There is at least one often-discussed system of axioms in which associativity is a theorem instead of an axiom.

There's no one right system of axioms.

The arithmetical axiom-system in which associativity is an axiom instead of a theorem is the widely used and cited one.

The existence of other axiom-systems is quite irrelevant to what I said.

To establish an implication like the one that I stated, it typically is not necessary to use every axiom in the system.

With the natural and obvious counting-number definition that I referred to, the use of the additive associative axiom establishes 2 + 2 = 4.

Michael Ossipoff

Of course defining the counting numbers in terms of the multiplicative identity makes use of the multiplicative identity axiom, but that was too obvious to mention.

Michael Ossipoff

Incorrect. Reincarnation is metaphysically-implied. There's an uncontroversial metaphysics that implies reincarnation. — Michael Ossipoff

I doubt there's such a thing as a noncontroversial metaphysics. Do you say you've got a hold of one?

If there's a reason why you're in a life (something that I've discussed here), and if that reason still obtains at the end of this life, then what does that suggest? — Michael Ossipoff

What do you mean "a reason why you're in a life"? Is this reason supposed to generate the implication you've singled out?

What do you mean by the phrase "in a life"? One reason I'm alive is that I was born. One reason I was born is that I was conceived. Is this the sort of reason you have in mind?

I’d said:

If an "if-then" proposition, an implication-proposition, might not be true, then it can't be called a fact. It's only a proposition.

You said:

And what is it that determines whether or not any proposition is true?

Abstract propositions such as mathematical theorems can sometimes be proved. Propositions about particulars in our physical surroundings, such as physical theories, can sometimes be verified by observation.

I’d said:

So then a fact can be defined as a true proposition, and so a true if-then proposition can reasonably be called an if-then fact.

A fact has also been defined as a state of affairs, or a state of affairs that obtains.

You said:

This is where your conflation lies. 'Fact' has two senses; a semantic sense, or the sense in which a fact is considered to be a true proposition, and a substantive sense, a sense in which substantive facts make propositional facts true. Only the latter kinds of facts are equivalent to states of affairs. The former kinds of facts are propositional descriptions of states of affairs.

"Paris is the capital of France' is not a state of affairs, it is a statement. Paris being the capital of France is a state of affairs.

You seem to be be saying that a proposition is a statement. That’s widely disagreed-with. A statement is an utterance of a proposition.

This distinction is the first thing you need to get clear. There is no such thing as an "if-then fact". Propositions, not facts, are in the form of 'if-then'. Tautologies don't count as facts either. You are distorting sensible usage in sophistical ways.
.
Consider Tarski's formulation: " 'Snow is white' is true iff snow is white"

A safe bet indeed.

; 'Snow is white' is a propositional
or semantic fact if it is a substantive fact that snow is white. For every propositional or semantic fact there is a corresponding substantive fact. Which is to say that for every (propositional) truth there is an actuality.

Of course that can reasonably be said about the things that you’re speaking of, and I have no objection to it being said. But you seem to be making a way of saying things into a doctrine about how things are, and a claim that nothing else can validly be said.

There are certain denominational-promoters who come to my door to tell me how things are, and that if I don’t accept and believe their version of how things are, then I’m uneducated and refusing to be educated. Is that what I’m hearing now?

What you’re saying sounds like a Facts-Dualism, and a bit unparsimonious, an unnecessary multiplication of entities.

If the truth of one proposition implies the truth of another proposition, that’s a state-of-affairs.

In other words, it’s a fact.

It’s an implication-fact.

It’s an if-then fact.

…even if that evidently isn’t the official way of saying it.

Then we hear, “If you don’t say things the official way, you can’t expect anyone to understand what you’re saying.”

Nonsense. I’m speaking English, as you are, and trying to say thing more parsimoniously.

You said:

Paris being the capital of France is a state of affairs.

I like that. I like that way of saying it. It’s a better description of experience.

You can change all of my statements of facts into that form. The state-of-affairs form best says what I mean by facts, and is the way I mean for facts to be worded.

For example, how about:

“A implying B”, with the same meaning as “Paris being the capital of France”.

