Why don't you give me some direction where the breakdown in communication happened? — ISeeIDoIAm

When you started posting.

As I explained previously, I only say how things appear to me. — 83nt0n

Another assertion.

I could assert that "my big toe is purple" is an axiom, but I need to demonstrate that it is true. — 83nt0n

No, you do not. Read the definition of "axiom" that you quoted again.

I did not say that as an assertion. And I AM NOT asserting this either. — 83nt0n

Two more assertions.

how does my inability to be a complete skeptic have any bearing on skepticism as a position? — 83nt0n

It is not so much your individual inability as the fact that no one can be a complete skeptic--again, such a position is self-defeating--so it then becomes a matter of which beliefs you adopt, just like anyone else.

This might not be the same as propositional knowledge, which is the knowledge that I am talking about. — 83nt0n

You and I are competent users of the English language. This is a true proposition that we both justifiably believe. Therefore, we both have propositional knowledge.

Overall, no offense, but I don't think you understand my position at all. — 83nt0n

Right back at you.

Sorry, I still have no idea what you are talking about or how it relates to my exchange with @83nt0n.

Again, I'm not asserting my statements/positions as definitely true. — 83nt0n

If you do not believe that anything you say is definitely true, including your assertions about your own beliefs, then how on earth is anyone supposed to have a meaningful conversation with you?

But the axioms that classical deductive logic employs are unsupported. — 83nt0n

There you go again, making an assertion. You need to stop doing that if you want to convince people that you are a genuine skeptic, but I guess you have no way of knowing whether you want to do that. Anyway, I suggest looking up the definition of "axiom."

Again, this is not an assertion. — 83nt0n

Now you are asserting that an assertion is not an assertion--self-defeating, just like I said.

This is how it seems to me. — 83nt0n

Asserting how it seems is still an assertion.

I have to utilize appearances, as I have nothing else to allow me to hold a conversation. — 83nt0n

See, you just used modus ponens. "If I have nothing else to allow me to hold a conversation, then I have to utilize appearances. I have nothing else to allow me to hold a conversation. Therefore, I have to utilize appearances."

It could be that someone does have knowledge, but as of yet I haven't found any. — 83nt0n

We have exchanged several posts now, all utilizing the English language. Unless you wish to claim that we have been throwing gibberish at each other, clearly you and I both have knowledge of the English language.

They are intrinsically true so long as the original presupposition rings true. — ISeeIDoIAm

What do you mean by "the original presupposition"?

I do not know this. — 83nt0n

How do you know that you do not know this?

I could be wrong. — 83nt0n

How do you know that you could be wrong?

However, skeptics like me do not assert as true what we're saying. — 83nt0n

This is your third assertion in a row that something is true.

We just explain how it appears to us to be able to hold a conversation. — 83nt0n

Why should I believe you?

Deductive logic (at least classical) seems unable to guarantee the conclusions it validates. — 83nt0n

The inference rules of deductive logic, including modus ponens, are intrinsically truth-preserving; if the premisses are true, then the conclusion is necessarily true. What deduction cannot guarantee is that the premisses are true.

I do admit that I probably cannot avoid employing it, but this has no bearing on whether it allows us to know. — 83nt0n

Another assertion that something is true. Do you not realize that thoroughgoing skepticism is self-defeating? In order to be consistent you would have to be just as skeptical about skepticism as you claim to be about everything else.

Continuous motion is a more fundamental reality that discrete positions in space and discrete instants in time. Put another way, space is not composed of dimensionless points any more than time is composed of durationless instants. That goes for a one-dimensional line, as well; the only points are the ones that we artificially mark for some purpose, such as labeling S, F, and as many Ms as you like in between. The arrow indeed will pass all the Ms that we actually mark, but that will be a finite number. In order to get from S to F, it does not have to take an infinite series of discrete intermediate steps--from S to M, from M to M2, from M2 to M3, and so on. It simply moves from S to F, and we describe its movement after the fact by means of those labeled positions.

My stance is that I do not know anything, but I (really) want to have knowledge, so I continue the search. — 83nt0n

How do you know that you do not know anything? How do you know that you want to have knowledge?

How do we establish a system of logic that allows us to know? — 83nt0n

We already have, but you claim not to accept it, even though you cannot avoid employing it.

By justified I mean entailing that a person can believe the proposition and stay rational. — 83nt0n

How do you justify that definition?

The justification is what connects a belief to whether or not it's true. — 83nt0n

How do you justify that assertion?

You might ask how something might be justified, I'd say I don't know. — 83nt0n

How do you justify that response?

I am a skeptic. — 83nt0n

How do you justify that self-characterization?

All methods of justification seem flawed to me, including deduction. — 83nt0n

How do you justify that judgment?

