From the proof process, we came to know that the assumption P -> Q is not true, which infers Q not true. This is a proof process, not Truth table. — Corvus

If you have P>Q and ~P, you got nothing about Q. Q can be either T or F.

if you have P>Q and ~(P>Q), you got nothing.

If you have ~(P>Q) then you may conclude that P is true and Q is false. That is, ~(P>Q)>~Q. Btw the word is implies, not infers; look it up. But this is not MT.

Since your book is misleading/confusing you, or itself wrong - which happens - I suggest you get another book. In any case it's usually good to have more than one book, one elucidating what another leaves dark.

Back to the PSR. I encounter it as from Leibniz as nihil est sine ratione, fairly translated as "nothing is without reason." But what does it mean? In absolute terms, nothing absolute, because it runs into the self-referential paradox that says reason, a something, must have a reason. And on this much time and effort could be wasted, but none of it to Leibniz's point.

An illustration: "The cows are in the high pasture (and they shouldn't be)." What might be the reason they're in the high pasture? The absolute presupposition here is that there is a reason, and that itself reasonable because it would be unreasonable to suppose them in the high pasture for no reason at all. And there is a plethora of possible reasons, or "constellations" of reasons. As reasons however none (yet) constitute knowledge.

And this clarifies Leibniz's program. Reasons the frames for testable hypotheses. And some testable by reason by itself: the cows neither flew nor quantum-tunneled their way onto the high pasture. And some by experiment and observation: maybe the fencing between the lower and higher pastures was breached, let's saddle up, or walk up, or take our helicopter, or use our drone, to take a look and see.

The idea being that reasons alone are in themselves at best useful tools for testing one way or the other. But that by themselves ground no knowledge of anything at all. To become knowledge, they require an added ingredient. Mere belief, however, does not require that ingredient, with the result that mere belief can never itself be knowledge of anything.

But some things in the real world also have metaphysical identities or essences. Even if we suppose that the world is merely physical, which means that everything supervenes on matter and energy, then at least matter and energy have identities (i.e. as matter is not the same thing as energy, they have different identities), and thus also have essential properties. E.g. matter has the essential properties of having a mass, volume, shape, etc. So, if a thing is made of matter, then it necessarily follows that it has a mass. — A Christian Philosophy

This doesn't imply that an object has "necessary" or "contingent" as an intrinsic property. You're still just equating a definition with essence, defining matter as an object that has volume, shape, mass, etc. Either an object has those properties (in which case it is "matter") or it does not (in which case it is not "matter"). Defining a term with some set of properties doesn't entail that any objects have an individual essence. And the only role of "necessity" in this is the necessity of having the defined set of properties in order to be classified with the term.

If I understand correctly, you say that the first cause's existence is necessary, but only because there is no prior cause and not because its existence is an essential property of its identity. — A Christian Philosophy

That's basically true, but it's based on the principle that contingency needs to be accounted for.

But then, how do you explain the fact that its existence is necessary, if not inherently? If this fact is left unexplained, then it violates the PSR.

If you use the simplest definition of the PSR, that everything (both necessary and contingent) has an explanation of its existence, then there would be an infinite vicious regress of explanations. Should we prefer 1) a vicious infinite regress, in order to keep the PSR intact, 2) entirely reject the PSR because of this, or 3) redefine the PSR to exclude something foundational?

I think the latter is the most reasonable option. There can be no explanation for the foundation of existence, and (as noted) it can't be contingent. Since it's not contingent, its existence is logically necessary: it can't not exist.. (i.e. if the foundation of existence didn't exist, there would be no existence at all; which is logically impossible because we obviously exist). You are free to assume this foundation = a god, although it obviously doesn't entail a god.

If you have P>Q and ~P, you got nothing about Q. Q can be either T or F. — tim wood

That was what I was pointing out to you. You seem to be totally relying on truth table for the value of Q. This is not a truth table case. It is a proof process based on the inference and reasoning.

From the introduction of P -> Q, we know ~P, hence the assumption P -> Q was not true.
Therefore we can make a logical inference ~Q.

This problem had been in discussion before a few times.
If you say, prove the ground is wet. You would bringing an assumption, If it rains, ground is wet.
But you find out, it doesn't rain. Therefore (from the assumption) you can prove the ground wet is not true. Why is it still true, the assumption is true? (when the antecedent is false?) This is what you are saying, and it is a rule. But if the rule doesn't make sense in real life, do you still have to go by the rule?