But it could also be said—and maybe it would be a better way of saying it—that an implication is one of the relations that are referred to when it’s said that a fact consists of some objects, and some properties &/or relations that that they have.

I’ll skip to your last paragraph first, and then answer your paragraph before it:

You said:

Consider this question: In all the time that you have been posting on forums have you found even one person who agrees with your 'argument'?

Yes.

Aside from that, I admit that the physicists Michael Faraday, Frank Tippler and Max Tegmark aren’t on this forum. But they’ve said things that agree with some of the main aspects of my metaphysical proposal.

You said:

…there is also the apparent lack of any cogent argument to support your "system".

Why not lay out your argument (if you have one) in a clear, concise form showing your major premise(s) and your conclusion, then you should be able to determine once and for all whether it is even a valid argument let alone a sound one.

Fair enough.

Premises:

1. We’re each, from our own point-of-view, in our experience, in a life in a physical world.

2. There are abstract facts, including abstract implication-facts. …at least in the sense that they can be stated.

Conclusion:

Premise #2 is sufficient metaphysical mechanism and explanation for premise #1.

Why the Premise Implies the Conclusion

In my statement of my metaphysical proposal, earlier in this thread, I told of how any fact about this physical world or about the physical events and relations in our experience, implies, corresponds to and can be said as an if-then fact. I told of how any fact in about this physical world is (at least part of) the “If “ antecedent of some if-then facts, and is the “then” consequent of other if-then facts.

As Michael Faraday pointed out, in 1844, there’s no observation or experiment that shows anything other than mathematical and logical structural-relation. There’s no evidence of anything other than that.

Among the infinity of complex systems of inter-referring abstract if-then facts, there inevitably is one whose events and relations are those of your experience.

There’s no reason to believe that your experience is other than that.

[That's the Premise, the Conclusion and their Implication-Relation]

Michael Ossipoff

As a precedent for my argument, I cited abstract if-then facts or implication-facts..

You might reply by standing by your previous statement that there are no if-then facts or implication-facts.

My answer to that would be: See my previous reply to you, above the part where I stated my premise, conclusion and argument. I said what I mean by "if-then facts" or "implication-facts".

So there are implication-facts, as I defined them.

But if you say that you can't understand anything that I say unless I use your own definitions (whether or not they're standard in philosophiy--and it's unlikely that they are, given what I've found at SEP.), then I'll say this:

I trust that we agree that there can be instances of one hypothetical proposition implying another hypothetical proposition. Call it what you want. Substitute that term everywhere that I said "abstract if-then-facts", "abstract implication-facts", abstract facts about hypotheticals" or "abstract implication-facts about hypotheticals".

Michael Ossipoff

My reply to your questions will be along this afternoon, tonight or tomorrow morning. I'll make my best effort to have enough computer-time to send it this afternoon or tonight.

My replies tend to be long, because I like them to be complete, and that can mean that they take a little longer.

Michael Ossipoff

Maybe we could have saved all that unnecessary quibble if I'd just said "Implication" instead of "Implication Fact" or "If-Then Fact".

Or maybe it should be "True Implication".

Obviously one or both of those would work fine as a substitute for "If-Then Fact" or "Implication-Fact".

But, just out of curiosity:

Yes, an implication-proposition is false if its antecedent is true and its consequent is false.

But what about just an implication? One would expect "Implication" to have a stronger meaning than "Implication Proposition". ...that an implication-proposition isn't a genuine implication if the proposition isn't true.

I don't know what the conventional definition says, but I guess it must be one of the following three:

1. Maybe "True" and "False" aren't applicable to an implication (as opposed to an implication-proposition), because something wouldn't even be an implication if it weren't true. P either does or doesn't imply Q. It does, or there isn't that implication. So "True" and "False" don't even apply to an implication.

2. Maybe an implication (as opposed to an implication-proposition) is said to be necessarily true, because otherwise it wouldn't be an implication. This differs from #1, in that, instead of not having truth-value, it necessarily has a truth value of "True", because an Implication (as opposed to just an implication-proposition) is taken to be a true proposition.

