There is no gap on the real number line.
— Relativist
If that was true then Aleph_1 was the largest cardinal number. — MoK

If you think there are gaps (discontinuities) in the real numbers, then you don't understand real numbers.

If you think transfinite numbers (which are not real numbers) somehow implies there are discontinuities in the real numbers, then you don't understand transfinite numbers.

c = "the cardinality of the continuum" = the cardinality of the real numbers. Consider what "continuum" means.

That you could having 'something' between two points of 'time'. — AmadeusD

It seems to suggest there's a duration of time between the discrete instants of time, which seems self-contradictory.

"Laws of Nature" just refer back to those causal regularities. — AmadeusD

That's a Humean account. More recent philosophers have developed an (arguably) superior account: law realism.

The notion is that there are actual, existing laws of nature. A law is a relation between universals.

Example: electron (-1 charge) and proton (+1 charge) are universals. It is a law that they will attract. Each particular electron instantiates the universal "electron" and each particular proton instantiates the universal "proton". They necessarily attract because the electron-proton pair necessarily instantiates the law.

Here's some sources:

https://www.jstor.org/stable/43153907

https://bates.alma.exlibrisgroup.com/discovery/fulldisplay?docid=alma991001713689708044&context=L&vid=01CBB_BCOLL:BATES&search_scope=MyInst_and_CI&adaptor=Local%20Search%20Engine&tab=MyInst_and_CI&query=sub,exact,%20Causality%20,AND&mode=advanced&offset=60

https://global.oup.com/academic/product/causation-9780198750949?cc=us&lang=en&

If there is no gap between two instants of time then they lay on the same point. Is this correct or not? — MoK

Not correct. There is no gap on the real number line. That's what it means to be continuous.

Isn't that the same as time passing in discrete moments (versus continuous points)?

How a thing remains the thing which it is through all sorts of changes as time passes, is unknown, as a mystery of the universe. — Metaphysician Undercover

Armstring's ontology accounts for it.

The principal alternative ontology. which you seem to be promoting, holds that every time a thing changes, it cannot still be the same thing because it is no longer 'identical" to the way it was before. From this perspective, each object must be created anew at each moment of passing time, ...This is a fine ontology, — Metaphysician Undercover

Perhaps, but it's not the ontology I've been trying to explain.

You say first, that every particular is an SOA. But then you say that an SOA consists of 3 type of things, and a particular is one of the three. So which is the true particular, the SOA, or the part of the SOA — Metaphysician Undercover

I did explain it, right here:

*Thin particular: Armstrong denies that SOAs (AKA existents; AKA particulars) are nothing more than bundles of properties. There is also particularity to which properties attach in a SOA. When we abstractly consider the constituents of an SOA, we therefore need to include "particular" as one of these constituents (the particular considered without the attached properties & relations). To distinguish the SOA's constituent particular from an SOA (also called a particular), he labels the constituent as a "thin" particular. — Relativist

You then responded:

This does not resolve the issue of which is the true particular, it simply creates an infinite regress. — Metaphysician Undercover

Then we haven't succeeded in communicating. I'll try this:
Every SOA is a "true particular" in the sense that we typically use the term. It is something that exists in the world, wholly and
independently (except that it may have relations to other particulars).

A "thin particular" is not a "true particular" - it isn't a thing that can exist wholly and independently. Here's how to conceive of a "thin particular": think about an object. Like all objects, it has intrinsic properties, and relations to other things. Now mentally subtract those properties and relations. What's left is the "thin particular".

(You may believe there's nothing left after you strip off the properties and relations. You'll need to set that aside and accept this as a stipulation of the ontology, at least for now).

The problem though, is that when you get to the base particulars (particles), which are necessary to assume to avoid infinite regress, identity is completely lost. One cannot be distinguished from another, and they are moving as time passes, so location is of not help. At this point, "strict identity" turns into no identity, and the entire ontology falls apart by proposing a fundamentally unintelligible universe. — Metaphysician Undercover

There's a distinction between "strict identity" and an "individual, perduring identity" (IPI, for short; my term, not Armstrong's, but corresponds to his usage). An IPI corresponds to our everday view of identity.

