Bumble

I'm inclined to agree with the comment from @DoubleKnot. If number represents a kind of logic, e.g. classical logic, 0 is zero-order (propositional) logic, and each successor is a higher order logic, then you could view the axioms as holding. But I don't see what you have achieved. What interesting properties are there that are true of every order of logic? What is the point of trying to use induction over them?

No, I am not assuming anything and I don't need to show that events are fundamentally random. I am not arguing for any such thing. And I am just repeating myself here, which is getting frustrating. I don't know and I don't care whether the universe is fundamentally random or not. What matters is that whether it is or it isn't we can describe events using probability distibutions. That is why the question is bad and the ticked answer is wrong.

As to whether there is some undiscovered deterministic theory of the universe under which every quantum event including radioactive decay has a single deterministic outcome, I can only repeat that we have no evidence whatever that this is so, no plausible candidates for it, and all the evidence we do have based on local effects is that this is not so.

@Syed If you allow that some events are caused, but not sufficiently, and those events are nondeterministic, then that's all we need to show that we can have probabilistic descriptions of events without assuming determinism. That is what the original question is asking and that is what the ticked answer gets wrong. You have moved away from what the question is asking and the issue of why I think it is not a good question.

Hence there is no additional information such that, if only we knew it, would allow us to predict when an atom will decay. Describing radioactive decay with a probability distribution is the best we can do. There is nothing else.

I'm not claiming that events are uncaused, only that they are not deterministic. Radioactive decay is caused by quantum tunnelling. Or perhaps one should say that radioactive decay is an example of quantum tunnelling. Either way, it is a nondeterministic process. It is fundamentally random. There is, as far as we know, no deterministic substrate of reality that determines when a radioactive atom will decay.

More generally, the wave function from the Schrödinger equation when squared yields a probability density function. This function agrees with experimental results to an extremely high degree of accuracy. This does not mean the universe is capricious and anything can happen at any time, but it does mean that the best we can do is describe the universe in a nondeterministic way. Maybe that changes in future, but that's how we stand at present.

That is why I objected to the ticked answer, which says, "The original question was: If a process is fundamentally random, how can it follow a probability distribution? The answer is: It won't." Yes it can. Random in this context means nondeterministic, it does not mean capricious or chaotic. Radioactive decay is nondeterministic and it follows a probability distribution.

We would not get far in statistics if we did not specify distributions, even when we are talking about the results of random experiments. Radioactive atoms decay according to some random process (as best we know). This does not mean that anything at all can happen. Something remains fixed. The probability of decay in a given time interval remains fixed. That is why we get a half-life. But it is still random. Random doesn't mean the universe is capricious and anything might happen.

@Syed I think we may have a misunderstanding, but I don't think it is about determinism. I suspect it is about the meaning of 'random'. To say some event is random does not mean it is uncaused or that there are no antecedent conditions that bear upon it. When we speak of a random selection we typically specify the distribution: we might say something like, select a number from 1 to 100 with a uniform distribution. Or, select a marble from this urn with a uniform distribution.

Maybe there is some mysterious feature of the universe that determines for each radioactive atom when it will decay. But if so we have not discovered it yet, we have no plausible candidates for it, and it is merely a hypothesis with no supporting evidence. Whether or not such a feature exists does not change the fact that we can describe radioactive decay using a probability distribution.

Picking up the example of radioactive decay, as far as we know nothing determines when a radioactive atom will decay. We know it will decay at some time in the future but not when. But if we have trillions of those radioactive atoms we can predict with a high degree of probability what proportion will decay in a given time interval. That is one of the things we mean when we say we can apply a probability distribution.

I am not trying to argue whether the universe is fundamentally indeterministic or whether there is some deterministic substrate to it. I am just pointing that we can, and do, use probability distributions either way. We do not need determinism to make probability work.