3. Maybe "Implication" is taken to just mean "Implication Proposition", in which case of course an implication could be true or false (Of course if it's false, that's because its antecedent is true and its consequent false).

---------------------------

I guess I could say that more briefly by saying that either:

1. An Implication ( as opposed to an implication-proposition) is considered a fact.

2. Just saying "Implication" (as opposed to saying "implication-proposition") is considered to mean "true implication-proposition."

3. An implication is considered to be a proposition that could be true or false (Of course if it's false, that's because its antecedent is true and its consequent false).

--------------------------------------------

Of course I was saying "Implication-Fact" because I wanted to clarify that was referring to a fact. .
--------------------------------------------

If the official interpretation is #1 or #2, then I could substitute "Implication" for "Implication-Fact" and for "If-Then Fact".

If the official interpretation is #3, then it would be necessary to instead substitute "True Implication" for "Implication-Fact" and for "If-Then Fact".

-------------------------.

Though I've read a bit about implications and facts, I don't remember an answer to this question about those 3 possible things hat might be said or meant in the official meaning of "Implication". Of course the standard definition of "Implication" that I read was the same everywhere where I found it. I just don't remember anything in that definition about #1 vs #2 vs #3.

I guess I was saying "Implication-Fact" in order to emphasize that I meant #1 or #2

..but, if if #3 is official, then I wanted to emphasize that I was referring to true implications.

You might say I should do some searching for the answer to that question before mentioning it here.

I guess I bring it up here as a matter of curiosity, because there's been such intense and emphatic objection to the phrases that I've been using, that the subject has already come up, in a big way.

Anyway, bottom-line:

If the official meaning of Implication is consistent with #1 or #2, then I could substitute "Implication" for "Implication-Fact" and for "If-Then Fact".

But if the official meaning of "implication" is consistent with #3, then it would be necessary to instead substitute "True Implication" for "Implication-Fact" or "If-Then Fact".

Michael Ossipoff

Obviously an implication, an implying of one proposition by another, is a fact, and is not a proposition.

And, because an implication is a fact, then an implication-fact is obviously the kind of fact that is an implication.

...a bit redundant, but clear and un-ambiguous.

And I'd long been saying that the if-then facts to which I was referring to were implication facts.

But, even of itself, "if-then fact" was unambiguous. Obviously it would refer to a fact that if one thing is true then another thing is true.

Anyway, even if there could have been misunderstanding before, that silly quibble was cleared yesterday when I said that i was referring to an instance of one proposition implying another, and then calling it an implication (or, unnecessarily and redundantly) a true implication.

Michael Ossipoff

(I don't use nested quotes, because they don't seem to work. I separately quote what I was quoted saying, and the other person's reply)

I'd said:

Incorrect. Reincarnation is metaphysically-implied. There's an uncontroversial metaphysics that implies reincarnation.

I doubt there's such a thing as a noncontroversial metaphysics. Do you say you've got a hold of one? — Cabbage Farmer

Yes. See my previous posts about it in this thread.

I'd said:

If there's a reason why you're in a life (something that I've discussed here), and if that reason still obtains at the end of this life, then what does that suggest?

You replied:

What do you mean "a reason why you're in a life"?

That's best answered by saying what I don't mean: I don't mean the reason in terms of physical causation in this world. I'm talking about a reason more fundamental and original than that.

Is this reason supposed to generate the implication you've singled out?

Yes.

What do you mean by the phrase "in a life"?

Nothing other than what you surely must interpret it to mean.

One reason I'm alive is that I was born. One reason I was born is that I was conceived. Is this the sort of reason you have in mind?

Of course not.

In terms of physical causation in this physical world, you're alive because you were conceived and then born. No one denies that.

But this is a philosophy forum, not a biology forum.

First, a brief summary of my metaphysics (which I describe and justify in more detail in previous posts in this thread):

In the metaphysics that I propose, and described and justified in previous postings in this thread:

There are infinitely-many complex systems of inter-referring abstract implication-facts (instances of one hypothetical proposition implying another).

Among those infinitely many such systems, there is inevitably one whose events and relations are those of your experience.