A "base particular" is an "Atomic State of Affairs". It's analogous to an elematary particle in physics. It exists at a specific set of spatio-temporal coordinates with it's specific set of properties and relations. It's strict identity ceases to exist at the next instant of time. But each point of time has a successor, with a slightly different set of coordinates, properties, and relations. Every member of this set of successors shares a single IPI.

I'll leave it there to see if this got across - and to see if you are sufficiently interested to continue. I acknowledge it's complex, and takes some work to try and understand it.

But, are universals themselves physical? I know David Armstrong says they are, but I think his is a revisionist account of universals shoehorned into a materialist framework and undermined by science itself — Wayfarer

Universals aren't "shoehorned". Armstrong wrote a book ("Universals: An Opinionated Introduction") where he lays out the case for his treatment of them. It's a stepping stone toward his comprehensive metaphysics (universals are integral), but it stands on its own.

For example, the Copenhagen interpretation suggests that quantum entities do not have definite properties until they are observed, which conflicts with Armstrong's view that properties (or universals) exist independently of perception and measurement) — Wayfarer

Quantum "entities" are quantum systems, and they evolve deterministically (per a Schroedinger equation)- irrespective of interpretation.

Measurements entail an interaction between the non-quantum (classical) world and a quantum system. The result of the interaction is probabilistic - repeated measurement will fit an expected probability distribution. Armstrong accounts for this as "probabilistic causation" (consistent with his account of laws of nature). As far as I can tell, this is consistent with any of the interpretations of QM.

In the SEP entry on Physicalism, cited above, there is a section on 'the problem of abstracta' ... So it's far from a settled question. — Wayfarer

There aren't many settled questions in philosophy. But Armstrong argues that the notion that abstractions have objective, independent existence seems unparsimonious - they are unnecessary additions to the "furniture of the world" (as he puts it).

Armstrong takes his case further: if objects depend on these abstract universals for their form, it entails a relation between the object and the the abstraction - so not only do abstractions add to the "furniture of the world", it also creates a the need for this relation. Immanent universals (universals existing exclusively in their instantiations) is more parsimonious and simpler.

As far as the abstractions that we mentally contemplate, Armstrong points to the "way of abstraction" (see this SEP article) which makes sense to me.

naturalism assumes nature' - it starts with the apparently self-evident fact of the existence of the empirical world, to be studied by science. But again, that apparently innocuous assumption always entails an implicit metaphysics and epistemology. — Wayfarer

I'd say that scientism (not science, per se) has to depend on the assumption that there is a compatible metaphysics underlying it all. I'm not aware of Feser ever acknowledging that. Instead, he criticizes scientism for its absence of accounting for a foundation of knowlege. Of COURSE it lacks that! But the physicalist metaphysics you consider entailed by it doesn't lack it.

His criticism also seems disingenuous by pointing out that the principle of scientism excludes the possibility of "knowledge" of some foundation for knowledge. If he's using "knowledge" in the strict sense, then the same thing applies to him and the Thomist metaphysics he embraces. It may be coherent, but it's not provably true.

I have no problem with his pointing out the fact scientism can't explain itself, but it would be more reasonable to point toward the need for a metaphysical model that fills the gap he identified. This ought to be accompanied with the observation that it is not actually possible to have "knowledge" of any metaphysical model.

An example is the status of objectivity: I've argued at length in another thread that objectivity is itself reliant on there being a subject to whom objects appear (per Kant). The fact that communities of subjects see the same sets of objects doesn't undermine that.

IMO, true epistemic objectivity is an unobtainable ideal, but we can pursue intersubjectivity.

And then, there's the observer problem in physics, already noted. And the objects of physics itself are essentially abstractions.

I addressed both points.

my arguments depend on the definitions. — MoK

And that is the reason your argument isn't compelling.

I see absolutely no reason to think there's a "gap" between instants of time, regardless of whether it's continuous or discrete.

They’re nearly always joined at the hip. Are there any advocates for ‘scientism’ who do not hold to physicalism? — Wayfarer

Both believe the physical world is all that exists, but Feser's objections to scientism do not apply to metaphysical physicalism. Someone who embraces scientism without a grasp of physicalism as a metaphysical system will be stumped by his assertions. So I can see them sort of joined at the hip, as long as we recognize that physicalism, but not scientism, is a metaphysical system.

