circular, ergo fallacious, deductive argument.

:up:

https://thephilosophyforum.com/discussion/comment/309821

As long as P does not change, then Q will keep necessarily following. — alcontali

Why? P, P -> Q | Q is just right because it follow from some rules. But these rules can change overnight, can they? So MP could be true today but false tomorrow. Imagine - overnight - our world becomes weird in the way that it becomes impossible to construct any implication P -> Q (~P v Q). I know it's hard to imagine, but I can just write it down and say: so be it from henceforward. Obviously in such a world you could not conclude anymore Q from P & P -> Q because it wouldn't be a wff at all. But somehow Hume and basically all philosophers after him disregard such a possibility.

Why? P, P -> Q | Q is just right because it follow from some rules. But these rules can change overnight, can they? — Pippen

It doesn't follow from a rule in the sense of a social construct. Something that people decided to do. It follows from the way human minds work. It's possible human minds change overnight, but we, being human minds, wouldn't notice.

So MP could be true today but false tomorrow. Imagine - overnight - our world becomes weird in the way that it becomes impossible to construct any implication P -> Q (~P v Q). I know it's hard to imagine, but I can just write it down and say: so be it from henceforward. — Pippen

The world cannot change logic, other than changing human minds. You can write it down, but can you actually believe it?

Kant got there before you, but made the same point. Take Hume's starting point that everything is, fundamentally, sensations and ideas, and then mathematics and logic rest on the same basis as chemistry and physics and whatever other special science you care to mention. Kant thought that this refuted Hume's position, since mathematics and logic are clearly not open to the same sceptical challenges as the inductive procedures that underpin those sciences. At least, that's what Kant thought. He was probably wrong, and there is a whole tradition of philosophy, pragmatism included in it, that really does see no difference in kind between logic and biology, just degrees.

Why? P, P -> Q | Q is just right because it follow from some rules. But these rules can change overnight, can they? — Pippen

Propositional logic is an axiomatic theory, derivable from 14 arbitrary, speculative beliefs with no justification. We have no clue as to why these axioms are the starting-point beliefs of this system. If we knew, then they would not be axioms.

In the Platonic view, we sense that these axioms may have some connection somehow to the physical universe. In the intuitionistic view, we argue that these axioms are somehow connected to our natural predisposition to believe them.

In every case, however, we commit to refrain from trying to justify axioms from within mathematics, unless we can replace them by more fundamental axioms. That would, however, still not change the axiomatic nature of the system.

Since we do not know why these axioms are there in the first place, we also do not know on what grounds they could change. Hence, your question is fundamentally undecidable.

Let me re-emphasize my thought-experiment: Suppose the world changes overnight so that it becomes impossible to model an implication (per se and of course especially for our human minds). It's hard to see why and how, but just bare with me. Wouldn't that mean that MP becomes impossible as well, in contrast to a day before where it was not only possible, but necessary? Doesn't that prove the induction problem for logic as well?

Again, I don't understand why the induction problem is never seen as problem for logic and math as well, there must be something I do miss since far wiser people than me dealt with this before.

Hume clearly differentiated between what he called 'matters of fact', ie facts about the real world, and what he called 'relations of ideas' ie abstract logical systems such as mathematics.
This differentiation separates induction from deduction, the real from the abstract.

[

Hume clearly differentiated between what he called 'matters of fact', ie facts about the real world, and what he called 'relations of ideas' ie abstract logical systems such as mathematics. — A Seagull

Your ‘i.e’ is yours alone, Hume never used such an expression. He distinguished analytical and empirical but never used the expression ‘the real world’.

Suppose the world changes overnight so that it becomes impossible to model an implication (per se and of course especially for our human minds). — Pippen

There is a reason why necessary truths are held to be true ‘in all possible worlds’. specifically, that no world in which such truths did not obtain could not exist. So your supposition is mere idle wordplay.

Which gives logic and math a kind of atemporal, aspatial quality. Which is odd, given that we inhabit temporal, spatial universe of change. — Marchesk

My takeaway is rather different: not that math and logic are atemporal and aspatial, but rather, that they are normative practises, techniques, employed and tailored for certain purposes, outside of which certain questions simply no longer make sense. Consider the OPs example: the idea that a definiendum no longer entails its definiens. If someone were to say this, the only possible response to make is that he or she does not understand what either or definiendum or a definiens is. That he or she does not understand how language is used in these cases.

Or else: if a 'circle is no longer round' - what could this mean? Either, on the side of 'language', that people no longer call what they used to call circles, circles (they call them 'kirkles' now). Or, on the side of 'things', that everything circular has changed shape (to decagons, say). But these two options exhaust the space of manoeuvre: circles just are round - not because of some deep, metaphysical necessity, but because that is how we use language. If someone says: 'but circles might not be round tomorrow' - the only response is: 'you don't understand what circles are'.

What prevents us from imagining that we all wake up tomorrow and apply other logical rules? What prevents us from imagining that we all wake up tomorrow and a circle is no longer round, because we find ourselves in a chaotic (inconsistent) world in which the definiendum no longer entails the definiens and vice versa? — Pippen

Nothing prevents us from imagining that.

Again, I don't understand why the induction problem is never seen as problem for logic and math as well, — Pippen

The problem of induction is about the justification of laws. It's a problem for knowledge internalists who insist that in order to know something (a law, for instance), one would have to have access to at least part of a justification.

I'm not a mathematician, so I don't know to what extent math is made of laws the way physics is. I'm not sure the method for determining the circumference of a circle is thought of as a law. But in imagining that circles might not be round tomorrow, the issue isn't math anyway. At first glance, it sounds like we're imagining that language use changed.

Which gives logic and math a kind of atemporal, aspatial quality. Which is odd, given that we inhabit temporal, spatial universe of change. — Marchesk

There's nothing mysterious here. Logic and math are our constructs, and as ideas that we entertain our minds, write down and communicate to each other, they originate in time, change and disappear - same as with our ideas about the world. But there is a fundamental disanalogy here between purely abstract ideas and ideas about the world: abstract ideas are not about anything, so they don't answer to the challenge of observation, which is what the problem of induction is all about.

ETA Or what said.

@StreetlightX But then there's the "unusual effectiveness of mathematics", particularly for physics. Also, space and time themselves might be emergent features and not fundamental.

abstract ideas are not about anything, so they don't answer to the challenge of observation, which is what the problem of induction is all about. — SophistiCat

I'm not sure what you mean by "abstract ideas." An abstract object is by definition a thing which various individuals note and can be wrong about. Mathematical objects are abstract objects. They do exist.

That Hume made a distinction does not mean he was entitled to. That was Kant's point really. Leading on from @Wayfarer's remark, what are Hume's famous "matters of fact" after all? His "system" had three fundamental types of things: sensations, ideas and relations we make between them. The difference between induction and deduction dissipates if that is all there is.

Make up your mind: are you talking about physics or mathematics? Physics, even mathematical physics, is about something - something over which we have no control and of which we can only judge on the basis of past observations - hence the problem of induction. Pure mathematics is not about anything, at least not about anything over which we don't have ultimate control - such as our own definitions and postulates.