There are several branches of logic but the science of logic as a whole is one coherent system. E.g. fuzzy logic is a branch that may be more suitable than other branches in some cases, but the different branches of logic do not contradict each other. — A Christian Philosophy

A logic system is built on axioms.

From The Foundations of Logical Reasoning: Axioms of Logic

Logic is the backbone of mathematical reasoning, providing the structure and rules that govern the validity of arguments and proofs. At the heart of logic are axioms—fundamental truths accepted without proof. These axioms serve as the foundational building blocks from which all logical reasoning is derived.

Axioms are assumptions taken to be true

From Wikipedia - Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

As any logic system is built on axioms, which are assumptions taken to be true, no one logic system exists necessarily.

However, the fact that they cannot both be said by the person at the same time does not imply that the person cannot have both ideas within one's mind at the same time. — Metaphysician Undercover

If it were possible to have two contradictory thoughts at the same time, then I could feel pain in my finger and not feel pain in my finger at the same time.

Language mirrors thoughts.

If it were possible to have two contradictory thoughts at the same time, then language would mirror this. For example, the proposition would be "I feel x in my finger", where "x" means feeling both pain and no pain at the same time.

However this is not the case. In language we say "one hour ago I felt no pain in my finger but now I feel pain in my finger".

Language by its very nature acknowledges that contradictory thoughts cannot be contemporaneous.
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Clearly people multitask, so they are thinking different ideas at the same time, required to do a number of different things at the same time, even though they cannot say everything that they are doing, all at the same time. — Metaphysician Undercover

A cyclist multi-tasks when they pedal and watch the road ahead at the same time. But thoughts about the road ahead should not be confused with the muscle memory of pedalling, which doesn't require thoughts.

A student multi-tasks when writing an essay and listens to music at the same time. But thoughts about what to write should not be confused with an instinctive pleasure in hearing music.

Musical pleasure and reward: mechanisms and dysfunction

Most people derive pleasure from music. Neuroimaging studies show that the reward system of the human brain is central to this experience. Specifically, the dorsal and ventral striatum release dopamine when listening to pleasurable music, and activity in these structures also codes the reward value of musical excerpts.

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How do you account for a person having many different ideas, in one's memory, all at the same time, which one cannot all say at the same time? — Metaphysician Undercover

I have many memories, none of which I am actively thinking about at this moment in time.
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Sure, you can state irrelevant conditionals, just like I can say that if I was not born yet, I would not be writing this right now, but such conditionals are not relevant to reality. — Metaphysician Undercover

Conditions in thought are essential to life:
If I don't eat, then I will die
If I cross the road now, then the approaching truck will run me over
If I don't apply for this job then it is unlikely that they will hire me
If an asteroid 15km in diameter hits the Earth, then most life may become extinct.
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Your if/then statement reveals nothing more than "if I was not born yet I would not be writing this right now" reveals. How do I get from this to believing that I was not born yet? And how do you get from your if/then statement to believing that determinism is the case? — Metaphysician Undercover

One cannot.

If I had not been born, then I would not be writing this post
I am writing this post
Therefore I was born

If Determinism is the case
then all thoughts are determined
I have the thought that my thoughts are not determined
therefore my thought that my thought has not been determined has been determined

then it should be called inevitablism, not determinism. Having determined something will happen is not the same as it being inevitable. — Barkon

Determinism seems to encompass more than Inevitabilism, and includes the concept of inevitability.

From Wikipedia Determinism

Determinism is the philosophical view that all events in the universe, including human decisions and actions, are causally inevitable

Wiktionary - Inevitabilism

The belief that certain developments are impossible to avoid; determinism.

Wiktionary - Determinism

The doctrine that all actions are determined by the current state and immutable laws of the universe, with no possibility of choice.

If determinism is true, and (it; who? What?) determines all our thoughts and actions — Barkon

OK. If Determinism is the case, and all our thoughts and actions are already determined, then your thought that you are free to choose is just another of those thoughts that have already been determined.

Determined by the nature of the Universe.

If determinism is true that all our actions are determined. That's all. It doesn't mean it's determined by causes external to our will. If it's determined that I will write this, then all that means is that it was probable that I would, thus it was determinable prior to the act. — Barkon

My understanding of Determinism is that your writing your post was inevitable, not probable.

From Wikipedia Determinism

Determinism is the philosophical view that all events in the universe, including human decisions and actions, are causally inevitable

From SEP - Causal Determinism

In order to get started we can begin with a loose and (nearly) all-encompassing definition as follows:
Determinism: Determinism is true of the world if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law.