Since your book is misleading/confusing you, or itself wrong - which happens - I suggest you get another book. In any case it's usually good to have more than one book, one elucidating what another leaves dark. — tim wood

I am trying to do some reading on Logic this holiday period, and try to brush up the practical side of Logic. I thought the Cauman book was quite good. It reads quite well. But perhaps I could get another 1 - 2 books to compare on these fuzzy points. Any recommendations?

There can be no explanation for the foundation of existence, and (as noted) it can't be contingent. Since it's not contingent, its existence is logically necessary: it can't not exist.. (i.e. if the foundation of existence didn't exist, there would be no existence at all; which is logically impossible because we obviously exist). — Relativist

Let's see... Me buying milk today was logically necessary, because I obviously did buy milk. That's not right, is it? Logic cannot imply existence or non-existence, necessity or contingency of anything outside its universe. The only way you can get the necessity of our existence as a logical conclusion is if you front-load it with postulated propositions, but that would be question-begging in the present context.

I think you confused logical contradiction with performative contradiction of denying your own existence.

Let's see... Me buying milk today was logically necessary, because I obviously did buy milk. — SophistiCat

That's not my reasoning.

Your buying milk was contingent iff the purchase was contingent upon something. If your purchase was the product of libertarian free will, then it was contingent upon your will making the decision.

OTOH, if determinism is true, then your choice to make the purchase was logically necessary, given your genetic makeup and history. There may be quantum indeterminism somewhere in the past that influenced your history or genetic makeup- in which case, your purchase was contingent upon the outcomes of those indeterminate events.

The first cause cannot have been contingent upon anything, because nothing is prior to it. So, whatever it actually was, it is metaphysically impossible for it to have been anything else.

we know ~P, hence the assumption P -> Q was not true. — Corvus

No, no, no. From the assumption that P>Q, and given ~P, you know that P is F. and with P being F, P>Q is always true, and that Q can be either T or F.

The proposition (e.g., and assuming it is true) is, if it's raining, then the ground is wet. Consider: it says nothing about whether it is raining (or not), and it says nothing about whether the ground is wet (or not). It says only that if it is raining, then the ground is wet. It does not say that if it is not raining, then the ground is dry - it could be wet for some other reason. Nor does it say, and this is a little more difficult, that if the ground is wet, then it is raining. The logic of if-then is a knot that takes some practice to get used to tying and untying. Until you get it, it can be a trap. When you get it, you will wonder what all the fuss was about.

Assuming P>Q is true, means only that the assumption applies to the whole expression. As to whether P is T or F, or Q is T or F, the assumption is silent. They can both be T; they can both be F. Or P is F and Q is T. You only cannot have P is T and Q is F. Whenever, then, P is F, the assumption is always T. And when Q is T, the assumption is always T. All this is clearer and less effort with truth tables.

It is a proof process based on the inference and reasoning. — Corvus

What this means in the context of this logic, I have no idea.

No, no, no. From the assumption that P>Q, and given ~P, you know that P is F. and with P being F, P>Q is always true, and that Q can be either T or F. — tim wood

Whenever, then, P is F, the assumption is always T. And when Q is T, the assumption is always T. All this is clearer and less effort with truth tables. — tim wood

Great explanation! :up: I appreciate that. Thank you.

That's not my reasoning. — Relativist

What follows doesn't resemble your reasoning above, either:

if the foundation of existence didn't exist, there would be no existence at all; which is logically impossible because we obviously exist — Relativist

In any case, your conclusion is incorrect:

The first cause cannot have been contingent upon anything, because nothing is prior to it. So, whatever it actually was, it is metaphysically impossible for it to have been anything else. — Relativist

Contingency and necessity are context-dependent. In ordinary usage (let's leave logic out of it - as I already explained, it does not apply here), this contextual meaning is usually unproblematic. We can recast it in terms of constraints: contingent events or choices are those that are not fully constrained by facts and assumptions that go into our reasoning, necessary ones are constrained to a single outcome, and impossible ones are ruled out. However, what those constraints are taken to be can vary widely, depending on discourse.

When it comes to the necessity or contingency of the world, the problem statement is so far outside ordinary usage of these words that it is not at all clear what is being asked. You interpret the question as being solely about event causation (without explaining your choice). Given such framing, a superficially plausible conclusion could be to say that the first cause must be contingent, since nothing constrains it - which is the opposite of what you concluded. But this too would be wrong.

The presumed absence of constraints on the origin of the world does not imply a multiplicity of possible outcomes, because there is no space of outcomes given to us. Note that I said "no space" - not an empty space and not a singleton space [consisting of a single possibility]. The latter is what you would need to make your conclusion of necessity, but assuming such a singleton space would beg the question. Assuming any space of possibilities would take you outside your original formulation, and so, the right conclusion is simply that contingency/necessity does not apply in this degenerate scenario.