Here's an example Feser gives (this was from the first "here" in the article you linked):

"Despite its adherents’ pose of rationality, scientism has a serious problem: it is either self-refuting or trivial. Take the first horn of this dilemma. The claim that scientism is true is not itself a scientific claim, not something that can be established using scientific methods. Indeed, that science is even a rational form of inquiry (let alone the only rational form of inquiry) is not something that can be established scientifically. For scientific inquiry itself rests on a number of philosophical assumptions: that there is an objective world external to the minds of scientists; that this world is governed by causal regularities; that the human intellect can uncover and accurately describe these regularities; and so forth. Since science presupposes these things, it cannot attempt to justify them without arguing in a circle."

A physicalist metaphysician has no problem addressing the philosophical questions he raises every bit as well as a Thomist like Feser. That science is a rational form of inquiry doesn't require a supernaturalist metaphysics to justify; the "causal regularities" he refers to can be accounted for as laws of nature (relations between universals).

I suspect Feser (who is prone to polemics) is being a bit disingenous with his criticism. Scientism isn't a metaphysical system, and that's why I bring up physicalism. No metaphysical theory (Thomist, Aristotelian, Physicalist...) is provably true, such that it can properly be labelled "knowledge" in the strict sense Physicalism is no less justifiable than Feser's Thomism ( physicalism is arguable MORE justified because it entails fewer ad hoc assumptions). So Feser is in no better position to claim actual knowledge than a physicalist.

You have demonstrated that you argument DEPENDS on assumptions. If I'm wrong, then recast your argument using my definition of time, events, discrete and continuous time.

Let's put it this way:if matter is pure extension, then it leads to certain paradoxes. If that's right, then it's a reason to at least consider alternatives.

We have made no progress from the point you said:

Let's see if we could agree on (2). — MoK

You seem to be making a number of specific metaphysical assumption that I disagree with, so it's pointless to continue.

The principle of infinity seems suspended within the concept of "the finite". It's like they are two sides of each other. The number 1 can be divided to infinity, but it's much more odd when doing this with spatial objects (because space has size). So we say "real spatial objects have much more to them then mathematical relationships to themselves. These other aspects make the mathematical sides appear distorted". — Gregory

I think we need to be careful when applying mathematics to reality. It may be less of a problem when applying reality to mathematics -because there are obvious mathematical relations between objects.

Xenos paradox is an example of a problem created by treating the mathematics of infinity as something that is instantiated in the actual world. Consider that there' a practical limit below which we can't divide accurately enough to actually conduct the scenario in the real world, but there may actually even be a real-world limit on the division, when we get down to the Planck length. The question should be asked: How does the mathematics map into the real world process? If that can't be described coherently, that's a clue that there is no such mapping. Mathematics is not ontology, albeit that there seem to be some mathematical relations among the actual objects of the world.

I mean a process in which all events occur at a single timeless point. — MoK

Change doesn't occur at a point of time. Change entails a passage of time.

You can also watch all the frames at a single point. — MoK

A single point...of what? You could watch a single frame, but time is passing while you look at it.

I get the impression that you are treating time as a metaphysical entity, which I don't agree with. I consider time to be a relation between states. So a passage of time entails transitioning from state to state, while each emerged state is an event.

Also, what is an event? I view an event as a state that was caused by a prior state.

I am not talking about the quantization of time in which time is made of indivisible units so-called Chronon. I am talking about the classical discrete time. — MoK

Describe it. I'll point out that as you make more assumptions, you weaken your case - because each assumption can be rejected (unless you can show it to be logically necessary).

*edit* I overlooked this portion of your post, since I didn't think it wasn't addressed to me. But it does seem relevant:

Properties are what we attribute, what we say about things. But in logic the object is represented as a subject, and we predicate. The predication is made of the subject, not the object, and there remains a separation between the subject with its predications, and any possible object which is represented in this way. This separation, makes the object completely separate from anything we say about it, even spatial-temporal location, it's reality is a possibility. This is what allows for the reality of mistake. — Metaphysician Undercover

In Armstrong's metaphysics, properties actually exist - they are not *just* what we attribute to things (and we often attribute characteristics to things that aren't actually properties). You seem focused on semantics, whereas Armstrong is focused on ontology. So I wonder if you're just treating individual identity as some semantical convention. That seems a defensible position, but it's not ontology - and it is ontology that Armstrong is dealing with.