This is a faulty argument because your designated time of "1pm" is completely arbitrary, and not representative of the true nature of time. As indicated by the relativity of simultaneity a precise designation of "what time it is", is frame of reference dependent.........................Do you agree, that by the special theory of relativity, event A could be prior to event B from one frame of reference, and posterior from another frame of reference? — Metaphysician Undercover

In my location, 1pm is simultaneous with my picking up a cup of coffee.
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As I explained in my last post, having two contradictory ideas at the same time is exactly what deliberation consists of. "Should I stay or should I go". — Metaphysician Undercover

This is why the words in the proposition "should I stay or should I go" are sequential. First one asks "should I stay" and then at a later time one asks "should I go".

Propositions, in that they mirror thoughts, are sequential.
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The problem here, is that you are treating a human subject as if one is a material object, to which the fundamental laws of logic (identity, noncontradiction, excluded middle), apply. — Metaphysician Undercover

This is the case in a Deterministic world.
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What's the point to even asking why matter obeys God, if you do not even believe that matter obeys God. — Metaphysician Undercover

It is not only a question of whether or not matter obeys God, it is also the question of is there a God.
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I know that I am free to choose, from introspection, analysis of my own experience. — Metaphysician Undercover

If Determinism is the case, and determines all our thoughts and actions, then your thought that you are free to choose is just another of those thoughts that have already been determined.

There can be thoughts not resulting in acts.

For example, I may think that Monet's "Water-lilies" is aesthetic or I may think that it is not aesthetic. Neither thought requires me to act on the thought.

Why does something being determined mean that the person has no control? Perhaps it's just predictable behaviour. — Barkon

If Determinism is true, then all our thoughts and actions are determined by causes external to our will. Our future is already written, and all our thoughts and actions are a consequence of preceding events.

In that sense, if all our thoughts and actions are determined, then it is true that we have no control.

However, in ordinary language, we do say things like "he was determined not to waste a single minute of his time" and "she was weak and the pain was excruciating, but she was determined to go home."

But the fact that a person is determined to do something, does not mean that their determination cannot be explained within Determinism.

If Determinism determines all our thoughts and actions, our being determined is just one of these thoughts, meaning that it is Determinism that determines our being determined to do something.

An argument against Free Will
At 1pm exactly I have the idea to pick up a cup of coffee.

Assuming free will, at T seconds prior to 1pm, it hasn't been determined whether at 1pm I will have the idea to pick up the cup of coffee or not to pick up the cup of coffee.

Suppose T is 1 second. If it has been determined at 1 second before 1pm that I have the idea at 1pm to pick up the cup of coffee then this is no longer free will.

Suppose T is ${10^{-100}}$ seconds. If it has been determined at ${10^{-100}}$ second before 1pm that I have the idea at 1pm to pick up the cup of coffee then this is no longer free will.

"T" can be any number

Therefore, free will only applies if I choose between picking up the cup of coffee and not picking up the cup of coffee at 1pm exactly.

But this means that at 1pm I have two contradictory ideas in my mind at exactly the same time. But this is impossible, meaning that free will cannot be a valid theory.

I have seen evidence that a person can have two contradictory ideas consecutively, but I have never seen any evidence that a person can have two contradictory ideas at the same time.
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I don't see how this is relevant. — Metaphysician Undercover

Having an idea is not evidence for free will if ideas have been causally determined in a causally determined world.
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Then you do not accept my explanation. — Metaphysician Undercover

You have described a world where things obey the laws of nature, but I don't see where you have explained why things obey the laws of nature.
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Free will is not about the thoughts, it concerns the acts. — Metaphysician Undercover

I thought free will referred to our being free to have whatever thoughts we wanted

This sounds more like instinct, in that I look at a bright light and instinctively close my eyes.
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That's what choice and deliberation is all about, having contradictory thoughts at the same time. — Metaphysician Undercover

I agree that a person can have two contradictory thoughts consecutively, but it would be impossible for a person to have two contradictory thoughts contemporaneously.
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You are equally free to reach out for the coffee, or to not reach out for the coffee. You are free to choose. — Metaphysician Undercover

How do you know that we are free to choose?

How do you know that we don't live in a causally determined world, where our actions have been causally determined?

Haven't you seen parts of your body start to move without being acted on by an external force? If the "reason" for movement is an immaterial "idea", then this is evidence of free will. Isn't it? — Metaphysician Undercover

No. Suppose a person has the idea to reach out for a cup of coffee.

On the one hand, assuming free will, a person can have the idea to reach out for a cup of coffee. On the other hand, assuming there is no free will, a person can also have the idea to reach out for a cup of coffee.

Having an idea is nether evidence for or against free will.
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I was the one who used "law of nature" — Metaphysician Undercover

I have never said that I want you to be using the term "law of nature" in a different way.

My point has been that I don't accept that a law of nature precedes an event and makes things act the way they do.

A "law of nature" in this sense necessarily precedes the event, because the laws of nature are what makes things act the way that they do. — Metaphysician Undercover

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The concept of "free will" does not involve self-causation. — Metaphysician Undercover

At 1pm a person has the thought to reach out for a cup of coffee.

Free will means that at 1pm that person could equally have had the thought not to reach out for the cup of coffee.