Deductive logic seems to require modus ponens to justify modus ponens. — 83nt0n

Again, that is why it is a leading principle or inference rule for deductive arguments in general, not an additional premiss (or infinite series of premisses) in each individual argumentation that employs it.

So I've come across a story (What the Tortoise Said to Achilles) that may pose some problems for deductive logic. — 83nt0n

It is not a problem for deductive logic so much as an observation about it: There is a difference between a premiss (such as A or B) and a logical leading principle (such as C and D and so on). The latter is also called an inference rule, and all other deductive inference rules can be derived from modus ponens once certain axioms are established. One must recognize that such rules are intrinsically truth-preserving in order to understand and employ deductive logic. The turtle is only compelled to accept Z if his goal is to adopt true beliefs.

Peirce recognized that even what we "know" is grounded in guesses, or if we want to use a more sophisticated term, retroductive hypotheses. That includes the notion of reality itself.

What is reality? Perhaps there isn’t any such thing at all. As I have repeatedly insisted, it is but a retroduction, a working hypothesis which we try, our one desperate forlorn
hope of knowing anything. — Peirce, 1898

Charles Peirce was an incredibly intelligent man...but he simply missed the boat on this issue. — Frank Apisa

I suspect that your view on this is much closer to his than you realize. If a purported belief makes no difference whatsoever in our conduct, then it is not a real belief, just empty words.

Obviously I disagree in total with Peirce. — Frank Apisa

Why do you disagree? And why would that be obvious?

As usual, I will offer something from Charles Sanders Peirce, in this case his somewhat famous definition of a belief as a habit of conduct.

And what, then, is belief? It is the demi-cadence which closes a musical phrase in the symphony of our intellectual life. We have seen that it has just three properties: First, it is something that we are aware of; second, it appeases the irritation of doubt; and, third, it involves the establishment in our nature of a rule of action, or, say for short, a habit ... The essence of belief is the establishment of a habit; and different beliefs are distinguished by the different modes of action to which they give rise. If beliefs do not differ in this respect, if they appease the same doubt by producing the same rule of action, then no mere differences in the manner of consciousness of them can make them different beliefs, any more than playing a tune in different keys is playing different tunes. — Peirce, 1878

As the old saying goes, "Actions speak louder than words"; what I truly believe is more reliably discerned from what I do than from what I say.

Lately I'm coming to understand what I take to be this postmodernist point of view. The suspicion of rationality itself. — fishfry

It is not rationality in general that is problematic, but the distinctly modern dominance of technical rationality, which is now widely treated as if it were the only legitimate form. If you are interested, I wrote three one-page columns about this for a structural engineering magazine several years ago: "Knowledge, Rationality, and Judgment" (here); "The Rationality of Practice" (here); and "Rationality and Judgment Revisited" (here). They all include suggestions for further reading.

I should be careful. — fishfry

No worries, I always appreciate your point of view on philosophical aspects of mathematical matters.

Philosophy is not about standing outside a given discipline and telling them they're doing it wrong. Philosophy has to be about explaining what practitioners are actually doing. — fishfry

I am not telling anyone that they are doing something wrong. I suggest that philosophy is (among other things) about explaining what practitioners are actually doing, regardless of whether they accurately recognize it themselves. I have written extensively about philosophy of engineering, my own discipline, and colleagues find it fascinating because they never otherwise think about it in the way that I explain it; they just do it. For better or worse, most practitioners are not reflective practitioners in that sense.

FLT is a statement about the natural numbers, and everybody knows exactly what they are. — fishfry

Yes, in accordance with a certain set of definitions and axioms. Since we cannot point at a natural number to indicate what it is, all we have is a hypothesis from which we can and do draw necessary conclusions (like FLT).

Mathematical truths are abstract things, so they have abstract existence. — fishfry

In Peircean terms, "abstract existence" is an oxymoron. Some abstractions are real, because they are as they are regardless of what any individual mind or finite group of minds thinks about them; but no abstractions exist, because they do not react with other things in the environment.

I didn't come up with the term absolute points in space so i'm not going to give an exact definition. — christian2017

If you do not know what the term "absolute point" means, then why do you keep using it? Why do you keep imposing it on me? In ordinary English, absolute is the opposite of relative; and again, coordinates are relative, not absolute. That is all I am trying to clarify.

Absolute points in space is the spot on my desk where my pencil is resting can in the future be occupied by another object and also possibly be the approximate center of another galaxy far far away at some point in the future. — christian2017

The spot on your desk is not an absolute point. It is always in motion due to the rotation of the earth about its axis, the revolution of the earth around the sun, the revolution of the entire solar system around the center of the Milky Way, and the movement of the entire galaxy through space. Motion is the reality, points and instants are our artificial creations.

Can two galaxies have ever occupied the same absolute points in space? You answered that question and i would like to thank you. — christian2017

I did not answer that question, because I said nothing at all about absolute points, because you still have not explained exactly what you mean by that term. Coordinate systems are arbitrary and artificial intervals marked off relative to an arbitrary and artificial origin, so there is nothing absolute about the points that they define.

