Thursday, 22 May 2014

Proof and Doubt: A Non-Technical Introduction







The notion of proof is over-used and little understood by the non-philosopher. Put plainly. He or she often demands proof when proof is impossible. Not only that, but he or she often doesn’t have a clear conception of what it is, precisely, that s/he wants.

Take the case of the law. In law we never really have proof that so-and-so is guilty or not guilty. All we have is that Mr X is guilty ‘beyond reasonable doubt’. Or that it is beyond reasonable doubt that Mr X was in vicinity Y on the night in question. But do we have a proof that Mr X was at that place? What would constitute such a proof? Even considered theoretically, we can see the problems here. For example, it may be the case that Mr X was caught on camera. But do we have proof that the man caught on video really is Mr X? Perhaps it is his twin brother.

Descartes, for one, believed that he had a proof of something if he could not possibly doubt its truth or existence. Thus he could not doubt his own thinking. He had proof that thought existed. More specifically, if his ‘ideas’ were deemed to be ‘clear and distinct’ then he had proof of their truth or existence.

Instead of relying on the Cartesian and psychologistic notion of doubt to establish proof, 20th century logicians and philosophers dealt with proof in terms of contradiction and non-contradiction. Put simply. Something has a proof if its negation constitutes a logical contradiction. Thus it is not the case that I cannot doubt that 
1 + 1 = 2

Or, rather, it may well be true that I cannot doubt this truth. However, what really matters, logically, is that its denial would constitute a contradiction. To say that 1 + 1 = 3 is to utter a statement that contradicts itself. But how, precisely, does the equation 1 + 1 = 3 contradict itself? It all boils down to definitions. If we define each symbol and number in the equation we will soon find that 3 is the wrong answer to the addition 1 + 1. That is, 1 + 1 must equal 2. Thus if:

i)                    ‘2’ df. = the number which doubles the number 1
ii)                  ‘1’ df. = the first positive number
iii)                ‘+’ df. = the sign for the addition of numbers
iv)                ‘=’ df. = the sign which signifies the identity, or equality, of both sides of an equation.

If 2 is the number which is double the number 1, then 1 + 1 states that it is the doubling of the number 1 and that it must therefore equal 2. Similarly, if the ‘=’ symbolises equality or identity, then if we have 2 + 1 = 4 we can see that we have ‘3’ on the left side and ‘4’ on the right. Clearly, then, 3 cannot be identical to 4. The ‘3’ of the ‘2 + 1’ contradicts the ‘4’ on the right side in the sense that it is not equal, or identical, to the ‘3’ on the left side of the equation.

We can apply the same kind of reasoning to non-mathematical analytic truths such as the well-known

All bachelors are unmarried.

This is also true because of the identity or synonymy of the words:

i)                    All bachelors are unmarried
ii)                  [bachelor] = [unmarried]
iii)                2 + 1 = 3
iv)                3 = 3

i) above is an analytic truth solely because of the synonymy of the words ‘bachelor’ and ‘unmarried man’. And like our ‘2 + 1 = 3’, we do not need empirical research to constitute its truth and that is why it too is analytic. Thus, again as with our ‘1 + 2 = 4’, the statement ‘All bachelors are married’ contradicts itself just as 1 + 2 = 4 does:

3 = 4

However, it is still true that in the Cartesian sense it is the case that we cannot doubt that 2 = 2 is true; though we can easily doubt, if only initially, the truth of a difficult or complex equation. So non-doubt may be of value for 2 + 1 = 3, but be of no relevance for difficult or complex equations.

We should doubt the Cartesian epistemic maxims:

i)                    non-doubt = truth
ii)                          doubt = falsehood

Not only that, despite the early reference to the logical notions of contradiction and non-contradiction,

i)                    What we cannot doubt (a psychological notion) may still be wrong.

More strongly:

ii)                  What we cannot doubt or deny without engendering what seems to result in a contradiction may still be wrong.

For example, it is no doubt (!) the case that very many people, both today and throughout history, can and could not doubt the existence of God. They found it impossible, psychologically speaking, to believe the statement ‘God does not exist’ (forget Plato’s beard!). Bringing God’s existence or non-existence into the equation shows us that doubt or non-doubt (belief) is clearly a psychological, not a logical, notion. We can say that non-doubt didn’t bring God into existence, as it were. Even universal and absolute non-doubt would not do so! More relevantly. We may find it hard or even impossible to doubt an equation or a truth of maths but it still may nevertheless be false. Similarly, we can easily doubt the truth of an equation which is in fact true. (It is worth stating here that many 20th century logicians and philosophers have questioned or even denied the Law of Excluded middle. They have questioned the truth of p or not-p, or more strongly, A or not-A. Some logicians, e.g., dialethic logicians, have even said that A and not-A is - or is possibly is - true in certain cases. [see G. Priest’s 2002])

Perhaps the introduction of the notion of doubt into logic, maths and philosophy itself was the prime vice and mistake of what Frege, Husserl and others took to be ‘psychologism’. Proof for these logicians is based on the notion of contradiction and non-contradiction which also determined a logical system’s validity and consistency. A contradiction in a system would make that system inconsistent and thus rejectable. In addition, the use of doubt, or even the reality of doubt, within maths and logic would make those disciplines far too psychologistic in nature for many late 19th century logicians. The laws and truths of logic were deemed by them, after all, to be mind-independent and not ultimately dependent on minds at all – even if it is minds that come to know and notate these abstract non-spatiotemporal truths and laws [see Frege’s 1884 and Husserl’s 1900].