It is not possible to have two contradictory thoughts contemporaneously, both to reach out and not reach out.

One the one hand, free will is equally free have either of two contradictory thoughts, but on the other hand, free will is equally free to choose to act on one of these thoughts.

It seems that if free will is equally free to act on the thought of reaching out rather than not reaching out, then it is equally free to act of the thought of not reaching out as rather than reaching out.

If free to make any decision, then there would be no reason to make any decision, leading to the inability to be able to make any decision at all.

demonstrate to me how introspection revealed to you that free will is an illusion, and you live in a deterministic world, — Metaphysician Undercover

I don't believe in particular that thoughts can cause themselves, and I don't believe in general in spontaneous self-causation.

One reason for my disbelief in spontaneous self-causation is that it is something I have never observed.

When I see a billiard ball on a billiard table start to move for no reason at all, then I may change my mind.
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No "reason why" is given for that law, it is stated as a descriptive fact — Metaphysician Undercover

Law of nature has more than one meaning.

It can be a description, as Newton's first law. From SEP - Laws of Nature

Science includes many principles at least once thought to be laws of nature: Newton’s law of gravitation, his three laws of motion, the ideal gas laws, Mendel’s laws, the laws of supply and demand, and so on.

It can be an explanation. As you wrote:

A "law of nature" in this sense necessarily precedes the event, because the laws of nature are what makes things act the way that they do. — Metaphysician Undercover

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However, I see no reason to discuss them if they are just proposed as reason to accept the illogical premise of contemporaneousness. — Metaphysician Undercover

One of the reasons I don't believe in free will is that it requires self-causation, where the thought one has is contemporaneous with the decision to have the thought.

Isn't FREE WILL time based ? — Corvus

At exactly 1pm I decide to press the letter "T" on my keyboard. If free will is the case, at exactly 1pm, I could equally decide whether to press or not press the letter"T". But at exactly 1pm I did decide press the letter "T".

By the Law of Contradiction, free will cannot be the case, as it would result in a contradiction. At exactly 1pm I can't equally decide to press or not press the letter "T" and decide to press the letter "T" at the same time.

Personally, I don't see too much point in discussing philosophy with someone who doesn't believe in free will. The entire discussion would then have to revolve around persuading the person that they have the power (free will) to change that belief. And this "persuading" would have to carry the force of a deterministic cause, to change that person's mind, which is contrary to the principles believed in by the person who believes in free will. This makes the task of convincing a person of the reality of free wil an exercise in futility. The only way that a person will come to believe in the reality of free will is through introspection, examination of one's own personal experiences. — Metaphysician Undercover

Free Will
A person hears an argument.

If that person has free will, then they are free to accept or reject the argument.

If that person has no free will, then it has been pre-determined whether they accept or reject the argument, and it is possible that they either accept or reject the argument.

Therefore, if I observe someone hearing an argument, my observing whether they accept or reject the argument is no guide as to whether or not they have free will.

Introspection
If a person has free will, through introspection they are free to reject the idea that they have free will, and conclude that they live in a deterministic world.

If a person has no free will, during introspection, it may have been pre-determined that they accept the idea that they have free will.

Introspection is no guide as to whether free will is an illusion or not.
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The Laws of Physics are the map (description), and the Laws of Nature are what is supposedly described by the map — Metaphysician Undercover

Two meanings of Law of Nature
It depends what you mean by "Law of Nature", because it has two possible interpretations.

Looking at Newton's First Law of Motion, possible meaning one is as a description, in that an object at rest will remain at rest until acted upon by an external force.

Possible meaning two is the reason why an object at rest will remain at rest until acted upon by an external force

I agree that there is a difference between a description of what happens and a reason why it happens

Looking at possible meaning two
Looking at why something happens, why an object will remain at rest until acted upon by an external force.

One question is, is the Law of Nature that an object remains at rest external and prior to the object or internal and contemporaneous with the object.

If this Law is external and prior to any particular object, and applies equally to all objects in space and time, then this raises the practical problem of where exactly does this Law exist?

If the Law is internal and contemporaneous within particular objects, and all objects in space and time follow the same Law, then this raises the practical problem as to why all these individual Laws, both spatially and temporally separate, are the same?

How exactly can there be a single Law of Nature that determines what happens to objects that are spatially and temporally separate?

As per the OP section "Argument in defence of the PSR", logic (and the PSR) are first principles of metaphysics. This means they exist in all possibe worlds, which means they have necessary existence. Thus, logic and the PSR exist necessarily or inherently. This is an internal reason which is valid under the PSR. — A Christian Philosophy

There are many different type of logic, suggesting that no one logic exists necessarily. For example, there is Propositional Logic, First Order Logic, Second order logic , Higher order logic, Fuzzy logic, Modal logic, Intuitionistic Logic, Dialetheism, etc.

Logical systems also change, also suggesting that no one logical system is necessary. For example, today few would maintain that Aristotle's logic doesn't have serious limitations.