The presumed absence of constraints on the origin of the world does not imply a multiplicity of possible outcomes, because there is no space of outcomes given to us. Note that I said "no space" - not an empty space and not a singleton space [consisting of a single possibility]. The latter is what you would need to make your conclusion of necessity, but assuming such a singleton space would beg the question. Assuming any space of possibilities would take you outside your original formulation, and so, the right conclusion is simply that contingency/necessity does not apply in this degenerate scenario. — SophistiCat

I disagree. I don't see any reason to call this a "degenerate scenario". Everything that exists, either exists contingently or necessarily: they are the converse of each other; there is no third option.

Take it as a premise that there is a first cause/foundation of existence- because that's what we're analyzing here. The context of the discussion is metaphysics- so the relevant modality is metaphysical possibility/necessity. Discussions like this often mistake conceptual possibility with metaphysical possibility. Just because I can conceive of a world grounded in a different first cause does not mean an alternative first cause is metaphysically possible.

I've proposed that it is a metaphysical axiom that contingency needs to be accounted for: X is contingent iff whatever accounts for X could possibly account for ~X. In the absence of such an account, X is metaphyically necessary. A first cause is not accounted for by anything else, therefore it cannot be contingent. This conclusion follows from my axiom

Why believe this axiom to be true? Because it's consistent with what we know about the world through physics. Events that are the product of classical physics are not contingent: causes necessitate their effects. OTOH, quantum mechanical events have a "space of possibilities" (a probability distribution), so the specific outcome is contingent (although the specific space of possibilities is necessitated by the quantum system - which evolves deterministically per a Schroedinger equation). So the laws of nature suggest the world is basically deterministic (outcomes are necessitated) with contingency present only where there is quantum indeterminacy (with a necessitated probability distribution of outcomes).

You also alluded to an "absence of constraints" applying (I assume) to a first cause. It is contrained to being whatever it was, conceptual possibilities notwithstanding.

It is a proof process based on the inference and reasoning. — Corvus

What this means in the context of this logic, I have no idea. — tim wood

You have explained the case of truth table application very succinctly in your previous post.  I understand exactly what you mean on all your points.

However, you seem to be in confusion for this particular case of the proof I have shown in my post.
What you are saying is totally based on truth table cases dealing with purely symbolic logic i.e. you don't know or care what the content of the antecedent or conclusion in the -> statement.  Of course in that case, you must take account of all the cases of Q, which could be T or F.  You don't know what the status of Q is.  You have explained that, and I agree with that.

In this case, we know the content of the antecedent and conclusion of the -> statement.P -> Q was introduced for an assumption.   We don't need to be worried about the case where Q is T or F.  Because Q will be totally dependent on P being T or F.

Think of an example.  If I was told to prove if I am a millionaire, I would start with an assumption If I won the lottery jackpot tonight, I am a millionaire. It is just an assumption introduced for the proof process.

If I won the lottery jackpot tonight, then I am a millionaire .I am a millionaire totally and solely dependent on the fact of the antecedent "If I win the lottery jackpot".  In this case, I don't need to worry about whether I am a millionaire is T or F.  Because it would totally depend on the antecedent "If I won the lottery jackpot".

So I introduce (discharge) a factual statement, I have not won the lottery jackpot, which proves (based on the antecedent which is found not true), I am not a millionaire (which is True), which proves the original statement Q (I am a millionaire) was False. Remember this is not a denial of the antecedent. It is a fact from the real world case.

Therefore your points are correct under the book and truth table application method.  But you are totally under the confusion in insisting that you don't know what the content of Q was, blindly thinking Q is just Q, not thinking at all about the fact the we know the content of Q. The content of Q was given out at the very first, prove that "I am a millionaire."

P ->Q is based on the assumption P (If I won lottery jackpot), which we also know the details of the content, and was found as F from the real world case.  In this case, you don't need to think about the case where Q is T or F.

You apply truth table when all you have are the symbols bereft of any content of the symbols like in the textbooks. Because you cannot verify the symbols with the real world cases you are trying to prove. However, when you know the content of the symbols, you don't apply all the cases in truth table. Because you can verify the P or Q from the real world observations, deductions or inferences.

Remember textbook truth table tells you how the symbols in the propositions gets T F value in all cases. They are not telling you anything about the the proof processes in the real life which you must take into account prior to examining the symbols.