Examples might be: (1) Two elastic bodies colliding. (2) The final resting position of a thousand marbles when a bucket containing the marbles is kicked over. (3) Tomorrow's weather at a particular location. (4) Radioactive decay. They are all different scenarios but in each of cases 2 to 4 we can use probabilities to describe the outcome.

I am not saying that a coin toss and radioactive decay are the same. (1) Some events have a high degree of predictability because they are close to being deterministic and are not chaotic. (2) Others are difficult to predict in practice because it is hard to specify the boundary conditions. (3) Others are very difficult to predict because chaotic dynamics have a significant effect on the outcome. (4) Others are impossible to predict because (as far as we know) they are not deterministic at all.

You claim once again that no physicist would say that a coin toss is not deterministic. That is not so. The real universe is not the same as our scientific theories about it. As far as our current best theories are concerned, the universe is not perfectly deterministic. And you do not seem to understand what a boundary condition is. Of course boundary conditions apply to coin tosses; they apply to everything.

@Syed You say that unpredictability is not what is at stake, but it is you who claimed, "...if we knew all antecedent conditions a second before a person tossed a coin, we would be able to predict whether it will land on heads or tails." I am pointing out that determinism and predictability are distinct things and an event may be deterministic and unpredictable.

I don't need to prove or provide evidence that the universe is fundamentally random. The universe is describable using probability distributions whether it is or it isn't. ψ-squared works as a probability density function either way. It is you who are claiming on the basis of no evidence whatsover that determinism is needed for probability distibutions to work.

As to QM, Bell's results show that there are no local hidden variables. There is no evidence of any deterministic substrate underlying our best account of how the universe behaves. Believing in determinism is just an irrational supposition based on no evidence. If future experimental results come along and show that determinism is true then well and good, but as of now there aren't any.

You have switched from claiming "consensus" to saying "most physicists" but it is still incorrect. Determinism does not entail predictability. When you have chaotic dynamics, which is almost everywhere, events can be deterministic and unpredictable. We cannot measure boundary conditions with infinitely fine precision and chaotic dynamics provides abundant examples of the butterfly effect whereby unmeasurably small differences in boundary conditions result in macroscopically different outcomes.

@Syed The coin may land heads or tails or on its edge or not land at all. The mechanics of real things does not follow any physical theory exactly. Even classical mechanics is not perfectly deterministic, there are sources of incompleteness, and classical mechanics only holds in the classical limit, i.e. it is an approximation. So you cannot make predictions with absolute certainty. This makes room for probabilities.

Also it doesn't matter whether QM is fundamentally random or not. In either case events such as radioactive decay can be described using probability distributions. The ψ-squared function is a probability density function either way. So the point remains that we don't need determinism to have probability distributions. And we have no evidence that QM is deterministic. We know from Bell's work that there are no local hidden variables. We have no plausible candidates for global hidden variables.

@Syed There is no consensus that dice rolls are deterministic. In our universe, classical deterministic physics only holds approximately, in the classical limit. And even if determinism were true, the mechanics of dice (and almost everything else) is unpredictable because of chaotic dynamics. It is impossible to measure any quantity with infinitely fine precision so we are unable to predict an outcome with absolute certainty. We use probability to describe the uncertainty.

@Syed You asked what part of the answer is incorrect: I told you. It is also incorrect to say that outcomes need to come from deterministic processes: I have no idea where you get that idea from. There is not unanimous agreement that dice rolls are deterministic, and even if they were they are still chaotic and unpredictable. Radioactive decay is describable using a probability distribution: that holds true whether or not quantum mechanics is deterministic or indeterministic.

@Syed The second paragraph of the first answer, i.e. the paragraph beginning, "In order to follow a probability distribution..." is incorrect. Outcomes of events do not need to be exclusive or independent; that is just a special case. The first and last paragraphs are also incorrect. Assuming that radioactive decay is fundamentally random, which is consistent with our best understanding, the time taken for a radionucleide to decay follows a probability distribution.

Let us continue this discussion in chat.

