WEDNESDAY 3rd 7PM
Topic Title: Is 2 + 2 = 4 a justified true belief?
Proposed by: Michael K
1. Some philosophers maintain that, although we have knowledge, the knowledge that we have is not known with certainty, and cannot be known with certainty. Such knowledge may be true but we have no way of knowing whether it is true. So all our knowledge is open to question, fallible, hypothetical, not much more than guesswork.
2. So the question is how can we square this philosophical position with some things that we regard as ' obviously true'? Obviously true! Ha! Ha! Surely the proposition that the earth is round rather than flat is an 'obvious truth'. And the proposition that 2 plus 2 equals 4 is, surely we might say, a 'certain truth'. Or is it? We may be certain of its truth, but what is the nature of this certainty? Is it no more than a feeling in the bones? If so it is mere subjective certainty.
3. And what about falsity.? Do we know that 2+2=5 is false? It seems that here we can refute this proposition by finding two pairs of solid objects, counting them, and not getting 5. This is a convincing refutation, and we can assume that our senses have not deceived us. Note here that 2+2=5 is a universal proposition and we have refuted it with a single instance.
4. And similarly, in logic, we can refute the argument from the singular to the general by the example of 'This pen is black; therefore all pens are black.' Yet, over there is a red and thoroughly non-black pen. It takes only one counter-example to refute an invalid form of argument.
5. And how in logic do we 'know' that certain forms of argument are valid? (Valid in the sense that if the premises are true, the conclusion must be true). An example is: 'All men are mortal; Socrates is a man; therefore, Socrates is mortal.' We know this is a valid form of argument because it has been used again and again, and no counter-example has ever been found. Nobody using this argument has reached a false conclusion from true premises.
6. As for Plato, he and many others, thought that knowledge had to be justified true belief. It had to be true. And it had to be justified. We all think that 2+2=4 is true, but where is the justification? There is evidence, yes. We can confirm individual instances of it by taking pairs of objects and adding up - assuming our senses are not misled - but we can never confirm the totality of its instances because 2+2=4 is a universal law and the instances of this law are infinite in number. So let us accept that our certainty about two plus two is a long way short of being justified. And it will not do to claim that the two plus two thing is proved deductively from the axioms of arithmetic because we would then need to justify the axioms of arithmetic. The justification of one proposition requires another proposition which also has to be justified. So to justify any claim to knowledge we need to justify an infinite number of underlying propositions. This is the infinite regress argument made by the ancient Greek sceptics who came after Plato and Aristotle. And their argument seems pretty good to me. The quest for justification just goes on - and on - and on.
7. So it is difficult to agree with Plato's view that knowledge is justified true belief. Our own human knowledge is what Plato called 'mere opinion' - unjustified, not always believed, and possibly not true. This seems to be the case with scientific theories, and scientific knowledge is, surely, the best knowledge that we have. Scientists often refer to their theories as the best guesses we yet have. They are right. Our knowledge is all hypothesis, but the best of these hypotheses have been tested against experience. When they have survived rigorous testing through observation and experiment they can be accepted as potentially true, but they could still, one day, be found to be deficient or plain false.
8. So the conclusion I suggest is that our knowledge is all hypothesis, never justified, never certain, possibly true and possibly false. We live our lives on the basis of infirm theories which have worked so far, and when they give us problems we try out new theories to replace them. And if you ask the question of how do we know that all our knowledge is hypothesis and conjecture, the answer is that we don't know. We simply guess!
The thinkers behind these thoughts include Xenophanes (6th C. B.C.), Pyrrho, Timon of Philus, Sextus Empiricus, Karl Popper, Joseph Agassi and David Miller.