Below is my original response (with a couple of *edit* comments added in italics):

I partially disagree: the parts of the car are still things, and can be a subject of discussion. I can refer to "my car's engine/steering wheel/tires" etc. — Relativist

These are predications though, your car is the subject, and you are saying that it has these things, as properties. At any time, such predications may be true or false. Therefore at sometime your car may not have any tires, then afterwards it might have tires which are different from the tires before. The swap in parts makes no difference to the identity of the car. — Metaphysician Undercover

(*edit* - the above reinforces my thought that you're dealing with semantic convention)

Again, I have not said the car has "things" as "properties". Rather, at a point in time, the car has a specific set of components. A swap in parts absolutely implies the resulting vehicle is not strictly identical to the car before the swap. I hope that is clear.

You claim that it makes no difference to the car's identity if some parts are replaced, but you haven't explained how that car's identity endures despite a change of parts. When the part-swapping process is completed, what has become of each original car's identity?

While I'm interested in hearing your view of how identity endures over time, don't lose sight of the fact that I'm describing David Armstrong's ontology. In Armstrong's terms, true identity is a strict identity. Below, I'll describe his concept of a personal identity perduring over time.

The point though is that there is nothing necessary and sufficient, because identity is the thing itself. — Metaphysician Undercover

This statement doesn't account for identity over time. What makes the car (or you) the same identity from one day to the next, or from one decade to the next? If you aren't accounting for it through essentialism, then how DO you account for it?

When you say "same car", you are designating a type of thing, "car", and that causes a problem because we might think that there are necessary and sufficient conditions for being "a car". I think the important point of the law of identity is that it makes identity distinct from anything we say about a thing, making it the thing itself. — Metaphysician Undercover

Considering "types of things" actually strikes close to Armstrong's account of identity over time, so I'll describe it now.

Remember that every thing that exists (i.e. a particular) is a State of Affairs (SOA), and every SOA has 3 types of constituents: a (thin) particular*, (intrinsic) properties, and relations to other SOAs (AKA extrinsic properties). Properties, relations, and "thin particulars"* do not exist independently; they exist only as constituents in a state of affairs. Strict identity means the exact same set of constituents.
-------------------
*Thin particular: Armstrong denies that SOAs (AKA existents; AKA particulars) are nothing more than bundles of properties. There is also particularity to which properties attach in a SOA. When we abstractly consider the constituents of an SOA, we therefore need to include "particular" as one of these constituents (the particular considered without the attached properties & relations). To distinguish the SOA's constituent particular from an SOA (also called a particular), he labels the constituent as a "thin" particular.
-----------------
Armstrong next defines a "State of Affairs Type (SOAT) - SOAs that have one of more properties/relations in common are the same SOAT. Electron is a SOAT. A specific electron located at some exact location is an SOA. Every SOAT is a universal: it can be instantiated multiple times. An SOAT can be a single property, or a set of properties+relations. As in the case of an electron, all electrons have the same exact properties (excluding location) - but they are different particulars (with distinct "thin particulars").

Identity over time is a loose identity (as opposed to the strict identity I've been discussing): it is a SOAT; it is a universal. An individual identity has "temporal parts": the actual SOA at each point of time. Each of these SOAs is temporally/causally connected to each other (directly or indirectly).

As I said, changing parts does not change the thing's identity, that's a matter of properties coming and going, what we express by having one subject with different predications at different times. Clearly two distinct things in the same place at different times, does not provide the temporal continuity required that it be one subject. — Metaphysician Undercover

You've indicated that personal identity is not identified by a set of necessary and sufficient properties. OK, then what does identify a specific personal identity, if it's not some subset of its properties that it holds throughout its existence? Are you, perhaps, referring to haecceity - treating identity as a primitive? *edit* or are you just treating individual identity as a semantic convention?

Defenders of physicalism will say:

1. The predictive power and technological applications of physics are unparalleled by those of any other purported source of knowledge.