The meanings of connectives in 3-valued logics cannot 'reduce' to 2-valued meanings. Reduction does not mean anything in this context. What connectives in 3-valued logics can do is include the values of 2-valued logic as particular cases. Kleene logic does just that. Consult the truth table for Kleene logic and you will see that it includes the classical 2-valued cases. As I mentioned above, if you want not(A and not-A) to come out true, you would do better to deploy the concept of supervaluation. You can make not(A and not-A) have the value supertrue.

3-valued logics are different from 2-valued classical logic. So the connectives cannot have the same meaning. What 3-valued logics can do is include the values of 2-valued logic as particular cases. Kleene logic does that: if you ignore the 1/2 value and look at the true and false values only, then it agrees with classical logic. You are welcome to choose not to use Kleene logic, or any logic. But you cannot prove that all 3-valued logics have not(A and not-A) as a tautology, because some don't.

You are not proving anything. You are simply discarding the logics that you don't like. In Kleene 3-valued logic, there are no tautologies. Even A → A is not a tautology because it evaluates to 1/2 when A is 1/2. Likewise for not(A and not-A). You are welcome to choose not to use Kleene logic if you don't want. But it is impossible to prove that all 3-valued logics have not(A and not-A) as a tautology because it just isn't so.

@JadeVanadium The truth tables are the same, but the logics are different, because they employ a different account of the logical consequence relation. In Kleene's logic, T is the designated value that is preserved in valid inference. In Priest's, both T and U are designated values.

Priest's logic of paradox is also a counterexample. Both the Kleene and Priest logics agree with two-valued classical logic but assign the third truth value to not(A and not-A) when A has that value. If you want not(A and not-A) to come out always as true, you would be better off taking a different approach using the concept of supervaluation. You can set up not(A and not-A) so that it is supertrue if it is true under all complete extensions.

I am not sure what you mean by 'denote' in this context. A proposition is, minimally, a truth bearer. It is also used in other related ways as I wrote in my answer to this question.

I might say, "Consider the proposition: All lions speak Turkish". Or I might ask a question. "Is it true that all lions speak Turkish?" Or, as in my example above, I might be looking for propositions to illustrate a valid syllogism. "All lions speak Turkish; some lions breathe fire; therefore, some Turkish speakers breathe fire". Or I might use it in the antecedent of a conditional. "If all lions speak Turkish then I shall be very surprised." All of these are examples of a proposition appearing in an unasserted context.

I can state the proposition, "All lions speak Turkish" without asserting it to be true. It is a proposition and its truth value is false. Frege distinguished 'thought' and 'judgment'. In modern terminology we might describe this as a distinction between a proposition and an assertion, or more generally between content and illocutionary force. A proposition is, minimally, something capable of being true or false. An assertion is a speech act with a propositional content.

Conventionally, stating a proposition and asserting it to be true are distinct things. This distinction goes back to Frege's use of the judgment stroke. We may demonstrate the validity of the following argument without commitment to any of its propositions being true: "All lions speak Turkish; some lions breathe fire; therefore, some Turkish speakers breathe fire".

But that is just the point. The truth values of Kleene-3 logic include the truth values of classical logic as a subset. So it meets your criterion. Nevertheless, not(A and not-A) is not a tautology: it does not come out true under all valuations.

The meanings of connectives in 3-valued logics cannot 'reduce' to 2-valued meanings. Reduction does not mean anything in this context. What connectives in 3-valued logics can do is include the values of 2-valued logic as particular cases. Kleene logic does just that. Consult the truth table for Kleene logic and you will see that it includes the classical 2-valued cases. As I mentioned above, if you want not(A and not-A) to come out true, you would do better to deploy the concept of supervaluation. You can make not(A and not-A) have the value supertrue.

3-valued logics are different from 2-valued classical logic. So the connectives cannot have the same meaning. What 3-valued logics can do is include the values of 2-valued logic as particular cases. Kleene logic does that: if you ignore the 1/2 value and look at the true and false values only, then it agrees with classical logic. You are welcome to choose not to use Kleene logic, or any logic. But you cannot prove that all 3-valued logics have not(A and not-A) as a tautology, because some don't.

