Comments

  • Godel's Incompleteness Theorems vs Justified True Belief
    ↪TheMadFool

    You are still conflating justification with truth, and consequently ascribing views to both of us that we did not state and do not hold.

    Proof does not establish truth, it establishes justification. — aletheist
    Proofs are sufficient, but not necessary, for justification.

    ... lack of counter-examples constitute a reason believe that something is true, or justified in belief. — god must be atheist
    Justification can be based on the absence of counterexamples.
  • All we need to know are Axioms
    A given proof is a sufficient condition to establish truth AND it's necessary too. I'm concerned about the latter aspect viz. necessity. — TheMadFool
    No, a proof is sufficient but NOT necessary. A true proposition is true regardless of whether humans ever construct a proof for it. One more time: a proof pertains to justification, not truth.
  • All we need to know are Axioms
    ↪TheMadFool

    No, my question demonstrates that a proof is not necessary for a proposition to be true. It is self-refuting to claim otherwise, unless you can provide a proof of that claim. Again, a proof is a form of justification, while truth is a property of propositions themselves.
  • Why the Euthyphro fails
    Your point? — Bartricks
    Stop being tedious. — Bartricks
  • Why the Euthyphro fails
    If God is an agent, why can't he change his mind? — Bartricks
    you saying being all powerful involves being able to do 'less' than I can do? — Bartricks
    You have the internet. Do some research. — Bartricks

    It is intuitively obvious that no mind values anything or prescribes anything of necessity. — Bartricks
    It appears self-evident to the reason of most that moral truths are necessary, not contingent. — Bartricks
    But just saying something doesn't make it so. — Bartricks
  • Why the Euthyphro fails
    ↪Bartricks

    I have tried having polite discussions with you in this and other threads today, but you have promptly and persistently resorted to baseless assumptions and corresponding insults, rather than sticking to the substance of each matter at hand. Why is that?

    I ask you to read charitably. — Bartricks
    Right back at you.
  • Aquinas, Hume, and the Cosmological Argument
    There wouldn't be a dent there if the ball wasn't on it. — Bartricks
    How do you know that, if the ball was always on it? Maybe that is just the natural shape of the cushion, and the ball has nothing to do with it.

    Suppose instead that a cushion is manufactured with an indent that happens to be just the right size and shape for someone to place the ball there years later. If I subsequently ask you what is causing the indent, what would you say?
  • Why the Euthyphro fails
    So, by 'God' you mean a mind who, if he is valuing X, is incapable of valuing Y? A mind whose attitudes are fixed - whose attitudes the mind itself is incapable of changing? — Bartricks
    Indeed, immutability is one of the standard attributes of God in classical theism. Of course, treating God as a "subject" and a "mind" is rather anthropomorphic.

    As well as appearing to be inconsistent with possessing omnipotence ... — Bartricks
    That God always wills and acts in accordance with His eternal and immutable nature is perfectly consistent with what omnipotence means in classical theism.
  • Paley, Hume, and the teleological argument
    ↪ModernPAS

    I think that the argument as formulated relies heavily on the definition of "design," which probably needs to be stipulated. If "design" is used in a relatively weak sense, then 2 is clearly false; but if "design" simply means "deliberate arrangement," then 2 is trivially true and 3 is the dubious premiss. Proponents of "intelligent design" arguments typically try to establish objective criteria for "specified complexity" that is considered to be evidence of design in this stronger sense, citing forensics and archaeology as examples of scientific fields that employ a similar approach to distinguish human intervention from natural causes. That is where Hume's objection about artifacts vs. universes comes into play.
  • Aquinas, Hume, and the Cosmological Argument
    The cushion is being indented. — Bartricks
    If the cushion has always been indented, then it is not being indented, it simply is indented.

    If we were to take the ball off the cushion, and someone asked us the cause of the indent on it, we would say "there was a ball on it" - yes? — Bartricks
    No, not if I knew that the cushion had always been indented; then I would say, "Nothing caused the indent, it has always been that way."