There’s no reason to believe that your experience is other than that.

Of course I can’t prove that the Materialist’s objectively, concretely, fundamentally existent physical world, and its objectively, concretely existent stuff and things don’t superfluously exist, as an unverifiable and unfalsiable brute-fact, alongside of, and duplicating the events and relations of, the hypothetical logical system that I described above.

I emphasize that, in this metaphysics, I regard the experiencer and his/her experience as primary.

So then, why are you in a life?

You’re in a life because there’s a life-experience possibility-story, consisting of a hypothetical logical system such as I’ve described immediately above, having as its protagonist someone just like you…you, in fact.

That’s who/what you are…the hypothetical protagonist in a hypothetical life-experience possibility-story.

You’re in a life, as that protagonist.

If, at the end of this life, you still have subconscious future-orientedness, feelings of incompletion in any regard, &/or any subconscious predisposition, desire or need for life or the things &/or experiences in life, then there will still be a life-experience possibility-story about you

…one that starts with someone like you are during the sleep during death, when there’s no waking-consciousness or knowledge that you were in a life, or knowledge of whether you’re coming or going, or knowledge of the distinction between unconsciousness at the beginning of a life and at the end of a life.

Unconsciousness (absence of waking-consciousness) occurs during the sleep at death. There will come a time when you don’t remember this life, but you retain your subconscious wants, needs, predispositions, etc. …and there’s a life-experience possibility-story that starts about someone like you are at that time during death’s absence of waking-consciousness.

So, if the reason for this life remains, at the end of this life, then what does that suggest?

According to Hindu and Buddhist teachers and writings, nearly everyone has those predispositions, needs, wants, incompletions, etc., that will bring them into a next life.

There are a very, very few people who are so life-completed and lifestyle-perfected, that they won’t have those subconscious needs, wants, drives, inclinations, predispositions, feelings of incompletion, future-orientation, etc. Their placid sleep at the end of their life won’t have any resemblance to the bewilderment and subconscious striving-emotion at the beginning of a life, and hence there won’t be a life-experience possibility-story about them; they won’t be in such a story.

It’s said in Hinduism and Buddhism that every one of us, after many lives, will reach that life-completion and lifestyle-perfection described above. Don’t count on it this time. It’s probably very far off, as it is for nearly everyone.

Michael Ossipoff

I do need to wheen myself of making a fool of you — MetaphysicsNow

Making a fool of yourself, actually.

"Wheen" yourself of that by all means, if you can. :D

Really? You are reduced to counting spelling mistakes? Metaphysicsnow has revealed his Scottish roots.:wink:
I've been away a while and just skim read through the various posts. Aren't we still waiting for your explicit statement of premises, one by one, giving us a valid argument that takes us from those premises to the conclusion that reincarnation happens? I asked you for that right at the beginning of this thread and you've waffled on quite a lot it seems, but logically valid arguments seem impossible to dig out of your words.

I humbly apologise for my spelling mistake - as it happens I am not Scottish at all, but I happen to have finished Wheen's biography of Marx not so long ago, so maybe that's an explanation of my inexcusable syntactic error. Do you think I should go through all of MO's posts looking for a spelling mistake to score a philosophical point? I didn't realise we were playing a game of ignoratio elenchi, but if we are I think I'd have to yield victory to MO - he is the undoubted champion.

Actually, not the undoubted champion - I'm forgetting that there are some strong contenders on other posts I've been involved in.

Eh? This sounds like you're using the word "metaphysics" in a New Age sense, rather than a philosophical sense

Really? You are reduced to counting spelling mistakes? — jkg20

...then shall I count run-on sentences too?

Look, the gross mis-spelling, and the run-on sentence in that post are just common sloppinesses that typically, and in MetaphysicsNow's case, accompany other instances of sloppiness, such as MetaphysicsNow's mis-statement that associativity only appears to be an axiom in some systems, and that to prove that 2 + 2 = 4, all of the axioms for the counting numbers are needed. .

...and that's just in one post.

And no, at least in Merriam Webster, "Wheen" isn't listed as meaning "Wean" in any language. Merriam-Webster lists it with an adjective meaning and a noun meaning.