Can a galaxy that is traveling through space at some point in time occupy the same space that another galaxy (there are many galaxies) used to occupy? I'm not saying i know the answer to this but i was wondering what your answer was? — christian2017

Since we invent points and instants as needed for any particular purpose, it depends on how we define them. If we set up a three-dimensional coordinate system for space only, then I suppose that the answer is yes--different things can occupy the same point at different instants. If we set up a four-dimensional coordinate system for spacetime (block universe), then I suppose that the answer is no--only one thing can occupy any individual point-instant.

I ask again: What exactly do you mean by "absolute points in space"?

aletheist actually confirmed my main issue. He agrees at least as far as i can tell that there is absolute points in space. If he doesn't agree with that then i guess i misinterpreted what he said. — christian2017

No, I do not agree with you. Here again is what I said.

What exactly do you mean by "absolute points in space"? My contention all along has been that space is not composed of points and time is not composed of instants. We artificially mark points in space and instants in time for purposes such as measurement. — aletheist

Points in space and instants in time are our creations. They do not exist apart from our designation of them; i.e., they are not real, which is what I take you to mean by "absolute" unless you clarify otherwise.

Doesn't speed/velocity require a standard unit of distance measurement over a standard unit of time? — christian2017

Not at all; the separation of space and time, with basic units assigned to each, is an arbitrary convenience. Presumably it reflects the fact that we can directly measure them more easily than velocity. Nevertheless, we could (and arguably should) instead treat velocity as primary and derive our units of distance and duration accordingly. In fact, that is how we get the Planck length and Planck time, the smallest measurable units of distance and duration.

I actually never heard Stephen Hawkings nor my Physics professor in college say there is not absolute points in space. — christian2017

What exactly do you mean by "absolute points in space"? My contention all along has been that space is not composed of points and time is not composed of instants. We artificially mark points in space and instants in time for purposes such as measurement.

Fixed, thanks.

If C is always the same that implies measurements are accurate and that there is absolute points in space (it least in most if not all senses or "levels") — christian2017

I do not see how that follows at all. The speed of light in a vacuum is constant regardless of whether or how we measure it, suggesting that continuous motion through spacetime is a more fundamental reality than discrete positions in space or discrete instants in time treated separately. A meter is an arbitrary unit of length, and a second is an arbitrary unit of duration. A material object traveling at half the speed of light relative to an observer would be measured by that observer as shorter than the same thing not moving at all relative to that same observer. The uncertainty principle is that it is impossible even in principle to measure both the velocity and position of a particle to the same precision at the same designated instant.

Our grasp of the state of things seems on some occasions and in some respects to change gradually and indefinitely, on other occasions and in other respects to change definitely and suddenly. — Cabbage Farmer

In my view, a definite change is an event, which is always realized at a lapse of time during which the change is strictly continuous. Some events are more abrupt than others; i.e., the lapse of time at which they are realized has a shorter duration, causing them to be perceived as more forceful. However, no event is truly instantaneous.

In what sense are possibilities "indeterminate"? — Cabbage Farmer

They are states of things that may or may not be realized (in the future). Only facts that have been realized (in the past) are determinate.

What are conditional necessities of the future? — Cabbage Farmer

Facts signified by conditional propositions in the subjunctive mood; e.g., "If state of things X were to be realized, then state of things Y would be realized."

If there are such things as conditional necessities of the future, might they imply conditional necessities of the past, not merely determinate actualities of the past? — Cabbage Farmer

I suppose that "conditional necessities of the past" are facts signified by counterfactual propositions; e.g., "If state of things X had been realized, then state of things Y would have been realized."

Objective matters of fact do not seem to depend on our thoughts about them. — Cabbage Farmer

That is exactly what it means for something to be real--it is as it is regardless of what any individual mind or finite group of minds thinks about it.

For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. — aletheist

I think that's limiting. It puts trivial conclusions derived from meaningless axioms on the same level as the deepest results. — fishfry

Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously.

No number theorist believes that Fermat's last theorem is merely a theorem that falls out of the axioms of set theory. Wiles proved that FLT is true. True in a way that transcends axioms. It's a truth about the natural numbers; not merely a truth about proofs in a formal system. — fishfry

It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers."

I think that when you do math, you tend to be a Platonist; but when you try to defend the activity rationally, you have to fall back on being a fictionlist. — fishfry

As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it. — Daz

I am more in agreement with @Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it.

Rationality's a fraud. — fishfry

Rationality is a tool, and like all tools it is only well-suited for certain purposes. It can get you from a set of premisses to necessary conclusions, but it cannot stipulate those initial assumptions. That goes for both theoretical and practical rationality--if you want to achieve X, rationality can help you identify means to that end, but it cannot specify X itself; that requires a deliberate choice on your part.