The Law of Identity "A is A" is one of the three Laws of Thought.

The Laws of Thought are axiomatic rules, taken to be true to serve as a premise or starting point for further reasoning and argument. In other words, taken to be true but not necessarily true.

The Law of Identity was described as fundamental by Aristotle, as primitive by Leibniz and to a certain extent arbitrary by George Boole.

"A is A" exists as a convenient axiom, not as a necessity.

The PSR states that everything has a sufficient reason. It is true that we use the Law of Identity "A is A" for a reason, but this is an external reason, in that it is convenient for further reasoning and argument. This is not a "sufficient reason" in terms of the PSR

We use the Law of Identity as an axiom as a convenience not because it has any internal necessity.

The Law of Identity, as an example of logic, is used for the external reason that it is convenient in further reasoning, not from any reason of internal necessity.

Yes, they have the freedom to do this. I don't believe that, do you? — Metaphysician Undercover

I think it is more likely that Free Will is an illusion than an actual thing.
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My usage was the latter sense of "laws of nature". — Metaphysician Undercover

In modern days we understand this as inductive reasoning, cause and effect, and laws of physics. This inclines us to think that these formulae are abstractions, the product of human minds, existing as ideas in human minds. And this is correct, but this way of thinking detracts from the need to consider some sort of "form" which preexists such events, and determines their nature. — Metaphysician Undercover

A "law of nature" in this sense necessarily precedes the event, because the laws of nature are what makes things act the way that they do. — Metaphysician Undercover

Laws of Nature
IE, you were referring to the Laws of Nature as "principles which govern the natural phenomena of the world" rather than "descriptions of the way the world is".

The question is, is it strictly true that "descriptions of the way the world is" are posterior to events and "principles which govern the natural phenomena of the world" are prior to events?

There is an overlap in Laws of Physics and Laws of Nature. For example, Newton's three laws of motion are described by the SEP article Laws of Nature as Laws of Nature and are described by the web site www.examples com as Laws of Physics.

By observing many times that the sun rises in the east, by inductive reasoning, I can propose the law that "the sun rises in the east". It is true that this law is posterior to my observations. But it is equally true that this law is prior to my observing the next sun rise.

When does a law become a Law of Nature?

If for hundreds of years hundreds of scientist have observed that F=ma, then this is sufficient for F=ma to become a Law of Nature.

But in principle the Law of Nature that F=ma is no different to my law that "the sun rises in the east", apart from the number of observations.

This Law of Nature is posterior to observations and prior to the next observation in exactly the same way that my law was posterior to my observations and prior to my next observation.

Whilst it is true that Laws of Nature are prior to the next event, they are also posterior to previous events.

The Law of Nature that F=ma is not the cause of the next event, it does not make the next event act as it does act and it does not determine the next event, but is a prediction about what the next event will be based on past experience.

Aristotle
Whereas for Plato Form is prior to physical things, for Aristotle's hylomorphic scheme, Form and Matter are intertwined. It may well be that Form is Matter, united by the Formal Cause.

As Form cannot exist independently of Matter, Form cannot exist prior to Matter but must be contemporaneous with it.

Free will allows a new, undetermined event to enter into the chain of causation determined by the past, at any moment in time. — Metaphysician Undercover

Some argue that Free Will is an illusion.

Since the prior forms are "idea-like" as immaterial, and the cause of things being the way that they are, in much the same way that human ideas cause artificial things to be the way that they are, through freely willed activities, we posit a divine mind, "God". — Metaphysician Undercover

The material and the immaterial

I can understand a God as being a prior cause to physical events, providing one accepts the possibility of a God.

I agree that human concepts can cause changes in the physical world, in that having the concept of thirst can cause a bottle of water in the world to move

The existence of Free Will is debated. Some argue that it is an illusion.

However, I don't agree that concepts in the mind and the Laws of Nature in a world outside the mind are immaterial, but rather that they are fully material.

As regards the particular Law of Nature that when there are regions of excess positive and negative charge within a cloud lightning occurs, there is nothing immaterial about this. The event is fully explainable as the deterministic behaviour of matter and forces between matter.

As regards concepts in the mind, as software exists within the hardware of a computer, concepts exist in the physical structure of the brain. If change the physical structure of the brain, then change the concepts within that physical structure.

No evidence has ever been presented of the dissociation of concepts from the brain, in that if a living brain moved from the living room to the dining room, no one would suggest the possibility of the concepts remaining in the living room.

IE, some may believe in God and Free Will, but it seems to me that they are not necessary as explanations of the relationship between mind and world outside the mind.

A common way of representing the difference between the two types of "form" are as the laws of physics (human abstractions), and the laws of nature (what the laws of physics are supposed to represent, which causes things to behave the way that they do). Aristotle provided much guidance for separating the two senses of "form", the causal as prior to events, and the human abstractions as posterior to events. — Metaphysician Undercover

There is a lot in your post, but taking your points one by one.