2. Therefore what physics reveals to us is all that is real. — Wayfarer

That's not phyicalism, it's scientism, which is:

... the view that science and the scientific method are the best or only way to render truth about the world and reality.

While the term was defined originally to mean "methods and attitudes typical of or attributed to natural scientists", some scholars, as well as political and religious leaders, have also adopted it as a pejorative term with the meaning "an exaggerated trust in the efficacy of the methods of natural science applied to all areas of investigation (as in philosophy, the social sciences, and the humanities)".

Physicalism doesn't entail believing physics necessarily reveals all that is real, or that science is the sole source of knowledge. Like any metaphysical theory, physicalism endeavors to provide a model that accounts for everything we know about the world. While that includes what we know about physics, it would be a poor metaphysical theory that depended on the prevailing scientific paradigms of the day. It's also fair game for a physicalist to account for things physics can't account for: e.g. foundations of knowledge; modal truths; or pretty much anything that a critic of scientism (like Feser) might raise.

You're applying the term "simultaneously" in an absurd way by claiming that event-A is "simultaneous with" event-A. — Relativist

No, I am talking about three different types of processes, namely discrete, continuous, and simultaneous. A simultaneous process is a process in which all the events occur at the same point. — MoK

OK, I see your point. However, that approach is vulnerable to objections based on special relativity (see this article). Since we're talking about the metaphysics of time in general, it usually makes more sense to consider the temporal evolution of the universe: the universe evolves from state S1 at time T1 to state S2 at time T2. T1 and T2 are points of time, and also correspond to events. On this global scale, there are no "simultaneous events". Does this work for you?

You're conflating the mathematical concept (of points) with a sequence of temporal durations. These durations are not actually divisible into smaller units - except abstractly, which is irrelevant because you're making an ontological claim. — Relativist

I am not conflating anything. If time is discrete then the points are points of time and the interval between two consecutive points is the smallest duration. — MoK

Yes, you are. Here's an excerpt from the Wikipedia article on the chronon:

"A chronon is a proposed quantum of time, that is, a discrete and indivisible "unit" of time as part of a hypothesis that proposes that time is not continuous. In simple language, a chronon is the smallest, discrete, non-decomposable unit of time in a temporal data model. "

You're trying to divide something that is indivisible, treating time as continuous (that's what you're doing when you consider the chronons divisible into points) - but events are merely advancing in stutter-steps. You can't have it both ways.

If you have a point, please make it. Biden's debate performance was terrible, suggestive of some degree of cognitive decline (probably a litte more than Trump's), but does not entail dementia (see this).

I expect you interpreted his performance from the perspective of your bias. My evaluation is based on the actual evidence. I'll grant that IF the GOP conspiracy theory is true (that Biden's people are hiding his dementia), then I'm mistaken. But there's no objective evidence this is the case.

Now that one cognitively declined candidate has dropped out, are you hoping the other one does? Imagine what shape he'll be in, in 4 years! If it was relevant for Biden, then if you're consistent- it should be relevant to Trump.

That is my point. If the distance/gap between two events is zero then events are simultaneous. — MoK

You're applying the term "simultaneously" in an absurd way by claiming that event-A is "simultaneous with" event-A.

Are you talking about the power set? It was a mistake on my part to write "c" instead of "R". If we define "R" as the cardinality of the real number lines then this number is the number of members of the set. This number is infinite but it is not the biggest infinity. Therefore, any small interval on the real number line no matter how small is divisible. — MoK

My point was simply that if time is continuous, it maps to the ordered set of real numbers:
there is a point in time for every real number, and there is a real number for every point in time.

The cardinality of the set is irrelevant to the mapping. As I said, cardinality is used only to compare two different SETS, and has no bearing on the mapping.

If time is discrete, then the smallest unit of time is a duration, and there's no correspondence to points. (More apples/oranges). — Relativist

There are points. The smallest duration/gap in fact separates points from each other.

You're conflating the mathematical concept (of points) with a sequence of temporal durations. These durations are not actually divisible into smaller units - except abstractly, which is irrelevant because you're making an ontological claim.