If anything we out to become more science oriented and focused on issues that can be dealt with by science. — Shawn

Science can't tell you whether to deliberately crash your economy in the expectation that not doing so would be even worse. — fishfry

Science can't tell us anything. Scientists can, but we should only listen to what they have to say about their relatively narrow areas of specialization. Economists purport to be (social) scientists, but only by assuming that the independent decisions of billions of individual humans are amenable to analysis in a manner analogous to laws of nature.

That decision can never be the output of any rational process. — fishfry

Exactly. I have been saying for years that science is an especially systematic way of knowing, while engineering is an especially systematic way of willing. There is no one "right" or even "optimized" solution to any given real-world problem, because tradeoffs are unavoidable and require the exercise of practical judgment, not a strict application of technical rationality.

If mathematicians say something has mathematical existence, then it does. There is surely no objective standard. It's a mistake to believe that there is. — fishfry

From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof.

Under what circumstances might the past be altered? — jgill

None, since the past is determinate.

I believe the future is determined regardless of what one believes. — Daz

Then you are indeed a hard determinist, as @Metaphysician Undercover stated; presumably also an eternalist, holding that the past, present, and future all exist. My view is more along the lines of the "growing block" theory, holding that the past and present exist, but not the future. The present is the indefinite lapse of time at which the indeterminate future is always becoming the determinate past.

Whether the so-called reality of past, present, future are either continuous or discrete? — 3017amen

I believe that time is real and continuous.

The bivalence issue , I thought, was in large part what you and MU were arguing relative to P and-P, generally speaking. — 3017amen

I believe that there are lapses of time during which a concrete thing (S) is changing from possessing an abstract quality or relation (P) to not possessing it, or vice-versa, such that neither "S is P" nor "S is not-P" is true; i.e., the principle of excluded middle is false with respect to the attribution of that predicate to that subject at that determination of time. Instead, either "S is becoming P" or "S is becoming not-P" is true.

And so, I thought one argument was that basically, time violated the laws of non-contradiction. — 3017amen

I believe that the principle of contradiction is maintained, because there is no instant or lapse of time at which both "S is P" and "S is not-P" are true.

Can I ask, on one summary point, has there been any consensus on the Reality of Time viz the bivalence/vagueness issue? — 3017amen

There has not been much consensus about anything in this thread, but please clarify exactly what you mean by "the bivalence/vagueness issue."

1. Reality is fundamentally flux, and permanency is constructed
2. Reality fundamentally is, and change is an illusion — Pneumenon

For clarity, please provide your definitions of the key terms here--time, reality, flux, permanence, constructed, change, illusion. Also, why not consider as a third option that the reality of time includes both permanence and change--enduring things and their varying qualities, not to mention the fixed but growing past and the constant but advancing present.

Time is not 'diversity of existence', all universe phenomenon are related. Humans are related to stars. Even if just time-wise. — xyzmix

This seems like a misinterpretation of what Peirce meant by "diversity of existence." Consider his other definition, which I also quoted in the OP.

Time is a certain general respect relative to different determinations of which states of things otherwise impossible may be realized. Namely, if P and Q are two logically possible states of things, (abstraction being made of time) but are logically incompossible, they may be realized in respect to different determinations of time. — Peirce, c. 1905

"Diversity of existence" simply refers to the reality that the same existential subject (enduring concrete thing) possesses different qualities and relations at different determinations of time.

Would that imply time is also "smooth?" Or space/time? — jgill

If by "smooth" you mean that any function with respect time is differentiable arbitrarily many times, then yes, that is my understanding (and Peirce's) of true continuity.

Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. — Peirce, 1908

Note that instants and instantaneous values are perfectly acceptable within mathematics as "the study of what is true of hypothetical states of things" (Peirce's definition, emphasis mine).

So you are suggesting that there is a judgement independent of human judgements, which is somehow more reliable than human judgements. — Metaphysician Undercover

No, there are states of things independent of human judgments--namely, facts. These are signified by true propositions, which are likewise independent of human judgments. Again, a judgment is a human decision to adopt a certain proposition as a belief.

You, have come to the point of realizing that the fundamental laws of logic appear to be applicable at some times, but at other times not. However, you refuse to take the analysis further, to determine which aspects of reality they apply to, and which they do not. — Metaphysician Undercover

On the contrary, I have consistently maintained that the principle of excluded middle applies to propositions signifying prolonged states of things (what you call "being"), but not to propositions signifying indefinitely gradual states of change (what you call "becoming"), both of which are only realized at lapses of time (not instants).

If you would take the analysis further, you would see that what is referred to by "S", varies depending on the circumstances of use. — Metaphysician Undercover

I stipulated from the very beginning that in my example, "S" denotes an existential subject, an enduring concrete thing; and "P" denotes one of the innumerable qualities or relations that it possesses at some determinations of time, but not at others. I have never been talking about any other possible referent of either term.