I observe a hundred times that when there are regions of excess positive and negative charge within a cloud then lightning occurs. I can ask why.

I can conclude that there is a Law of Nature such that when there are regions of excess positive and negative charge within a cloud lightning occurs.

There are two aspects to a "Law of Nature": as it exists in the mind and as it exists in a world outside the mind.

As regards the Law of Nature as it exists in a world outside the mind

Am I right is thinking that an Aristotelian Form and Law of Nature are analogous?

My question is, is it in fact the case that a Law of Nature precedes the event it describes, or is the Law of Nature contemporaneous with the event it describes. My belief is the latter.

If the Law of Nature is contemporaneous with the event it describes, and if the Aristotelian Form may be thought of as a Law of Nature, then the Aristotelian Form will also be contemporaneous with the event it describes.

Suppose there is a cloud with regions of excess positive and negative charge. This is not a prediction of a future lightning strike, but immediately gives rise to a lightning strike. The Law of Nature determines what does happen not what will happen.

It therefore seems that an Aristotelian Form, as with a Law of Nature, rather than pre-existing an event, can only be contemporaneous with an event.

IE, the Laws of Nature as abstractions in the mind are a posteriori to events, but the Laws of Nature in a world outside the mind, and by analogy the Aristotelian Forms, must be contemporaneous with events.

Plato thought that since things exist as types, then the form, or type, idea, must be prior to the thing itself, to cause it to be the type of thing that it is — Metaphysician Undercover

There is a particular lightning strike, and being a particualr instance is a token. Several lightning strikes would create a class of events, The Lightning Strike, which would be a type.

In practice, can anyone give any explanation, other than in the mind of God, where a Lightning Strike could exist prior to a lightning strike?
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Aristotle showed that since a particular thing has a form unique to itself, which must be prior in time to the thing itself to account for it being the thing that it is and not something else, forms must be prior to material things. — Metaphysician Undercover

A particular lightning strike has a particular form, which is unique to itself, and is different to the form of tree.

Suppose the form of the lightning strike existed 1,000 years before the actual lightning strike happened. Then by the same logic, to account for the form of the lightning strike being the thing that it is rather than something else, the form of the lightning strike must have existed prior to 1,000 years before the actual lightning strike, ad infinitum until the beginning of existence.

Therefore, the form of the lightning strike must have existed at the beginning of existence. Similarly the form of every event must have existed at the beginning of existence.

In other words, according to Aristotle, the form of this post, which has a form unique to itself, must have been determined at the beginning of existence, 13.7 billion years ago, which is a scary thought.
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This indicates that there must be something similar to ideas, forms, which are prior in time to material existence, therefore outside of human minds. — Metaphysician Undercover

Expanding to minds rather than just human minds, as as I am sure that the dinosaur had a mind.

I have the idea that lightening strikes are terrifying.

In practice, prior to minds, what was terrified by the idea of a lightning strike?
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Isn't "force" just a concept? — Metaphysician Undercover

Yes. That's the problem.

But every word in language refers to a concept, in that "fundamental" is a concept, "particle" is a concept, "and" is a concept, etc.

It can also be argued that every word in language should be taken as a figure of speech rather than literally. For example, George Lakoff and Mark Johnson wrote the book Metaphors We Live By 1980. In science, Andrew May in Science 2000 argued that even Newton's second law, F = ma is a metaphor.

But concepts don't exist outside the mind.

Therefore, the problem is that language is using concepts which only exist in the mind to describe a world that exists outside the mind, where such concepts don't exist.

I agree that I am using the concept of "force", which exists in my mind, to describe something in the world, even though the concept "force" doesn't exist in the world.

And this is true for every word in language.

Language as a whole is using concepts, including the colour red and number, to describe a world where those concepts don't exist.

One possible conclusion would be that the world we observe exists in the mind, not outside the mind, in a Kantian kind of way.

How could a wavelength of 700nm exist in the world? — Metaphysician Undercover

You are right. How can I say on the one hand that "numbers exist in the mind but not the world" but on the other hand say that "science tells me that a wavelength of 700nm exists in the world", when a wavelength of 700nm depends on the existence of numbers.

All I can say is that in order to communicate my ideas I can only use language.

My premise is that ideas only exist in the mind. This would lead to the paradox that if I am able to successfully communicate my ideas using language, then it follows that, as language exists outside the mind, these ideas now exist outside the mind, thereby negating my original premise.

All I can conclude is, as ideas only exist in the mind, and language exists outside the mind, it is impossible to communicate my ideas using language.

This sounds a bit like 6.54 of Wittgenstein's TLP, where he wrote

My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.)

However, saying that, as an Existence Nihilist, I don't believe that wavelengths exist in the world. Hawthorne and Cortens (1995) spoke for the nihilist thus: “the concept of an object has no place in a perspicuous characterization of reality”.