They are not telling you anything about the the proof processes in the real life which you must take into account prior to examining the symbols. — Corvus

They are telling you everything about the process. Recourse to facts simply is independent of the logic. This the tension between validity and truth. You assumed P>Q. Then you note the ~P is the case. From that you conclude that also ~Q is the case. But what you have not done is specified that if P is not the case, then Q is not the case. Which, if you specified it, would yield ~Q.

(P>Q) ^ (~P) ^ (~P>~Q) => ~Q.

And without which, you have left the logic behind, not having proved ~Q, but simply having asserted it.

And without which, you have left the logic behind, not having proved ~Q, but simply having asserted it. — tim wood

.I am a millionaire totally and solely dependent on the fact of the antecedent "If I win the lottery jackpot". — Corvus

You could already be a millionaire prior to the lottery drawing.

I find it interesting that the PSR and P of Causality seem to overlap regarding mathematical causality chains. Clearly sufficient reason and sufficient cause are there, whereas reason is more logic and cause is more physical. Just thought I'd go back to the OP.

(P>Q) ^ (~P) ^ (~P>~Q) => ~Q.

And without which, you have left the logic behind, not having proved ~Q, but simply having asserted it. — tim wood

Why make it more complicated? When you can

P -> Q
~P
Therefore ~Q

Truth table is for the classical logic, and has serious limitations. If you read some First Order Logic books, you would see they use axioms and real life case verifications in the proof rather than Truth tables.

You could already be a millionaire prior to the lottery drawing. — EricH

In which case, Q would have been proved without the proof process.

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.

P -> Q
~P
Therefore ~Q
Truth table is for the classical logic, — Corvus

Which you apparently do not want to do. And you do not have to. But you appear also to want your "argument' to stand, somehow, and absent other premises, it doesn't. Denying the antecedent is a basic and elementary mistake. You appear to know this, but that you dismiss it means you also do not understand it. You need to understand it.

Suggestion, fwiw. A two-circle diagram, overlapping as in a Venn diagram. One circle P, the other Q. In order to illustrate P>Q, black out the portion of the P circle not in Q. Pictorially, then, if there is an X in P, then it has to be in Q also. Now if you deny Q, then it should be clear that there can be no P. Instead, however, deny P. The diagram should make clear that denying P leaves plenty of Q, and says nothing about Q other than if there is a Q, then it is not also a P.

The diagram should make clear that denying P leaves plenty of Q, and says nothing about Q other than if there is a Q, then it is not also a P. — tim wood

You are still totally dismissing the fact P was verified as ~P from a real life event. When P is ~P, then it can be inferred ~P -> ~Q proving ~Q. It is not a groundless denial of the antecedent, but an assertion verified from the fact.

I have agreed that your points makes sense too, but only from the Classic Logic point of view. You seem not applying the proof methods using the Axioms, inference, implications and real world events in the proof process.

You are still totally dismissing the fact P was verified as ~P from a real life event. When P is ~P, then it can be inferred ~P -> ~Q proving ~Q. — Corvus

Ok. Prove it.

Ok. Prove it. — tim wood

I thought it was proved and explained already in the previous posts, but you seem to disagree, or haven't read it at all. That's fair enough. Maybe you still have points on the issue, or missed what I was saying. I am not saying that you are totally wrong here.

I am saying your point seem to be coming from the Classic Logic theory which is mainly based on utilizing Truth Table ignoring the fact that the modern logical proofs are done using the Axioms and real life events.

I am reading some Logic books now, and trying to expand my knowledge on the subject. If I find anything relevant and interesting to our topic in discussion, I will get back to you. Thank you for your feedback.

I thought it was proved and explained already in the previous posts, — Corvus

Your proposition as I have understood is, (P>Q) ^ (~P) => ~Q. If this is it, it wasn't proved because it is not provable. Above I briefly described how this appears in a modified Venn diagram.

Now I think it's time to stop. You've taken a turn from Mistaken Dr. onto Fool Ave., and that a wrong turn to make, a waste of time for everyone. Return as seems best to you, but if you insist on yours, I insist you provide a proof.

I submit that maybe what you're doing is adding premises/facts and then supposing you can do without them, and you can't.

Now I think it's time to stop. You've taken a turn from Mistaken Dr. onto Fool Ave., and that a wrong turn to make, a waste of time for everyone. Return as seems best to you, but if you insist on yours, I insist you provide a proof. — tim wood

You seem to keep writing contradictory posts. You suggest to stop, but at the same time you insist to provide a proof. Isn't it a real waste of time? I suggest you to read some First Order Logic books. If you cannot see from P -> Q, ~P, therefore ~Q, then you are not reasoning at all. Ok, I will leave you to it. All the best.