But you refuse to get out of the muddled mess you are in, holding that the same substance is both subject to the law of excluded middle, and not subject to the law of excluded middle, depending on what time you look at it, while insisting that there is no valid markers in time to determine the one time from the other. — Metaphysician Undercover

The principle of excluded middle does not apply to subjects at all, it applies to propositions; and what matters in this context is whether the state of things signified by a given proposition is prolonged (unchanging) or gradual (changing). These are necessarily realized at different determinations of time for the same subject.

This is a blatant conclusion by equivocation. You are not adhering to how I defined "instant". — Metaphysician Undercover

The pot is calling the kettle black.

Utter nonsense. — Metaphysician Undercover

I have the same opinion of your responses at this point, so maybe it is time (no pun intended) for us to call it quits.

"S is P", and "S is not-P" are human determinations, propositions, they are judgements made by us, and these fundamental laws of logic are there to guide us in those judgements. — Metaphysician Undercover

There is an important difference between a proposition and a judgment. A true proposition signifies a real state of things--i.e., a fact--so it is true regardless of what anyone thinks about it. A judgment is one person's belief that a certain proposition is true, which is fallible.

To make the assertion that there is something about the object which we are judging, rather than accepting the likelihood that there is something deficient in our capacity to make the judgement, which is responsible for this situation, is not only completely unwarranted, but it also kills the desire and inspiration required for further analysis of the object to determine the precise time when S is P becomes S is not-P, through the assertion that there is no such precise time. — Metaphysician Undercover

This cuts both ways. To make the assertion that there is a precise time when S changes from being P to being not-P, rather than accepting the likelihood that a state of change is only ever realized at a lapse of time rather than at an instant, is not only completely unwarranted, but it also kills the desire and inspiration required for further analysis of the reality of time, through the assertion that there must be such a precise time.

Therefore it is impossible that time is composed of instants. — Metaphysician Undercover

Indeed, and if time is not composed of instants, then it must be continuous. Instants in time, like points on a line, are artificially imposed.

Do you accept that "S is P", and "S is not-P" are terms of representation? — Metaphysician Undercover

Of course, they signify two different prolonged states of things that are incompossible; i.e., they cannot be realized at the same determination of time.

Notice the difference between what "S" refers to when "S" is a representation of the past, and when "S" is a representation of the future. The future event has no real existence. — Metaphysician Undercover

No, "S" does not refer to an event at all, it denotes an existential subject; i.e., an enduring concrete thing. An event is the gradual state of things when a change is realized, which is signified by "S became not-P" (past) or "S is becoming not-P" (present) or "S would become not-P under such-and-such circumstances" (future). That is why neither "S is P" nor "S is not-P" is true during the lapse of time at which the event is realized.

But I think that this is irrational, because "S" signifies an intelligible object, a subject, it does not signify a physical object. — Metaphysician Undercover

On the contrary, it is an assertion like this that is irrational and unintelligible. "S" denotes an enduring concrete thing, which is an intelligible object.

And, to say that neither P nor not-P is applicable to S is to render S as unintelligible, which is inherently contradictory. — Metaphysician Undercover

It is perfectly intelligible once we recognize that the enduring concrete thing denoted by "S" is changing from possessing the quality or relation denoted by "P" to no longer possessing that quality or relation, or vice-versa.

That is fundamentally irrational, because "S" as subject refers to something different when talking about a past object, from what it refers to when talking about a future object. — Metaphysician Undercover

Nonsense, in this context "S" denotes the same enduring concrete thing. What changes over time are the qualities and relations that it possesses, such as the one denoted by "P."

Right, an "instant" is not a real part of time, and that is why time cannot be continuous. — Metaphysician Undercover

This is backwards; time would only be discontinuous if a discrete instant were a real part of it.

Human beings have not been able to determine real instants in time, therefore there are no real instants, and time is continuous. Since time is continuous, there are no real instants, therefore human beings ought not seek to determine the real instants in time. — Metaphysician Undercover

The fallacy here is that of a straw man--neither of these sentences accurately expresses any assertion or argument that I have actually offered.

If time is truly a trichotomy, as you say, then "the present", as the limit, and a third distinct thing between future and past, would prevent the two adjacent portions, future and past, from having anything in common. — Metaphysician Undercover

No, my whole point is that the present is a third portion, not a limit at all. The past and present are not adjacent portions, because the present is another lapse of time between them, not an instant.

So, the problem with your proposal is that you want to inject a third distinct thing, into time, between future and past, a lapse of time when neither "S is P" nor "S is not-P" is true, and despite the positing of the intermediary, you want to claim that time is continuous. Clearly, if this third thing which separates future from past has real existence, then time is not continuous. — Metaphysician Undercover

In my terminology (and Peirce's), a determination of time is not a thing and it does not exist. Time is a real law that governs existents, like the enduring concrete thing denoted by "S." The present is not distinct from the past and the future, it is an indefinite moment such that we directly perceive the continuous flow of time.