When you speak of the identity of a part, then you are not talking about the identity of the whole, and vise versa. So, I think you have produced an example which shows that these two are incompatible. If "my car" is the object referred to, then the supposed individual parts cannot have a distinct identity, because the part's identity is subsumed as it is "a part" of the whole. — Metaphysician Undercover

I partially disagree: the parts of the car are still things, and can be a subject of discussion. I can refer to "my car's engine/steering wheel/tires" etc. I think your issues are a tangent, because states of affairs do not have a mereological composition: a part can be a constituent in multiple states of affairs.

I was only trying to show that "enduring indentity" is a problematic concept - because it depends on essentialism: the notion that there is something that is both necessary and sufficient to an individual identity. I probably clouded the matter by referring to "my car"; the real question is whether it can be considered the "same car" (an enduring identity).

This can be understood as a matter of what is our subject here, the part or the whole. If the car (the whole) is the subject, then the part is a property of that subject. When the part is removed, that property is negated from that subject, and the subject maintains its status as the subject, without that property. So it's just a matter of affirming and negating properties It really does not matter which properties come and go. — Metaphysician Undercover

My point is that it's arbitrary, and not of much ontological signficance- it's more of a semantic convention. Consider this snapshot from one day to the next:

Day 1: I purchase a car and park it in my driveway (=Car1)
Day 2: I replace a tire on that car (=Car2)

Car1 is not strictly identical to Car2, but there is a temporal/causal link between Car1 and Car2: Car1 is a material cause of Car2.

Focus instead on humans: you are not strictly identical to the person you were yesterday: the sets of elementary particles that comprise the respective bodies are somewhat different, and you now have one more day of memories. Get more extreme: compare today-you to infant-you on the day you were born. There is no identifiable set of necessary & sufficient conditions that today-you shares with infant-you - so what would be your ontological basis for claiming you're the same person as infant-you? This is the problem with endurantism: it requires essentialism, the notion that there is some core of you that endures throughout your existence. If you're a theist, you might consider this your "soul", a substance that is assumed to never change -but good luck on proving such a thing exists.

The important thing to notice is that whether a thing is a "part of" something else, is never an essential property, — Metaphysician Undercover

Two problems with this:
1) In Armstrong's ontology, a "thing" (AKA an existent AKA a state of affairs) is not a property. Instead, we might define a complex state of affairs as a set of things connected through relations of some sort. (an "atomic" state of affairs is not composed of other states of affairs - it's just a thin particular+intrinsic properties+relations). So a car consists of parts that are connected to the other parts to form a functional whole. (I'll defer explaining the technicality of how a thing's identify perdures over time).
2) you're referring to something being "essential", while seemingly ignoring the fact that nothing can be identified as essential (both necessary and sufficient).

Lebniz' law is not the same as the law of identity. — Metaphysician Undercover

I am referring to the conjunction of:
(the identity of indiscernibles) & (the indiscernibility of identicals).

Some refer to this conjunction as "Leibniz law" (see this). But whether or not it's a correct label is moot. The point is that strict identity entails an identical set of properties. We likely agree that personal identity is not the same thing as strict identity, but Armstrong's ontology makes sense of the distinction, without essentialism.

I believe that "causally connected" is an unwarranted assumption here, which only complicates things. We can simply say that there is temporal continuity between the thing in your driveway yesterday, and the thing in your driveway today, which would allow us to represent it as a subject for predications, and "causation" is left as a distinct and unnecessary conception. — Metaphysician Undercover

The temporal continuity of the car depends on each version of the car being a material cause of the next version. That is warranted. Compare the completed process of gradually swapping car parts to simply swapping complete cars on day 1. The latter provides no basis for claiming the car I now possess is the same car as before.

If the distance between two immediate points on time is absolutely zero then these points are simultaneous. — MoK

If there is 0 distance, it is the same point.

Moreover, the number of points on the real number line is known to be "c" so-called the cardinal number of the continuum.

Apples/oranges. The cardinality of the set of real numbers is not a member of the set of real numbers. Transfinite math is only relevent to comparing sets (e.g. the set of natural numbers to the set of real numbers). It has zero bearing on the discussion.

Yes, each point of time corresponds to an indivisible duration. But that is not what I am talking about. I am talking about two consecutive points on time. — MoK

If time is continuous, there are no "consecutive" points of time (there are no consecutive real numbers- just a "less than"/"greater than" relation.