Wean is derived from an Old English word, "Wenian", to accustom or wean.

I've been away a while and just skim read through the various posts. Aren't we still waiting for your explicit statement of premises, one by one, giving us a valid argument that takes us from those premises to the conclusion that reincarnation happens?

No. Janus asked me to state, in regards to my argument for my metaphysics, a premise, conclusion, and to tell how the premise implies the conclusion.

His question was about the metaphysics ("your system"), and wasn't about the matter of reincarnation.

Reincarnation isn't part of my metaphysics. It's merely implied by it. And I've clarified that, when I say that, I mean "implied" in the ordinary weaker sense, not in the logic sense. I don't claim to have a "valid argument that takes us from...premises to the conclusion that reincarnation happens."

I've already told how my uncontroversial metaphysics implies reincarnation. ...in the sense of plausibly and convincingly suggesting it. Proving it? I don't say that.

I asked you for that right at the beginning of this thread and you've waffled on quite a lot it seems, but logically valid arguments seem impossible to dig out of your words.

When you asked about the metaphysics, I stated my metaphysical proposal, and my justification for it,

When Janus asked for an argument for my metaphysics, with clearly-labeled premise, conclusion, and statement of how the premise implies the conclusion, I posted that.

I've long been inviting people to specify a mis-statement or un-supported conclusion in my metaphysical proposal.

Of course that invitation still remains.

All the objections and disagreements that I've been getting are quibbles about terminology, which I've now amply answered.

Michael Ossipoff

Eh? This sounds like you're using the word "metaphysics" in a New Age sense, rather than a philosophical sense — gurugeorge

Can you be more specific about what I said that suggests that?

You mean reincarnation?

From as soon as I began participating in the discussion of reincarnation here, I've been clarifiying that reincarnation isn't part of my metaphysics. I've merely said that reincarnation is implied by my metaphysics. And I've also been clarifying that, when saying that, I don't mean "imply" in its logic sense, but only in its weaker ordinary sense...of strongly or convincingly suggesting.

Michael Ossipoff

And no, at least in Merriam Webster, "Wheen" isn't listed as meaning "Wean" in any language. Merriam-Webster lists it with an adjective meaning and a noun meaning.

Calm down, I didn't say that "wheen" meant what "wean" means. There is a word used in Scotland "wheen" which means "small amount of something" - I think, I stand to be corrected on its meaning, but its existence as a word I'm sure of. I thought maybe MN's being acquainted with that word accounted for his mispelling. Anyway, MetaphysicsNow's own explanation for the spelling mistake makes more sense - he meant wean but wrote "wheen" because he cannot get an author out of his head.

I didn't mean that you were only making fool of yourself via your gross mis-spelling, and your run-on sentence.

I was referring also to your mis-statements in that post, regarding associativity only appearing to be an axiom in one system, and your silly statement that all of the arithmetical axioms are needed in order to prove that 2+2=4.

...and that's just in one post,.

Michael Ossipoff

No. Janus asked me to state, in regards to my argument for my metaphysics, a premise, conclusion, and to tell how the premise implies the conclusion.

No, I asked you for premises and a sound argument (and all sound arguments are logically valid ones by the way), and you said you would provide at least the premises of the argument:

Presumably the idea is that there is a sound argument with metaphysical premises (i.e. premises which concern existence) which are acceptable to all and that has as for its conclusion that reincarnation happens. — jkg20


Yes, well said. That's what I mean.

So, Michael Ossipoff, over to you to lay out the premises one by one so we can subject them to scrutiny.


Will do.

Michael Ossipoff — michael ossipoff

You still haven't done that.

Consider this question: In all the time that you have been posting on forums have you found even one person who agrees with your 'argument'? — Janus

Though I answered that, I'd like to also say this:

One thing that I haven't found in reply, is a specific substantive and valid disagreement with what i said in my metaphysical proposal, or the specification of an incorrect statement or un-supported conclusion in that proposal.

You asked me for an argument for my metaphysical proposal, lising premise, conclusion, and a statement of how one implies the other..