I believe a world outside the mind exists, but not a world of objects, whether chairs or wavelengths, but rather a world of fundamental particles and forces existing in space and time.

As regards language, it can be argued that language, including the language of science, is more metaphorical rather than literal.

For example, metaphors are commonly used in science, such as: evolution by natural selection, F = ma, the wave theory of light, DNA is the code of life, the genome is the book of life, gravity, dendritic branches, Maxwell's Demon, Schrödinger’s cat, Einstein’s twins, greenhouse gas, the battle against cancer, faith in a hypothesis, the miracle of consciousness, the gift of understanding, the laws of physics, the language of mathematics, deserving an effective mathematics, etc

In this sense, I am using the expression "numbers exist in the mind but not the world" literally and the expression "science tells me that a wavelength of 700nm exists in the world" metaphorically.

Unfortunately, it is in the nature of language to mix literal and metaphorical expressions, and it is only the context that enables the reader to distinguish between the two.

Colors we project mentally are compatible to what exists in nature it seems, we know or assume others are projecting that color as well...animals, plants included living things adapting to environment and survival instincts have developed with time. — Kizzy

The OP is about the ontological status of ideas. It goes on to ask "So, chairs exists and numbers subsist? Is that a common understanding?"

I agree when you say:
1) "Colors we project mentally are compatible to what exists in nature it seems"
2) "Vision is an important sense that humans/animals have."

As regards:
3) "we know or assume others are projecting that color as well.......animals, plants included living things adapting to environment and survival instincts have developed with time."

I agree that it is more than likely that when I see a postbox emitting a wavelength of 700mn, my subjective experience of the colour red is the same as everyone else's, all things being considered equal. After all, life has evolved over 3 billion years in synergy with its environment. However, it is not something that I can ever know in the absence of telepathy.

When I observe a postbox, I know that the colour red exists in my mind, and science tells me that a wavelength of 700nm exists in the world.

When I observe the world, I see the colour red, meaning that either i) a wavelength of 700nm is the colour red in the absence of any observer, which I find hard to accept or ii) I project my subjective experience of the colour red onto the world, which I find easier to accept.

As with colour, similarly with number, such that numbers exist in my mind but not the world. When I observe the world and see numbers, I have projected my subjective experience of numbers onto the world.

As colour exists in the mind but not the world, numbers exist in the mind but not the world.

Regarding the word "exist, as unicorns exist in literature but not the world, colours and numbers exist in the mind but not the world.

The word "subsist" is unnecessary.

Logic has a reason for existing, as provided in the OP under section "Argument in defence of the PSR". — A Christian Philosophy

Is a logical necessity a sufficient reason or a Brute Fact.

I agree that:
1) The Unrestricted PSR states that everything has a sufficient reason
2) "Logic is part of the fabric of reality"
3) Logically A = A

However, what is the reason that A = A?
What is the reason for the existence of logic?
If logic has no reason, then logic is a Brute Fact.
Brute Facts are unexplainable and uncaused.
That logic is part of the fabric of reality is not sufficient reason why logic is part of the fabric of reality.
If, for example, A = A is a brute fact, then the Unrestricted PSR is no longer valid.

As regards your argument that "logic has a reason for existing"

I agree that
1) Truth means conformity with Reality
2) We discover Truth using Reason
3) Reason uses Deduction and Induction
4) An example of Deduction = i) the sun rises in the east, ii) therefore tomorrow the sun will rise in the east
5) An example of Induction = i) for the past 100 days the sun has risen in the east, ii) therefore the sun rises in the east.
6) If reason can find truth, then reason must mirror reality.

However, I don't agree that
1) "deduction is equivalent to the principle (or laws) of logic, in that deduction is based on an axiom (the sun rises in the east) that may or may not be true.
2) "induction is equivalent to the principle of sufficient reason", in that induction is assuming the regularity of nature, which may or may not be true, as pointed out by Hume.
3) "Reason finds truth", in that there is no logical necessity that either deduction or induction find the truth.

IE, we have no reason to think that logic has a reason for existing.

If logic has no reason for existing, then logic is a brute fact.

If logic is a brute fact, then the Unrestricted PSR is not valid.

Clearly sufficient reason and sufficient cause are there, whereas reason is more logic and cause is more physical. — jgill

An example of logic would be the Modus Ponens. i) IF P THEN Q ii) P iii) THEN Q

Let P and Q be physical facts, such that P = it rains and Q = I get wet

In Ordinary Language:
The reason I get wet is because it rains.
The cause of my getting wet is because it is raining.

Reason and cause can be purely logical:
The reason for Q is P providing that IF P THEN Q.
The cause of Q is P providing that IF P THEN Q.

Such logic may then be applied to physical facts.

Reason and cause are treated the same.

The fact that we are discussing something is not the evidence for existence of something. We can discuss about the unicorn or a flying pig. Does it mean the unicorn or flying pig exist? — Corvus

They exist in literature and in the mind as thoughts.