I am a millionaire totally and solely dependent on the fact of the antecedent "If I win the lottery jackpot". — Corvus

In which case, Q would have been proved without the proof process. — Corvus

I thought I was clear, obviously not. I've bolded the key phrase. As you acknowledged, this is simply not the case - your being a millionaire is clearly NOT dependent on winning the lottery. For your example to work, it needs to be re-phrased. I can think of two options:

1 - Add an additional qualifier: If I was not previously a millionaire, then etc etc . .
OR
2 - Get rid of the "totally and solely dependent". E.g., If I win the lottery jackpot then I will be a millionaire

Maybe the millionaire example was not clear. Try with these example cases.

If X=0, then X+1 = 1
X = 2
Therefore X+1 = 3

The statement X=2, tells that
X=0 was not true.
Until the value of X is known, nothing is known in the premise.
X=2 decides the variable X and T or F of P and Q.

It also tells X+1=1 is false too.
X+1=3 is true.


If John is in Tokyo, then John is in Japan.
John is in Paris (not in Tokyo). <=== A fact from real life situation.
Therefore John is not in Japan.
P -> Q
~P
Therefore ~Q

You don't need more complication in the proof here.
You use axioms and facts in reality as the verification statements instead of the antique Truth table.

What you're not seeing - I don't know why - is that you're making two different arguments.

If john is in Tokyo, then John is in Japan. John is not in Tokyo. Maybe he is in Osaka or Yokohama.

But you're argument really is, If John is in Tokyo then John is in Japan. John is in Paris, therefore he is not in Japan. In this argument is the extra premise.

You can conclude John is not in Japan not because he is not in Tokyo, but because he is in Paris.

We're going on circles. — Clearbury

Very well. Then we can drop this topic.

You haven't answered the question. What explains it [logic]? — Clearbury

I have provided the reason why we know that logic exists on the epistemology side. Then the reason why logic exists on the metaphysics side is because, being a first principle of epistemology, it is also a first principle of metaphysics; i.e. logic is part of the fabric of reality. With that, the existence of logic is explained inherently (reason type 1 in the OP section "PSR in Metaphysics").

Is logical necessity a brute fact? — RussellA

No, nothing is a brute fact under the unrestricted PSR. Logic has a reason for existing, as provided in the OP under section "Argument in defence of the PSR". The section explains why logic is a first principle of epistemology. Then, the reason why logic exists on the metaphysics side is because, being a first principle of epistemology, it is also a first principle of metaphysics; i.e. logic is part of the fabric of reality. With that, the existence of the laws of logic is explained inherently (reason type 1 in the OP section "PSR in Metaphysics").

is the logical necessity that "A triangle has three sides" a brute fact? — RussellA

No, this is a logical necessity only because it describes its own definition, which is man-made. "A triangle, defined as a shape that has 3 sides, has 3 sides". A=A.

We have been through this already. — Fooloso4

Very well. Then no need to repeat the conversation, and we can leave this topic here.

When natural explanations cannot explain why there is anything at all you resort to a super-natural explanation even if you do not use that word. — Fooloso4

I am unclear on what you mean by "natural" vs "super-natural". How do you define those two terms?


"And my conclusion is that a thing whose existence is essential is necessary to explain the existence of all other contingent things"— A Christian Philosophy

All other contingent things? Something whose existence is necessary is not something that is contingent. — Fooloso4

What I meant is, "And my conclusion is that a thing whose existence is essential is necessary to explain the existence of all other things, which all happen to be contingent."

Why must there be a reason for what is? Positing a principle that there must be is circular and question begging. — Fooloso4

The PSR is defended in the OP under section "Argument in defence of the PSR".

This doesn't imply that an object has "necessary" or "contingent" as an intrinsic property. — Relativist

Hmmm. I'll have to think some more about that one. But it can wait as I believe it is not critical to the main discussion.

Should we prefer 1) a vicious infinite regress, in order to keep the PSR intact, 2) entirely reject the PSR because of this, or 3) redefine the PSR to exclude something foundational? I think the latter is the most reasonable option. There can be no explanation for the foundation of existence — Relativist

There can be an internal explanation: the existence of the first cause is explained inherently if its existence is part of its essence. In other words, the proposition "the first cause, whose existence is part of its essence, exists" is a tautology, and tautologies are necessarily true, and their negations are necessarily self-contradictory. In this way, the PSR is kept intact.

As I understand, you are willing to redefine the PSR because you deny essences. But why deny essences? Note, I acknowledge that most things do not have essences, but I believe that some things do.