If time is discrete, then the smallest unit of time is a duration, and there's no correspondence to points. (More apples/oranges).

Philosophim Well, we just disagree. — Sam26

Philosophim provided a well-reasoned rebuttal, in his post, and your response is simply that you disagree. I hope you can see that he was spot on, when he said:

Sam, I'm reading your future posts to others after our discussion ended, and an observation is that you don't address the criticisms people are levying against your points
... — Philosophim

All events lay on the same point if there is no gap or the gap length is zero. — MoK

Non-sequitur, and confused.

If time is continuous, it maps to the real number line. There are no "gaps" in the real number line.

If time is discrete, then each point of time corresponds to an indivisible/unmeasureable duration (relative to the real number line) - each unit abutting the next. Still no "gap", as you've described it.

I don't have a problem with that. I have a problem with your assumption there's a "gap" between points of time.

I can agree that: change iff time elapses. An event is any point in time (excluding an initial state, if there was one).

The gap exists in the discrete time as well as the continuous time. The gap however is arbitrarily small in the continuous time. If the gap is zero then all points of time lay on the same point therefore there cannot be any change in time.

If time is discrete, it still doesn't entail a gap, so it's an unsupported assumption. — Relativist

If time is discrete then it entails a gap. That is true since time exists on a discrete set of points with an interval between which there is nothing. — MoK

Sorry, I don't buy it. It seems a contrivance to lead to some desired conclusion, or the product of naivetee. But of course, I haven't yet seen your argument that shows it metaphyisically necessary that a gap exists. Got one?

My understanding is that Einstein believed in a non-personal, non-anthropogenic "law giver", and denied there existing a life after death. I suspect he felt this way because he could think of no other way to account for laws of nature. Metaphysics has advanced beyond that, so that doesn't sound as reasonable now as it might have back then.

There have been occasions on which I though that it was possible such a god existed, although it is didn't seem likely because it depends on the rather unparsimonious assumption that such a being just happens to exist. Even if it did exist, it would have no relevance in anyone's life - so it would be irrelevant.

Therefore, X and Y must lay on two different points of time. This means that there is a gap between X and Y. By gap I mean an interval that there is nothing between. But the substance in X cannot possibly cause the substance in Y because of the gap. That is true since the substance in X ceases to exist right at the point that the gap appears. Therefore, a single substance cannot undergo a change. — MoK

If time is continuous, there's no gap. If time is discrete, it still doesn't entail a gap, so it's an unsupported assumption.

What is "substance"? If the world is a quantum field, evolving over time consistent with a Schroedinger equation, what is the "substance"?

Let's see if we could agree on (2). We can move forward if we agree on (2). — MoK

Looks like we can't move on.

When I think of "spritual" I relate it to my childhood Christian faith. I was told to interpret various feelings a certain way, and I bought into it. So I think I have some grasp of what you're saying, but I know longer accept that paradigm.

Quantum indeterminacy can be interpreted as probabilistic causation: effects are still the consequences of causes, and the range of effects still predictable while the specific outcome (the eigenstate) is only preditctable probabilistically from among that range and the associated probability distribution.

1) Change exists — MoK

Actually, change occurs. What exists is the present, and its propensity to change - arguably because of laws of nature.

2) A single substance, let's call this the first substance, cannot undergo a change

What's your basis for claiming there is such a thing?

3) This means that we need another substance, let's call this the second substance, to cause a change in the first substance

Clearly, you have some metaphysical paradigm in mind, but you're only giving vague references to it. Maybe (just maybe) it's coherent, but you need to show why this paradigm should taken seriously, while explicitly defining it
...
The rest of your argument depends on the above.

What about matter creating the spiritual — Gregory

Why should I think "spirtual" refers to something that exists?

Being is the unity of what subsists and for him thoughts are being. The world is becoming, but our thoughts are eternal. — Gregory

I'm a materialist, and can't accept that a thought (nor abstraction) is truly a part of the furniture of the world. I don't insist everyone agree; I'm just defending the coherence and plausibility of materialism, based on Armstrong's materialist metaphysics.