All change all the time is "an indefinitely gradual state of change", so it doesn't make sense to say that S is P or S is not-P are applicable in any real way. — Metaphysician Undercover

Again, any particular existential subject (S) is always changing with respect to some of its qualities and relations (P), but not all.

If "S is P" is true at some time, and then ceases to be true, then "S is not-P" is true at this time. — Metaphysician Undercover

Only if one assumes that the principle of excluded middle is always true.

You might posit a time in between, during which the human being is incapable of determining which of these is true, but this is not the same as saying neither is true, it's a case of saying that we haven't the capacity to determine it. — Metaphysician Undercover

No, I am positing that a state of things is realized during an event-lapse that is objectively indeterminate, such that the existential subject S neither determinately possesses the quality or relation P, nor determinately does not possess the quality or relation P. Consistent with the definition of "real," it has nothing to do with human capabilities.

The problem with your position is your claim that "S is P", and "S is not-P" are sound propositions at some times, but are not sound at other times. — Metaphysician Undercover

There are no sound propositions, only sound (or unsound) arguments and true (or false) propositions. Again, "S is P" and "S is not-P" can each be true at different determinations of time, but neither can be true at a determination of time when S is changing from possessing P to not possessing P. If this change is not realized at some determination of time, then it cannot be realized at all.

S is not P follows directly in time, after S is P. They are separated by the same boundary which separates one instant in time from the next. — Metaphysician Undercover

As you know, I deny that time is composed of instants; but even if it were, there could be no "next" instant after any given instant, just as there is no "next" rational or real number after any given rational or real number. Otherwise, there would have to be an arbitrary and finite number of instants within any measured interval of time.

There is a necessary boundary, an instant, when "S is P" ceases to be true, and neither "S is P" nor "S is not-P" starts to be true. — Metaphysician Undercover

This is a clever bit of sophistry, because it distracts from the primary issue of when states of things are realized (at determinations of time) to the subordinate issue of how states of things are represented (by propositions).

Any time you replace the instantaneous boundary with an intermediary description, you still have the boundary which marks the end of S is P and the beginning of the intermediary, so you need to posit another intermediary at that boundary, and so on ad infinitum. — Metaphysician Undercover

No, the only real boundary is between the lapse of time at which a particular determinate state of things (signified by "S is P") is realized and the lapse of time at which an incompatible determinate state of things (signified by "S is not-P") is realized. Our disagreement boils down to whether that real boundary can be an individual determination of time (instant) or must always be another general determination of time (lapse). In other words, is instantaneous change really possible, or does real change always require a lapse of time?

"Being" and "becoming" are distinct and incompatible aspects of reality, and this is the basis of Aristotelian dualism. — Metaphysician Undercover

What you call "being" and "becoming" are simply two different classes of states of things (prolonged vs. gradual) that are realized at different determinations of time, involving the same enduring existential subject (denoted by S) and one of its innumerable qualities and relations (denoted by P). Paraphrasing Peirce, the being of the quality/relation as form lies wholly in itself, the being of the existential subject as matter lies in its opposition to other things, and the being of the fact as entelechy lies in its bringing qualities/relations and existential subjects together.

I realize that this definition does not match up exactly with the way that the human being experiences the present, but the human being is always experiencing a part of the future along with a part of the past, and we may not be capable of apprehending the instantaneous division. — Metaphysician Undercover

It is precisely because "the human being is always experiencing a part of the future along with part of the past" that the present cannot be an "instantaneous division" between them. Anything that we are experiencing (present progressive tense) is, by definition, in the present.

Now there are great difficulties in the way of supposing that we are immediately conscious of time, and therefore of the past and future, unless we suppose it to possess ... continuity, so that we can be immediately conscious of all that is within an infinitesimal interval from any instant of which we are immediately conscious, without its thereby following that we are immediately conscious of all instants. — Peirce, c. 1894

The consciousness of the present, as the boundary between past and future, involves them both. — Peirce, 1899

The only important thing here is our metaphysical phenomenon, or familiar notion, that the past is a matter for knowledge but not for endeavor, that the future is an object that we may hope to influence, but which cannot affect us except through our anticipations, and that the present is a moment immeasurably small through which, as their limit, past and future can alone act upon one another. — Peirce, c. 1900