    The problem with appealing to what is supposedly "intuitive" is that it is not necessarily the same for everyone.
  • Aquinas, Hume, and the Cosmological Argument
    What, right now, is causing the cushion to be indented? — Bartricks
    I honestly thought that we were addressing the question, "What caused the cushion to be indented in the first place?" I am not seeing how this other question is relevant to the OP.

    I ask you to read charitably. — Bartricks
    I always try to do so, but admittedly do not always succeed. My point was simply that the argument does not require an uncaused event; i.e., I was (gently) trying to correct the mistake in the OP.
  • Aquinas, Hume, and the Cosmological Argument
    ↪Bartricks

    I agree that the argument presented in the OP establishes the need for an uncaused (substance) cause. However, the very next paragraph makes the statement that I found problematic.

    if there can be no infinite regress of causes, then there must be an event that is uncaused. — ModernPAS

    No, this would imply that every event is caused by another event, which is not what the first premiss asserts. The conclusion is that there must be a first cause that is uncaused, not a first event that is uncaused. — aletheist

    That is why I politely asked you to read more carefully.
  • Aquinas, Hume, and the Cosmological Argument
    if I showed you the ball on the cushion and asked you the cause of the indentation, you - and everyone else possessed of reason and not in the grips of a theory - would agree that the cause was the ball. — Bartricks

    Because I would (understandably) assume that the cushion was previously undeformed, until the ball caused the indentation. Upon being informed that the ball and cushion were eternally in that configuration, I would then deny that anything caused the indentation. Only changes (events) require causes in the sense that is relevant here.
  • Aquinas, Hume, and the Cosmological Argument
    ↪Bartricks

    Here is what you said that started us down this road.

    The OP is correct in what they say. There must be an event - so, an occurrence, a happening - that is uncaused. — Bartricks

    What is required to avoid an infinite regress is an event that is caused by a substance, rather than another event--not an event that is uncaused.
  • Aquinas, Hume, and the Cosmological Argument
    ↪Bartricks

    What are you talking about? I not only follow the argument, I agree with it. Indeed, "there must also be 'substance causation,'" but that is exactly the opposite of claiming that there are uncaused events.

    Premise 2 of the argument is highly intuitively. — Bartricks
    Which argument? The one in the OP? I agree with that premiss, and even offered a supporting argument for it.

    Rejecting it becusae it doesn't fit with one's favourite worldview is incompetent. — Bartricks
    My worldview has nothing to do with it. If the indentation has always been present, then nothing caused it.
  • Aquinas, Hume, and the Cosmological Argument
    That's just what an event is — Bartricks
    Right, the result of a cause is just what an event is. An "uncaused event" is a self-contradiction.

    That's counter-intuitive. — Bartricks
    Lots of true propositions, especially in philosophy, are counterintuitive.

    Finding out that the ball has always been on the cushion does not call that into question. — Bartricks
    On the contrary, it entails that nothing caused the indentation, since there was no event of changing its shape.
  • Aquinas, Hume, and the Cosmological Argument
    There must be an event - so, an occurrence, a happening - that is uncaused. — Bartricks
    That is an unwarranted assumption that is not even part of the argument as presented in the OP. In fact, it directly contradicts its very first premiss--"Every event has a cause."

    I showed that the assumption that all causation requires a cause that is prior to its effect is false. — Bartricks
    No, you did not. If the indentation has always existed, then nothing caused it--not the ball, not the cushion, and certainly not the indentation itself.
  • Why the Euthyphro fails
    So, not addressing the problem in the OP. — Bartricks
    Right, I was instead addressing the problem named in the thread title. And the original was about "the pious" and "the gods," so the common version that I presented simply updates the terminology for a monotheistic context.

    make that subject God and tell me which premise you're denying. — Bartricks
    The first one, obviously. If moral values are the values of God, then they are necessary, not contingent, since they must be consistent with His eternal and unchanging nature.
  • Why the Euthyphro fails
    ↪Bartricks

    I am not addressing your argument at all, just describing the actual Euthyphro dilemma as it is commonly set forth by contemporary philosophers. It is about the relation between goodness and God, not the relation between "moral values" and "Reason."
  • Aquinas, Hume, and the Cosmological Argument
    ↪Bartricks
    Here is the actual exchange.

    if there can be no infinite regress of causes, then there must be an event that is uncaused. — ModernPAS
    No, this would imply that every event is caused by another event, which is not what the first premiss asserts. — aletheist

    I was correcting the mistake in the OP. You and I are in agreement here--it is false that every event is caused by another event.