I assume my car has an individual identity. Suppose my neighbor has a identical make and model, and we gradually start swapping parts. Eventually, the car in my driveway has none of its original parts and all of my neighbors parts. Is it now the neighbor's car? If so, how many parts had to be replaced to constitute the transformation?

Leibniz's law:
if, for every property F, object x has F if and only if object y has F, then x is identical to y.

This means identity implies identical in every way. Any other definition of identity depends on an arbitary set of necessary and sufficient properties that persist over time - or the assumption that identity is some metaphysical thing that could take on any form (your identity could exist as a cat, a stone, a quark, or a gust of wind.)

Under strict identity, the car in my driveway today is causally connected to the car that was there yesterday so I can claim it as my car from day to day. There's no metaphysical core that makes it so, it's just the way I choose to identify "my car"

Identity (i.e. true identity, consistent with Leibniz' law) doesn't endure over time. Rather, we can identify a perduring identity, as a causally connected series of temporal parts.

Yes and no. I am saying that there is an ontological relation that is being identified, but not claiming the semantic conventions are relevant.

I'll add that there IS a bit of arbitrariness to what we identify as a "state of affairs" (i.e. an existent) in terms of what we choose to consider. We could rightly say the universe is a single existent, or we could say it consists of the set of all galaxies, or the set of all stars, or the set of all quarks, leptons, etc. But any object(s) we identify is still a state of affairs - something that exists, and it fits the framework (a thin particular+intrinsic properties+relations to other objects).

¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")
— Relativist

is not true. — Lionino

The statement "It is not the case that ("all bluebirds fly" implies "Fred is a duck") IS true. But you're right that it's not equivalent to :
¬(A→B)

But why isn't it? It's because there is no material implication. The formula (A→B) cannot be used in all semantic instances of "if A then B".

I don't think I ever realized this before. When I took sophomore logic (50 years ago!), we concentrated on formulaic proofs. But the mapping to semantics is critical.

Thanks for this - I haven't read Heidegger. I did find this description of his use of the terms:

The ontic concerns concrete properties and characteristics of an entity, in contrast to the ontological which pertains to the specific way an entity of a certain kind has its characteristics.

I was not using the terms this way. I used "ontic" and "ontological" interchageably to simply identify something as actually existing in the world. The chair I am sitting on actually exists. A concept or perception of a chair (or anything else) does not exist in the world. So I would have said the chair I occupy is ontic or ontological, but my mental concept of the chair is not. Heidegger's distinction doesn't seem to apply to Armstrong's metaphysics, but to avoid confusion, I'll just "existent" or "existing in the world".

You can infer A from ¬(A→B) by De Morgan.
¬(A→B)
¬(¬A∨B) (definition of material implication)
¬¬A∧¬B (de Morgan)
A∧¬B (double negation) — Lionino

I concede your point, but what you have proven is that:
¬(A→B)
Implies A
(Which I confess seems counterintuitive - see below*).

You had said: If A does not imply B, and B is false, A is true

That second premise(¬B) is superfluous to the conclusion (A).
--------------------------------------

*Now suppose we apply the logic to these statements:
A=All bluebirds fly
B=Fred is a duck

¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")... which is certainly true because the antecedent has no bearing at all on the consequent
(¬A∨B) = "not all bluebirds fly" or "Fred is a duck"
...
A∧¬B: All bluebirds fly and Fred is not a duck

Problems:
Despite the fact that ¬(A→B) is a true statement...
1) it is NOT true that all birdirds fly (hatchlings don't fly),
2) My pet duck is actually named Fred.
But the logic conclusion says otherwise.
Something ain't right. I had to dig out my 1973 Logic textbook to understand the problem, but I'd like to see if anyone can identify the problem on their own.

If the system we live in isn't just physical determinism, but physical determinism + soul determinism (or whatever independent realm he thinks the mind exists in), that's just... determinism. — flannel jesus

Pretty much, except that under physical determinism, it is (in principle) possible to predict all future decisions given perfect knowledge of initial conditions and laws of nature (set aside quantum indeterminacy). Not so with soul determinism: God isn't algoritmically figuring out what choices will be made, he just "knows" by magic.

Why the mental gymnastics? : to rationalize various theistic beliefs. IMO, these tortured rationalizations are good reasons to reject the nonsense.