Of course, if there is no such thing as an absolute instant, there is nothing absolutely present either temporarily or in the sense of confrontation ... The present moment will be a lapse of time, highly confrontitial, when looked at as a whole, seeming absolutely so, but when regarded closely, seen not to be absolutely so, its earlier parts being somewhat of the nature of memory, a little vague, and its later parts somewhat of the nature of anticipation, a little generalized. It contains a central part which is still more present, still more confrontitial, but which presents the same features. There is nothing at all that is absolutely confrontitial; although it is quite true that the confrontitial is continually flowing in upon us ...
Another plain deliverance of the percipuum is that moment melts into moment. That is to say, moments may be so related as not to be entirely separate and yet not be the same. Obviously, this would be so according to our interpretation. But if time consists of instants, each instant is exactly what it is and is absolutely not any other. — Peirce, 1903

It is logically impossible for a state of things to be realized in the present instant and not to be at all so in the past and future. Were the instants independently actual, as they are in the Time of the analysts, memory would be a perpetual miracle. — Peirce, c. 1904

An event is a gradual state things involving change, so it is logically impossible for it to be realized in the present instant and not be at all so in the past and future. The minimum of time at which any possible state of things can be realized is an indefinite (mathematically infinitesimal) moment.

Your pathway forward is misleading because you have assumed a subjective present, while I am seeking to understand the objective present. — Metaphysician Undercover

Your paragraph summarizing our disagreement is basically accurate, except this last sentence that sets up a false dichotomy. Our phenomenological experience of the present is what calls for an explanation in the first place, and only a hypothesis that adequately accounts for our observations of both its internal ("subjective") and external ("objective") aspects should be considered plausible.

In one period of time S is P is true, and in the next period of time S is not-P is true. The "instant" is the boundary between the two periods of time. There is no time at that "instant", it has no temporal extension, like a dimensionless point in space, except in time, or a line with no breadth, separating one side from the other. — Metaphysician Undercover

This is the fundamental problem with your model. Since there is no time at an instant, an instant is not a real part of time; since there is no space at a point, a point is not a real part of space. We artificially introduce discrete and dimensionless instants and points into continuous time and space for purposes such as marking and measuring.

Then you simply misunderstand what a limit is. — Metaphysician Undercover

No, as usual we just have different technical definitions of a particular term.

Let's say that future and past form a dichotomy, all time must be either future or past ... — Metaphysician Undercover

This is precisely what I deny. The present is neither future nor past; time is a trichotomy, not a dichotomy.

Saying that S is P is no longer true is equivalent to saying S is not-P. — Metaphysician Undercover

Only when the principle of excluded middle is true. Again, it is false during an indefinitely gradual state of change, when S is in the process of becoming not-P after previously being P, or vice-versa.

Therefore it doesn't make sense to speak of S in terms of P, at any time, if at any time S is neither P nor not P — Metaphysician Undercover

This does not follow, since existential subjects undeniably have different qualities and relations at different determinations of time. If "S is P" is true at an earlier determination of time, and "S is not-P" is true at a later determination of time, and both propositions cannot be true at the same determination of time (principle of contradiction), then there must a determination of time in between at which neither is true.

To say S is neither P nor not-P implies that it is a category mistake to say that P might be predicated of S. — Metaphysician Undercover

This mixes up the modalities of actuality and possibility. To say that neither "S is P" nor "S is not-P" is true does not entail that "S may be P" is false.

If S is either P or not-P at some times, then what is it about S which would make it suddenly be neither P nor not-P? — Metaphysician Undercover

The fact that S is in the process of changing from being P to being not-P, or vice-versa. The only alternative is to claim that such negation is instantaneous, which requires S to be both P and not-P at the same determination of time, thus violating the principle of contradiction.

Care to explain how you think marking could be other than arbitrary? — Metaphysician Undercover

I already did--it is not completely arbitrary, given our purpose of distinguishing the lapses of time at which two incompatible prolonged states of things are realized by marking off a lapse of time between them, during which an indefinitely gradual state of change is realized. There is some leeway, since whatever is realized at any given instant is also realized at other instants beyond all multitude that are "near" it, within its immediate neighborhood or "whenabouts."

If the present is an "indefinite lapse of time", then that length, or time period which comprises the present is completely arbitrary. — Metaphysician Undercover

"Indefinite" is not synonymous with "arbitrary."

The present might be a nanosecond or it might be a billion years. Each of these is "present to the human mind". What would make one of these more "the real present" than the other? — Metaphysician Undercover

This is a valid point from a strictly mathematical standpoint, which comes into play as soon as we introduce measurement. According to Peirce, Josiah Royce suggests in The World and the Individual that the present is a "time-span" of two seconds for humans, and "points out that a consciousness for which the events of a millionth of a second should exceed the time-span and another consciousness for which the events of a million years should be present at a glance would both be so utterly unlike our own that we should not easily recognize them as conscious beings at all." However, Peirce himself takes a very different approach. On the one hand, the present cannot mathematically be an instant with zero duration:

The true conception is, that ideas which succeed one another during an interval of time, become present to the mind through the successive presence of the ideas which occupy the parts of that time. So that the ideas which are present in each of these parts are more immediately present, or rather less mediately present than those of the whole time. And this division may be carried to any extent. But you never reach an idea which is quite immediately present to the mind, and is not made present by the ideas which occupy the parts of the time that it occupies. Accordingly, it takes time for ideas to be present to the mind. They are present during a time. And they are present by means of the presence of the ideas which are in the parts of that time. Nothing is therefore present to the mind at an instant, but only during a time. The events of a day are less mediately present to the mind than the events of a year; the events of a second less mediately present than the events of a day. — Peirce, 1873

On the other hand, the present also cannot mathematically be a lapse with finite duration:

It has already been suggested by psychologists that consciousness necessarily embraces an interval of time. But if a finite time be meant, the opinion is not tenable. If the sensation that precedes the present by half a second were still immediately before me, then, on the same principle, the sensation preceding that would be immediately present, and so on ad infinitum. Now, since there is a time, say a year, at the end of which an idea is no longer ipso facto present, it follows that this is true of any finite interval, however short. — Peirce, 1892

Instead, the present must mathematically be a moment with infinitesimal duration:

But yet consciousness must essentially cover an interval of time; for if it did not, we could gain no knowledge of time, and not merely no veracious cognition of it, but no conception whatever. We are, therefore, forced to say that we are immediately conscious through an infinitesimal interval of time. — Peirce, 1892

The problem is that the mathematical term "infinitesimal" implies "too small to be measured," while our phenomenological experience is such that the present moment is not amenable to measurement at all. Measurement requires comparison with an established standard, and there is obviously nothing else with which we can compare our ongoing experience of the present, since it is sui generis. That is why it is better to characterize it as "indefinite"--a qualitative description, rather than a quantitative one.

If there are boundaries within a thing, then the thing which has boundaries within it (time in this case), is not continuous. I don't see why you have so much trouble understanding this fact. — Metaphysician Undercover

I understand it just fine, but it is not a fact, unless we define a boundary in this context as a discontinuity--which you evidently do, but I do not. Again, there are no real boundaries within a true continuum; we artificially introduce them for various purposes, including marking and measuring.

... I conceive that a Continuum has, IN ITSELF, no definite parts, although to endow it with definite parts of no matter what multitude, and even parts of lesser dimensionality down to absolute simplicity, it is only necessary that these should be marked off, and although even the operation of thought suffices to impart an approach to definiteness of parts of any multitude we please.*
*This indubitably proves that the possession of parts by a continuum is not a real character of it. For the real is that whose being one way or another does not depend upon how individual persons may imagine it to be. It shows, too, that Continuity is of a Rational nature. — Peirce, c. 1906

Specifically, real time has no boundaries (instants) or parts (lapses) except those that we create, and we can mark off as many such limits and corresponding portions as we want or need. For some purposes, those boundaries and parts are quite arbitrary; examples include designating one particular revolution of the earth around the sun as year 1, designating one particular rotation of the earth about its axis as January 1, dividing the lapse required for each rotation into 24 hours, designating one particular hour as 1:00, dividing each hour into 60 minutes, and dividing each minute into 60 seconds. For other purposes, there are constraints.

Under this assumption, at one moment S is P is true, and at the next moment, S is P is false. The "instant" acts as a boundary of separation between these two such that one is before and the other is after, and the law of non-contradiction is not violated. — Metaphysician Undercover

Under this assumption, which proposition is true at that instant? If only "S is P" is true or only "S is not-P" is true, then that instant is obviously not the boundary between the two moments; it is within one or the other. If both "S is P" and "S is not-P" must be true--as I maintain, since I understand a limit to be what two adjacent portions have in common; an immediate connection, rather than a discontinuity--then the principle of contradiction is violated. But you deny this; is it your view, then, that neither "S is P" nor "S is not-P" is true at that boundary-instant? This would amount to McTaggart's assumption--events/changes are only realized at discrete instants, rather than continuous lapses--which (he argues) entails the conclusion that time is self-contradictory, and thus unreal.

So when we talk about "becoming", the activity which is change in the world, it makes no sense to use propositions like S is P, because we are talking about something which is categorically different from "being", which is the type of thing that S is P refers to. — Metaphysician Undercover

On the contrary:

... we may speak of the state of different things at the same time as well as of the states of the same thing at different times and, of course, of different things at different times and of the same thing at the same time. At different times a proposition concerning the same things may be true and false; just as a predicate may at any one time be true and false of different things. — Peirce, c. 1904-5

[Time] is certainly a law. It is simply a unidimensional continuum of sorts of states of things and that these have an antitypy is shown by the fact that a sort of state of things and a different one cannot both be at the same time. — Peirce, 1908

The principle of contradiction is indispensable, but the principle of excluded middle is not. Because classical logic insists on enforcing the latter, it indeed cannot handle the reality of temporal change (becoming).

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