I've posted those.

Now then, do you disagree with one or both of my premises?

(For "if-then fact", we can substitute "implication" (or true implication, if you prefer) or any other term that you like, for an instance of one proposition implying another..)

...or do you find an incorrect statement or unsupported conclusion in my discussion of why my conclusion is implied by my premises?

Michael Ossipoff

Now then, do you disagree with one or both of my premises?

I think the point is that nobody can really tell what your premises are. I suppose one of them must be
"There are if-then facts"
But when you give examples of what these if-then facts are supposed to be, you just give tautologies, and then refuse to engage in a discussion about the distinction between logical truth and substantive truth that risks being conflated when identifying tautologies as a kind of fact.

It's far from clear what you're saying in that post, but:

I've complied with Janus's request that I post, as an argument for my metaphysics, premise(s), conclusion, and as statement regarding how the former imply the latter. I did that

And, before that, I posted, in this thread, a complete statement of my metaphysical proposal and its uncontroversial justification.

It's easy for you to say "You haven't done that."

...but can you or can't you specify an incorrect premise, an incorrect statement, or an unsupported conclusion in my metaphysical proposal, or my argument for it?

As for reincarnation, let's get clear about this: I don't claim to have proof of it, or a logically-sound argument for it. I've merely told how my metaphysics implies it (but not in the strong sense that "imply" has in logic).

Nor is reincarnation part of my metaphysics, though it's implied by it.

Michael Ossipoff

The proofs that I've seen of 1+1=2 do require all the axioms of the number system being used, since the axioms together define what a natural number is and then certain of the axioms are reused in the recursive definition of addition. (You also need to make some symbolic definitional equivalences of course so that, for instance, "2" is defined as "1' " and so on). If that is true for 1+1=2, then its true for 2+2=4 as well. I don't see what mistake you believe MetaphysicsNow is making when he says you need all the axioms to prove 2+2=4. What system of axioms are you using, and what is the proof of 2+2=4 in that system?

"2" is defined as "1' " — jkg20

???

Three question marks - does that indicate that you are not aware of the proof for 1+1=2 within Peano arithmetic? Sorry. Here's a straightforward presentation which will give some context to the remark I made about the definition of the symbol "2".
http://mathforum.org/library/drmath/view/51551.html

First, definition of some counting-numbers:

Let "1" mean the multiplicative identity.

Let "2" mean 1 + 1
Let "3" mean 2 + 1
Let "4" mean 3 + 1

By those definitions, 2+ 2 means (1+1) + (1+1).

The additive associative axiom says:

a + (b + c) = (a + b) + c.

Applying that axiom:

(1+1) + (1+1) = ((1+1) +1) + 1

Because, by definition, 1 + 1 = 2:

((1+1) + 1) + 1 = (2+1) + 1

So (1+1) + (1+1) = (2+1) + 1

Because, by definition, 2 + 1 = 3:

(2+1) + 1 = 3 + 1

So (1+1) + (1+1) = 3 + 1

Because, by definition 3 + 1 = 4:

(1+1) + (1+1) = 4

Because, by definition, 1 + 1 = 2:

2 + 2 = 4 QED

(As I said before, the use of the multiplicative identity implies acceptance of the multiplicative identity axiom.)

Michael Ossipoff

I didn't notice the apostrophe.

I've heard of Peano arithmetic, but that wasn't my topic.

In answer to your question, the system of axioms that I'm using is the one that is usually used, stated and cited. ...the one in which, for one thing, associativity is an axiom, not a theorem.

Michael Ossipoff

Now then, do you disagree with one or both of my premises?

I think the point is that nobody can really tell what your premises are. I suppose one of them must be
"There are if-then facts".

Yes. And, because some didn't like the term "if-then" facts, I invited Janus to substitute some other term for an instance of one proposition implying another.

Then i suggested substituting "implication" for "if-then fact".

Or, if you think that an implication is a proposition then substitute "true implication".

We've been all over that topic, and I've amply answered that terminology-objection.
— jkg20

But when you give examples of what these if-then facts are supposed to be, you just give tautologies

As a particularly simple example of an implication, I gave the Slilthytoves syllogism.