    The example shows that one thing can cause another without preceding it. — Bartricks
    Okay, but it still does not demonstrate that something can cause itself. Besides, if the ball and cushion "have always existed in that arrangement," then nothing caused the indentation, since the cushion was never in any other shape.
  • Why the Euthyphro fails
    And it is the Euthyphro problem that I am interested in here. — Bartricks
    The Euthyphro is usually posed as a question: Is something good because God wills it, or does God will it because it is good? The first option makes goodness an arbitrary choice by God, while the second subordinates God to an external principle of goodness. Like many philosophical dilemmas, it sets up a false dichotomy; something is good because it is consistent with the eternal and unchanging nature of God, and everything that God wills is consistent with His eternal and unchanging nature.
  • Aquinas, Hume, and the Cosmological Argument
    What we can conclude is that every event has a first uncaused cause or causes. — Bartricks
    More accurately, every series of events has a first uncaused cause.

    The whole point of the argument is to establish that not all causation can be by events. — Bartricks
    Which is exactly what I said; please read more carefully.

    What I do not understand is why something cannot be the cause of itself. — Bartricks
    ↪ModernPAS
    already addressed that--"then it would have to exist prior to itself and this is impossible.”

    Well, it is still true that the ball is causing the indentation, even though there was no time prior to the indentation when it was caused. — Bartricks
    This is not a counterexample, because the indentation is not causing itself.
  • Aquinas, Hume, and the Cosmological Argument
    if there can be no infinite regress of causes, then there must be an event that is uncaused. — ModernPAS
    No, this would imply that every event is caused by another event, which is not what the first premiss asserts. The conclusion is that there must be a first cause that is uncaused, not a first event that is uncaused. The overall claim, of course, is that God caused the first event.

    Thus, the claim in the second premise the there can be no infinite regress of causes is simply an unsupported assumption. — ModernPAS
    As I understand it, the supporting argument is that there cannot have been an actual infinite series of causes, because it never would have been completed by reaching the present.
  • Godel's Incompleteness Theorems vs Justified True Belief
    Transliterated or translated? — god must be atheist
    Transliterated from the actual Greek work αλήθεια.

    Noble conviction you have. — god must be atheist
    Thanks; but then, the denial that there is such a thing as (absolute) truth is self-refuting.
  • Godel's Incompleteness Theorems vs Justified True Belief
    By-the-by: what does your moniker mean? — god must be atheist
    Thanks for asking. The Greek word for "truth" is transliterated aletheia, so I call myself "aletheist" because I believe that there is such a thing as (absolute) truth.
  • Godel's Incompleteness Theorems vs Justified True Belief
    @TheMadFool (et al) this is a theorem by Godel, not a proof. — god must be atheist
    No, it is called a theorem because Gödel provided a proof; otherwise, it would be called a hypothesis or conjecture. Fermat's conjecture came to be known as a theorem because he claimed to have a proof, which no one ever found; Andrew Wiles finally came up with one in 1994.
  • Godel's Incompleteness Theorems vs Justified True Belief
    I sensed that there wasn't the required level of correspondence between Godel's incompleteness theorems (GIT) and the justified true belief (JTB) of philosophy. The shared characteristic between the two I was hoping to emphasize was the need for proof to establish truth. — TheMadFool
    Proof does not establish truth, it establishes justification. However, since mathematics is the science of drawing necessary conclusions about hypothetical states of affairs (Peirce), there is a sense in which mathematical justification is equivalent to mathematical truth. A sentence is "true" within a consistent formal system as long as it does not contradict the underlying assumptions (axioms). A sentence is "undecidable" within that same system if it can neither be proved nor disproved on the basis of those particular axioms.