If either the unicorn or flying pig didn't exist somewhere, then you couldn't have written your post.
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"Numbers and colours exist somewhere"? Somewhere is like saying nowhere. — Corvus

The expression "exists somewhere" does not mean "exists nowhere".

Where about in the brain do you see numbers existing in physical form? — Corvus

P1 - Numbers and colours exist somewhere otherwise we couldn't be discussing them.
P2 - We know that numbers and colours exist in the mind because we are aware of them.
P3 - We observe the colour red even though colours don't exist in the world. What exists in the world is the wavelength 700nm. We mentally project the colour red onto the world that we are observing.

C1 - Similarly, it is possible that even though we observe numbers in the world, numbers need not exist in the world, as we can mentally project numbers onto the world.
C2 - As it is possible that numbers need not exist in the world, there need be no metaphysical angst about numbers existing in the world.

P4 - A calculator can manipulate numbers within the physical structure of the calculator in a deterministic and explainable way.

C3 - There needs be no metaphysical angst about numbers existing in a calculator.

P5 - Similarly, there is no reason why the brain cannot manipulate numbers within the physical structure of the brain in a deterministic and explainable way.

C4 - Similarly, there needs be no metaphysical angst about numbers existing in the mind.
C5 - Any metaphysical angst about numbers is unnecessary.

Logical necessity is a type of sufficient reason. It is reason type 1 in the OP section "PSR in Metaphysics". — A Christian Philosophy

If you accept an unrestricted form of the Principle of Sufficient Reason (PSR), you will require an explanation for any fact, or in other words, you will reject the possibility of any brute, or unexplainable, facts. (SEP - PSR)

Is logical necessity a brute fact?

For example, is the logical necessity that "A triangle has three sides" a brute fact?

A Brute Fact has no explanation.

I am not clear on the distinction between logical necessity and brute fact.

For example, there is something that has three sides and is named triangle.

In what sense is "triangle" an explanation of "something that has three sides"?

If not an explanation, then it is a brute fact that a "triangle" is "something that has three sides"

In that case, the Unrestricted PSR is no longer valid.

You apply the thoughts onto the physical world i.e. typing, measuring, hammering, drilling, and driving ... etc. You have ideas how to use and manipulate the physical objects. But the ideas are in your head, not in the world. — Corvus

Ideas are in the head, but ideas can change the world.

If ideas in the brain exist as part of the physical structure of the brain, then this would explain how an idea can change the physical world.

In the same way that an idea physically exists within the brain, numbers, being ideas, would also only exist within the brain in physical form.

I had thoughts, but I wouldn't say the thought existed. You cannot use "exist" on the abstract concepts. — Corvus

If thoughts didn't exist, then how can a thought affect the physical world, such that the thought of pressing the "t" key on the keyboard turns into actually pressing the "t" key on the keyboard.

That sounds like a categorical mistake. It is not matter of real or unreal. It is matter of knowing or not knowing — Corvus

Thoughts exist, otherwise you couldn't have written your post.

Thoughts exist in the physical brain which exists in the physical Universe.

Thoughts must be real otherwise it wouldn't be possible to write posts on the Forum.

That is not what I asked. I asked which version says that it is contingent on our knowing that an event has occurred.............Then you reject every version of the PSR that does not explicitly state that the principle only applies to events we know of.........................It makes an ontological claim. — Fooloso4

There are no definitive versions of the PSR. The PSR is a family of principles.

The PSR is, in fact, a family of principles.......................Variants of the PSR may be generated not only by placing restrictions on the relata at stake (both the explananda and the explanantia), but also on the notion of the relation at stake. (SEP - PSR)

The PSR may be factive, true in actual or possible worlds, or regulative, guiding our study of nature.

Another distinction can be drawn between a factive, as opposed to merely regulative, version of the Principle. A regulative version of the PSR would consider it as a condition for intelligibility (on a par with the Law of Non-Contradiction) and thus as guiding our studying of nature. The factive version simply states that the Principle is true in actuality (or even in all possible worlds). (SEP - PSR)

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How do you know that? — Fooloso4

Principles exist in the mind, not outside the mind.
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Until recently we did not know it existed. We now know it does. According to the PSR it must have a reason for existing. That reason was not created by our discovery of it. — Fooloso4

I can understand a principle that states that we know that things we know about must have a reason, but I find it hard to accept a principle that states that we know that things we don't even know about must have a reason.

Do you know of any argument justifying how we can know that something that we don't even know about must have a reason?

You mean all the science fiction books are real stories? Or merely exist in the authors' minds. — jgill

If a science fiction adventure is being talked about, the science fiction adventure must exist somewhere, whether as print in a book, in the mind of the author or reader or in a world outside any book or mind.

Is a thought in the mind any less real than something in a world outside any mind?

And which of those versions says that it is contingent on our knowing that an event has occurred? — Fooloso4

There is no one definitive version of the PSR.