As an additional example of an implication, I mentioned that the additive associative axiom and the multiplicative-identity axiom imply 2 + 2 = 4.

(...because the multiplicative identity is used in my definitions of counting-numbers).

But, in my statements of my metaphysical proposal, I spoke of various implications and propositions about implications.

As an example of how any fact about our physical world implies, corresponds to, and can be said as, an implication, including ordinary facts in our experience, I gave the example:

There being a traffic-roundabout at the intersection of 34th & Vine.

A fact that if you go to the intersection of 34th & Vine, you'll encounter, there, a traffic roundabout.

Also, I mentioned that a set of hypothetical physical quantity-values, and a hypothetical relation among them (called a "physical law") together comprise the antecedent of an implication...except that one of those hypothetical physical quantity-values can be taken as the consequent of that implication.

(It goes without saying that a proposed physical law starts out as a theory or hypothesis, and isn't called a law until it has been found to apply with sufficient consistency.)

I mentioned that a proved mathematical theorem is an implication whose antecedent includes (at least) 1 or more mathematical axioms.

So I've mentioned a number of kinds of implications, some of which relate to my metaphysical proposal.

(Here I've been using "implication" as a substitute for "if-then fact". I consider an implication, an implying of one proposition by another, to be a fact. ...in contrast to a proposition about an implication. But if you don't think an implication is necessarily a fact, then substitute "true implication".)

, and then refuse to engage in a discussion about the distinction between logical truth and substantive truth that risks being conflated when identifying tautologies as a kind of fact.

I'm speaking honestly when I say that I have no idea what you're talking about.

I've thoroughly discussed and answered the objections to my use of "if-then fact", and I've obligingly substituted another term, "implication". ...and, if that isn't suitable to you, then I invite you or anyone to choose a term for referring to an instance of one proposition implying another, and substitute it for "implication", the term that I use in this post.

By "substantive truth", do you mean an alleged objective, fundamental, concrete "existence" for our physical world and its things and its stuff?

I've admitted that I can't prove that Materialism's objectively fundamentally, concretely existent physical world, and is concretely and objectively existent stuff and things don't superficially exist, as an unverifiable and unfalsifiable brute-fact, alongside of, and duplicating the events and relations of, the hypothetical complex system of inter-referring abstract implications about hypotheticals, that I've been referring to.

Michael Ossipoff

In answer to your question, the system of axioms that I'm using is the one that is usually used, stated and cited.

Well, I don't want to turn this into a thread about mathematical logic, but Peano arithmetic is one of the standard ways of defining the natural numbers, so I'd have to challenge you to provide a more "usually used, stated or cited" system of axioms for doing that.
In any case, you cannot define "1" this way (as you do):

Let "1" mean the multiplicative identity.

without already having defined multiplication (recursively) over the natural numbers, which means that the natural numbers need already to have been defined within your system, which is what the Peano axioms do. Perhaps you have some non-standard set of axioms to capture what a natural number is supposed to be? But in any case, you will not be able to infer 2+2=4 without all those axioms. MetaphysicsNow is right about that and you are wrong.

By "substantive truth", do you mean an alleged objective, fundamental, concrete "existence" for our physical world and its things and its stuff?

Now we are getting somewhere. Idealists and anti-realists more generally can (and have) made the distinction between substantive truth and logical truth, so no, one does not need to be a materialist in order for the distinction to make sense. You seem to be a fan of online philosophy encyclopedias - look up "logical truth" and see how complex a notion it is and how various philosophers have tried to distinguish it from substantive truth. As far as I can tell, every example of if-then fact that you introduce is an example of a logical truth, but substantive truths are the ones that concern the empirical world (whether that world be independent of our coming to know it or not). Since we, as sentient beings, are in the empirical world, it is substantive truth that will have a bearing on whether or not we can be reincarnated, not logical truth.

Now, if you want to argue that there is no genuine distinction between logical truth and substantive truth, that might be an interesting discussion to have. Take a look at Quine's "Two Dogmas of Empiricism" for example.