    What Gödel proved is that it is possible to formulate an "undecidable" sentence within any sufficiently powerful formal system. However, there are still many formal systems--include standard first-order propositional and predicate logic--that are both consistent and complete, such that it is not possible to formulate an "undecidable" sentence within them.

    As I have pointed out in other threads, classical logic does not require proof to establish truth, which is why "proof" by contradiction (reductio ad absurdum) is allowed. By contrast, constructive systems such as intuitionistic logic do require positive proof, such that the Law of Excluded Middle (every proposition is either true or false) does not apply.
  • All we need to know are Axioms
    What was the sufficient proof for the Godel statement "this statement is true but unprovable"? — TheMadFool
    That would be Gödel's proof of his incompleteness theorem, and the correct term is not "unprovable" but undecidable.

    Also, it seems that you're denying my claim: truth of a proposition necessarily requires proof. — TheMadFool
    In this context, I am not so much denying it as pointing out that it does not apply to all systems of formal logic. Informally, the claim is obviously false, since lots of propositions are true without ever having a formal proof. It seems like you may be confusing truth with justification.

    That means it's possible for a proposition to be true and without proof. Can you name one such truth? — TheMadFool
    Can you provide a proof that the truth of a proposition necessarily requires proof? If not, why do you claim that?
  • All we need to know are Axioms
    On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction? — TheMadFool
    No, the two Wikipedia quotes are not contradictory. The second one only affirms that a proof is sufficient for the truth of a proposition; it does not state that a proof is necessary for the truth of a proposition.
  • Known Valid Argument Forms - Is the system complete.
    You mean it's possible to create an axiomatic system that is complete and consistent as long as it doesn't involve arithmetic? — TheMadFool
    I mean exactly what I said, quoting the Stanford article--Gödel's incompleteness theorem only applies to formal systems "within which a certain amount of arithmetic can be carried out" (emphasis mine).

    Do you know why this is the case? — TheMadFool
    Because there are minimum requirements for a formal system to be able to generate the kind of undecidable sentence that Gödel's incompleteness theorem requires.
    ↪alcontali
    gave an example of a formal system that can do some arithmetic, but not enough for the theorem to apply.
  • All we need to know are Axioms
    If a proposition P is true then necessarily that a proof must exist for P being true. — TheMadFool
    This sounds like an endorsement of constructivist logic, such as intuitionistic logic, which requires a positive proof in order to affirm any proposition and accordingly denies the Law of Excluded Middle (LEM, either A or not-A must be true). By contrast, classical logic affirms LEM, which is why it allows double negation elimination (not not-A implies A) and proof by contradiction (reductio ad absurdum).
  • Known Valid Argument Forms - Is the system complete.
    One thing I'd like to know is whether logic - the entire system - is complete or not. I vaguely remember reading somewhere that logic is a complete system. — TheMadFool
    As
    ↪tim wood
    already noted, it depends on which system of logic you have in mind, since Gödel's incompleteness theorem only applies to formal systems "within which a certain amount of arithmetic can be carried out" (Stanford). For example, there are well-established proofs that the now-standard systems of first-order propositional and predicate logic are both (deductively) complete and consistent.
  • Zeno and Immortality
    ↪Gregory

    Suit yourself, but I will go with the mathematicians on this. Cheers.
  • Zeno and Immortality
    ↪Gregory

    Density is irrelevant to multitude, and in any case the whole numbers are of the same multitude as the odd numbers.

    For any collection A that has n subjects, the collection B of all the possible combinations of A's subjects has 2^n subjects, and 2^n > n for any value of n (whether finite or infinite). Therefore, B is always of greater multitude than A; it is commonly called the "power set" of A. The real numbers correspond to all the possible combinations of the rational numbers, so the collection of real numbers is of greater multitude than the collection of rational numbers.
  • Zeno and Immortality
    If I can find all the even numbers but line all the odd numbers with all the whole numbers, why can't I do this with all the real numbers? — Gregory
    Because the real numbers correspond to all the possible combinations of rational numbers, and therefore are necessarily of greater multitude than the rational numbers themselves--which are of the same multitude as the natural numbers, along with the even numbers, the odd numbers, the whole numbers, the integers, etc.