You interpret the PSR as saying that every event, known or unknown, must have a reason.

It is not a contradiction. An event is something that happens. According to the PSR there is a reason for it happening. Our knowledge of something happening is not a requirement for it to happen. The Webb telescope has detected the earliest known galaxy, JADES-GS-z14-0, which formed about 290 million years after the Big Bang. There is a reason for it happening, whether we know it happened or not. — Fooloso4

I don't believe that the PSR can logically be formulated to apply to unknown events.
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We can now see events that occurred millions of years ago, how does our seeing it now but not previously change what occurred or why it occurred? — Fooloso4

I agree that our seeing an event that occurred millions of years ago doesn't change the original event

It might be that the original event occurred for a reason, or it might be that the original event occurred for no reason. EG, if an event happens where two single objects come together to form two objects, in what sense can the reason for two objects be two single objects.

The Principle of Sufficient Reason is a principle, and principles only exist in the mind.

When the original event happened, the event wasn't following the principle that it could only happen if there was a reason.

As principles only exist in the mind, the mind can only apply principles to things it knows about, meaning that the mind cannot apply principles to things it doesn't know about.

The original event wasn't determined by a Principle. It is only the mind that can determine whether an event followed a Principle, and these can only be events known by the mind.
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We cannot say anything about an event we know nothing about — Fooloso4

I agree when you say that we cannot say anything about an event we know nothing about.

But then you say we can say something about an event we know nothing about, ie, that it must have a reason.

I asked you.................Whose version of the PSR are you relying on?: — Fooloso4

There is no one version of the PSR. There are different formulations. The PSR is a family of principles (SEP - PSR).

For Leibniz, God knows all events whether known or unknown by humans. I am making the case that in the absence of a God, it wouldn't be sensible to apply a PSR to unknown events.

Is there any argument that could explain how we can know something about an unknown event, such as the unknown event having a reason?

So, then, if the first even prime greater than 100 didn't exist I couldn't be writing about it? — Art48

Yes, if the first even prime greater than 100 didn't exist, you couldn't be writing about it.

But you are writing about the first even prime greater than 100, so it must exist.

If something doesn't exist, it is not possible to write about it. If something is being written about, then it must exist somewhere.

Similarly, in the expression "it's said that numbers don't exist", then if something doesn't exist, then how is it possible to write about it.

But you are writing about numbers, so they must exist somewhere, otherwise you couldn't be writing about them.

What is missing in the above is the location of the existence, whether in the mind or in a world outside the mind.

I've seen some YouTube videos where it's said that numbers don't exist. — Art48

If numbers didn't exist, then you couldn't be writing about them, so they must exist somewhere.

I don't propose it. I cite it. — Fooloso4

There are different formulations of the PSR. You cite one version of it. See SEP - Principle of Sufficient Reason.
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The principle is not based on our ability to know the reason, but rather states that there must be a reason. I do not know that there is a reason or that there is not a reason for everything — Fooloso4

A principle that cannot be justified shouldn't be used.

My argument is that if you accept the PRS then you must accept that there is a reason for everything whether that reason is known to us or not — Fooloso4

You propose a formulation of the PSR that states that for everything, whether known or unknown, there must be a reason.

You must feel that there is a justification for this particular formulation.

But do you know of any argument justifying that there must be a reason for things not even known about?

If not, then why accept this formulation?

Since the PSR states that every thing must have a sufficient reason, no exception, then both 2) and 3) would be deniers of the PSR — A Christian Philosophy

You defend the PSR, in that for any thing that exists or is true there is sufficient reason.

Principle of Sufficient Reason (PSR): For any thing that exists or is true, there is a sufficient reason for it to exist or to be true.............We then defend its validity as a first principle — A Christian Philosophy

You also give the examples of things that don't have a sufficient reason, but rather logical necessity, such as "All bachelors are unmarried".

Not all explanations are external to the thing explained. Here are examples of things that are explained by an internal reason, that is, out of logical necessity or inherently. — A Christian Philosophy

Your position seems to fit in with item 2).

We cannot say what that reason is if the thing or event is unknown, but it must have a reason whether we know it or not. — Fooloso4

For Leibniz, God knows all events whether known or unknown by humans.

Can you justify your statement above, in that if an event is unknown to humans then we as humans know that it must have a reason, even if we as humans don't know what the particular reason is.

Is your argument based on the existence of a God?

Are you arguing against the PSR? — Fooloso4

No, I am arguing that the PSR cannot be applied to unknown events

Therefore the PSR cannot be applied to the unknown. — RussellA

If the PSR is valid it should hold for all events whether known or unknown — Fooloso4

I am arguing that it is not possible to know about something that we don't know about, including any reason for the something that we don't know anything about.

There is a reason for it happening, whether we know it happened or not. — Fooloso4

My question is, how is it possible to know the reason for something that we don't know about?