Yes I agree, there is only a rough isomorphism between the mathematics of binary logic and the real world; binary logic seems to describe a limit that reality approaches but never reaches.

We should consider that the mathematics of binary logic are the limiting case of probability theory; it is probability theory where the probabilities may only take the values of 0 or 1. Probability theory can do everything that logic can but it can also handle those real world cases where the probability of knowing something is something other than 0 or 1, as is the usual case with scientific knowledge.

Yes, good point. Classical physics, dealing with macroscopic objects, predicts definite (non-probabilistic) measurement outcomes for both the first and second measurements.

The point I was (poorly) aiming at is that while quantum theory is inherently probabilistic even it sometimes predicts specific results as certainties.

I guess the important point for me is that while theories may predict certainties they are always falsifiable; the theory itself may be wrong.

Thanks for the link to Gettier's paper.

It seems he considers that the statement 'S knows that P' can have only two possible values, true or false. This may have been a historical tradition within philosophy since Plato but it seems to rule out many usual usages of 'knowledge' such as 'I know a little about that'.

As noted by Edwin Jaynes Bayesians usually consider knowledge in terms of probability:

In our terminology, a probability is something that we assign, in order to represent a state of knowledge.

In his great text on Bayesian inference, Probability theory: the logic of science, he demonstrates that Aristotelian logic is a limiting case of probability theory; The results of logic are the results of probability theory where the value of probabilities are restricted to only 0 and 1. I believe this probabilistic approach provides a richer context for knowledge in that there are degrees of certainty. My reworking of Plato's definition attempted to transition it to this context.

Pick your favorite case of a scientific theory that was once well supported by the evidence, but turned out to be false. Back when available evidence supported it, did scientists know it was true?

Perhaps those scientist from the past should have said it had a high probability of being true. I may be misunderstanding you but I do not believe science can produce certainty and this seems to be a common view. I quote wikipedia.

A scientific theory is empirical, and is always open to falsification if new evidence is presented. That is, no theory is ever considered strictly certain as science accepts the concept of fallibilism.

It may be interesting that although all measurable results in quantum theory are in the form of probabilities there is at least one instance where this theory predicts a certain result. If the same measurement is immediately made a second time on a quantum system the second result will be the same as the first with probability 1. In other words the state of the quantum system revealed by the first measurement is confirmed by the second measurement. It may seem odd that the theory predicts the result of the first measurement as a probability distribution of possible results but predicts only a single possible result for the second measurement.

Wojciech Zuruk considers this as a postulate of quantum theory (see his paper quantum Darwinism ). (sorry for the typo in the quote).

• Postulate (iii) Immediate repetition of a measurement yields the same outcome starts this task. This is the only uncontroversial measurement postulate (even if it is difficult to approximate in the laboratory): Such repeatability or predictability is behind the very idea of a state.

If we consider that information exchange took place between the quantum system and the measuring device in the first measurement then we might view the probability distribution implied by the wave function as having undergone a Bayesian update on the receipt of new information. We might understand that this new information moved the quantum model to predictive certainty regarding the result of the second measurement.

Of course this certainty is only certain within the terms of quantum theory which is itself falsifiable.

I have some skepticism about absolute certainty. Logic deals in certainties but it seems unclear if it absolutely describes anything in the real world. I am not sure if observed evidence plays a role in logic. If all men are mortal and if Socrates is a man then Socrates is mortal appears to be true. If we were to observe Socrates being immortal the syllogism would still be true but one of the conditional premises that all men are mortal or that Socrates is a man would not be true.

In science at least where evidence plays a decisive role there is no certainty; scientific theories must be falsifiable, there is always some possibility that an experimental result will not agree with theory.

The examples I gave are true by virtue of logical relationships such as if all A are B and all B are C then all A are C. In this vein it might seem certain that if something is here it cannot be there, however this is not true for quantum systems; due to superposition a quantum entity can be said to be both here and there.

Another interesting approach to this problem was taken by David Deutsch. He considers that any mathematical proof is a form of calculation and all calculation is physical just as all information has a physical form. Thus mathematical proofs are no more certain than the physical laws invoked to calculate them. All mathematical proofs require our mathematical intuition, the intuition that one step of the proof follows logically from the other. Undoubtedly such intuition is the result of our long evolutionary history that has built knowledge of how the world works into our brains. Although these intuitions are formed from principles encoded in our genetics they are no more reliable than any other hypothesis supported by the data; they are not certain.

One example in classical logic is the syllogism where if the premises are true then the conclusion is by necessity true:

Socrates is a man

All men are mortal

therefore it is true that Socrates is mortal

Another example is mathematical proofs. Here is the Wikipedia presentation of Euclids proof from 300 BC that there is an infinite number of prime numbers. Perhaps In your terms this proof provides 0% confidence that we will observe the largest prime number.

Take any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then, q is either prime or not:

1) If q is prime then there is at least one more prime than is listed.

2) If q is not prime then some prime factor p divides q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but as we know, p divides P + 1 = q. If p divides P and q then p would have to divide the difference of the two numbers, which is (P + 1) − P or just 1. But no prime number divides 1 so there would be a contradiction, and therefore p cannot be on the list. This means at least one more prime number exists beyond those in the list.

This proves that for every finite list of prime numbers, there is a prime number not on the list. Therefore there must be infinitely many prime numbers.