You are not proving anything. You are simply discarding the logics that you don't like. In Kleene 3-valued logic, there are no tautologies. Even A → A is not a tautology because it evaluates to 1/2 when A is 1/2. Likewise for not(A and not-A). You are welcome to choose not to use Kleene logic if you don't want. But it is impossible to prove that all 3-valued logics have not(A and not-A) as a tautology because it just isn't so.

@JadeVanadium The truth tables are the same, but the logics are different, because they employ a different account of the logical consequence relation. In Kleene's logic, T is the designated value that is preserved in valid inference. In Priest's, both T and U are designated values.

Priest's logic of paradox is also a counterexample. Both the Kleene and Priest logics agree with two-valued classical logic but assign the third truth value to not(A and not-A) when A has that value. If you want not(A and not-A) to come out always as true, you would be better off taking a different approach using the concept of supervaluation. You can set up not(A and not-A) so that it is supertrue if it is true under all complete extensions.

Mathematical induction does not allow you to go from a proof of P(n) for each n to a proof of (∀n)P(n). The 'proof' on the question you linked is bogus, as the answers to it point out. Read the article in Wikipedia on omega-consistency or google the omega rule. en.wikipedia.org/wiki/%CE%A9-consistent_theory

It doesn't follow because it is a built-in assumption of standard logic that sentences are of finite length and proofs are of finite length. If you want infinitely long sentences and proofs you need to use an infinitary logic. Read the article in SEP on infinitary logic. plato.stanford.edu/entries/logic-infinitary Most logicians don't use infinitary logics because they typically lack completeness, which is a desirable property.

And even with an infinitary logic, quantifier logic is not reducible to propositional logic. As I said before, proving P(n) for each natural number n is not the same as proving (∀n)P(n). You need the omega-rule to move from the first to the second. That is why there is a difference between 'consistency' and 'omega consistency'. It is possible for a theory to include P(n) for each natural number n and also include ¬(∀n)P(n). In which case it is consistent but omega inconsistent.

Unless you are using an infinitary logic then it is a built-in feature of logic that you cannot have infinitely long formulas or infinitely long proofs. It does not follow from the fact that if A is a wff and B is a wff that A ∧ B is a wff that a formula can be infintely long. Infinitely long formulas are expressly ruled out in standard logic. Sentences and proofs need to be finite in order to verify them.

Also, it is not true in general that if you can prove P(n) for each natural number n then you can prove (∀n)P(n). You need a separate rule called the omega rule to bridge the gap. That is why there is a difference between 'consistency' and 'omega consistency'. A theory can be consistent without being omega-consistent.

Logics are standardly finite because of the need to write proofs down and because of the relationship between proof and computability. An infinitely long sentence, or an infinitely long proof, would be impossible to verify, even for a computer. One of the advantages of propositional logic is that you can verify a proof by constructing a truth table. You cannot do that if the table is infinitely large.

You are indeed extending propositional logic, and not in a conversative way. What you propose creates a quantified propositional logic. Writing the string "..." or "and so on" in a formula is just another way of writing the quantifier "for all". So what you are doing is not reducing quantifier logic to propositional logic. There are infinitary logics that permit infinite conjunctions and disjunctions, but again, these are non-conservative extensions of propositional logic.

What you are writing there is simply not propositional logic. Writing "from y=0 to y=infinity" is not a well-formed formula in propositional logic. There is no "..." in propositional logic; there is no "infinity"; there is no "from y=0 to y=infinity"; there isn't even a way to write expressions with variables such as "P(x)". All of those are features of quantifier logic that you are trying to smuggle into propositional logic.

Not only that, but even with a finite domain, e.g. with three individuals named a, b and c, writing "P(a) ∧ P(b) ∧ P(c)" does not express the information: "and that's all there is," which is expressed by (∀x)P(x), so they are not equivalent.