    Nothing has been settled to be countable or uncountable at that point yet — Gregory
    "Countable" is defined as being of the same multitude as the natural numbers, and thus applies to the rational numbers, the even numbers, the odd numbers, the whole numbers, the integers, etc. "Uncountable" is defined as being of a multitude greater than that of the natural numbers, and thus applies to the real numbers.

    By the way, I am using Peirce's terminology by referring to the "multitude" of a "collection," rather than the standard terminology that refers to the "cardinality" of a "set."
  • Zeno and Immortality
    ↪Gregory

    Mathematicians are well aware of it, and it is not a problem at all. The real numbers are of greater multitude than the rational numbers, but the even and odd numbers are of the same multitude. We would never "run out" of even numbers to pair with the odd numbers in a one-to-one correspondence. We would never even "run out" of even numbers to pair with the integers, despite the fact that there are only half as many of them on any finite interval. Again, we cannot reason about an infinite collection in the same way as a finite collection, and we also cannot reason about a true continuum in the same way as an infinite collection.
  • Known Valid Argument Forms - Is the system complete.
    Is our list of valid argument forms complete? — TheMadFool
    It depends on what you mean by "argument forms." As
    ↪alcontali
    just pointed out, what you seem to be seeking is an axiomatization of classical logic, which typically involves a few primitives and an inference rule. The "existential graphs" of Charles Sanders Peirce are an innovative diagrammatic alternative.
  • Zeno and Immortality
    ↪Gregory

    Again, all the different combinations of subjects of any collection--including any infinite collection--is of greater multitude than the collection itself; i.e., there are not "enough" subjects to be put into one-to-one correspondence with their combinations. The real numbers correspond to all the different combinations of rational numbers, so the real numbers are of greater multitude than the rational numbers; i.e., there are not "enough" rational numbers to be put into one-to-one correspondence with the real numbers. Put another way, the real numbers are a "larger" infinity than the rational numbers.
  • Zeno and Immortality
    You can put a one to one correspondence between any infinite set, because infinite sets have units. — Gregory
    No, the collection of all combinations of the subjects of a collection--even an infinite collection--is always of greater multitude than that collection itself. The integers and the rational numbers can be put into one-to-one correspondence with each other, but not with the real numbers, because those are of the next greater multitude. There is another multitude greater than that, and another greater than that, and so on endlessly--which is why an infinite collection of any multitude can never be "large" enough to qualify as a continuum.

    Likewise, unless you are speaking of process philosophy, an object must have parts. These can be divided endlessly, so it is neither discrete nor continuous. — Gregory
    That depends on what you mean by "parts." The portions of a continuum are indefinite, unless and until they are deliberately marked off by limits of lower dimensionality to create actual parts. For a one-dimensional continuum like a line or time, those limits are discrete and indivisible points or instants that serve as immediate connections between portions, but the portions themselves remain continuous--which is why they can always be divided further by inserting additional limits of any multitude, or even exceeding all multitude.
  • Zeno and Immortality
    ↪TheMadFool

    The mistake in the OP, going all the way back to Zeno, is thinking that discrete dimensionless positions in space and discrete durationless instants in time are real. Instead, it is continuous motion through continuous spacetime that is real, while positions and instants are useful fictions that we create for the sake of description and measurement.

    The problem is that mathematics is a way that we think about relations. The world isn't required to match that. — Terrapin Station
    Indeed, mathematics is the science of drawing necessary conclusions about hypothetical states of things, which may or may not match up with any real states of things.

    If we take time to be on a number line how many points of time are there between 1976 and 2019? — TheMadFool
    Again, a continuous line or interval of time does not consist of discrete points or instants at all, but we can mark any multitude of points or instants along it to suit our purposes. In other words, contrary to Cantor, there is a fundamental difference between a continuum and a collection.

    there are infinite numbers between 0 and 1, but it is intrinsical that there are more numbers between 0 and 2 — Filipe
    What Cantor got right is that there is likewise a fundamental difference between an infinite collection and a finite collection, such that we cannot reason about them in the same way. The multitude of real numbers between any two arbitrary values is the same, because they can be put into one-to-one correspondence with each other.
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