We can also go on to say that it is possible to doubt just about anything if we so desire. And this was precisely the kind of radical or global scepticism that Descartes reacted against - thus he came up with his famous Cogito. For example, we can doubt that we are truly sane; that we are not actually awake but asleep; that we are not a brain in a vat. And so on. Descartes, again, reacted against global scepticism by using doubt itself to defeat doubt. His ‘method of doubt’ was Descartes’ attempt to prove that we cannot doubt our own doubting or thinking. Modern logicians, however, simply dispensed with the whole notion of doubt vis-à-vis logic and maths.

As the late Wittgenstein [1969] argued, radical and global doubt is not possible anyway. Radical doubt, if actualised, would render all communication - and even thought itself - impossible or senseless.

In tandem with the notions of contradiction and non-contradiction in the domains of logic and maths, we also see the importance of definitions, as we said earlier. We have already said that it is ultimately definitions that constitute the nature of proof and contradiction. Thus, from our explicit definitions of numbers or logical principles and logical components we derive logical laws and truths and ultimately whole logical systems. We can see, then, that definitions in this sense are the axioms of logical systems. And the same is the case with geometry and its definitions.

On the other hand, contradiction and non-contradiction are of little or no value when we come to empirical statements and those of science as a whole. For example, it is an established scientific fact that the earth revolves around the sun. This truth of science has been recognised for over 400 years. However, if I were to say that the sun revolves around the earth this would not be to utter something that is a contradiction of the reverse scenario:

i)                    the sun revolves around the earth.

does not contradict the statement that

ii)                  The earth revolves around the sun.

The former does not contradict ii) in the strict logical sense we have covered. It is simply the case that i) is a scientific falsehood, not a contradiction of ii).

There is also another sense in which proof in empirical matters or science is different to proof in logic. Take our examples again. We do not really have a proof that the earth revolves around the sun. Strictly speaking, we have no proof that this is so. In fact it is difficult to even imagine what could constitute a proof of that scientific fact. It could be the case that all our scientific theories are false. Indeed, historically speaking this has been the case. It can also be argued that proof in science would prove to be counter-productive in the long term. That is because it was the eventual refutations and rejections of well-established scientific theories that pushed science, as a whole, forward. If scientists really believed that they had iron-clad proofs of their theories (which couldn’t then be called by that name!), then new theories would never have been accepted if they negated the established ones. Thus science would have atrophied at the time of Copernicus or even at the time of the pre-Socratics. Instead, scientists have come to accept that science, or its theories, are all fallible. Scientists are, or should be, fallibilists. They should thus reject or even do without any strict notions of proof in science. Proof is the game which only logic and maths play. Science should not play that game. We can conclude positively that a lack of scientific proof of a theory is no reason to throw it overboard. There is no proof that Darwinian evolution [see Popper’s 1974] has and is the case in the natural world. On the other hand, our inability to disprove a theory in no reason, on its own, to accept it. We can say that the Big Bang theory has not been disproved (how could it be?). However, this negative compliment is no reason – on its own! – to accept this theory. Quite simply, no scientific theory can be proved to be true. But we don’t need proof in science. And if proof is unobtainable because of the incompleteness of a theory, or its lack of a 100% truth-content, is no reason to reject a theory. For a starts, an incomplete theory is better than no theory at all. That theory may have much to tell us - even if incomplete. Similarly, a 90% truth-content, or even only a 50% truth-content, is better than no truth-content at all. We can say, as with Popper [1963/72], that Newton’s theory of gravitation has less than a 100% truth-content. And yet this theory has proved to be immensely important and successful in scientific history and has provided us with countless benefits in technology. And if we see Einstein’s theory of relativity as its successor, it may be, or is, still the case that this theory doesn’t have a 100% truth-content either - even if the percentage is higher than that of Newton’s theory. Perhaps a 100% truth-content is impossible in principle. Indeed what would that mean? What would constitute its 100% truth-content?

Similarly, an inability to prove something wrong is no reason to accept that something. As we said earlier, global scepticism cannot be conclusively refuted, but that is no reason, on its own, to accept it because if we did, communication - and even thought itself! - would be rendered impossible or senseless. More particularly, we have no strong or good reason to really believe that we are brains in a vat, or dreaming when we think we are awake, or that other people have no minds. As Wittgenstein [1969], and Hume [1739] before him, argued, we just can’t help but believe these things anyway [see Strawson’s 1985]. And, again, if we did deny other minds, etc., we would need to become - or be - solipsists. In practice that would prove (!) to be impossible too.

We cannot prove God’s non-existence. We cannot prove His existence. But proof alone doesn’t give us a strong or good reason to either believe or disbelieve in God’s existence. I cannot prove that a teacup floats around a distant planet, as in Russell’s example [Russell]. And neither can I disprove the assertion ‘that there is an invisible pink elephant dancing on this book in front of me’ [Baggini, 2002]. Just as Wittgenstein’s Tractatus tautologies do not even allow the possibility of their negation or refutation, so many sceptical scenarios cannot be disproved in the strict logical sense [Wittgenstein, 1921]. Indeed, many sceptical hypotheses are designed by philosophers to be irrefutable or beyond disproof (and proof!) [see A. J. Ayer’s 1936]. What would make today living as brains in vats differ from yesterday when we were not brains in vats? What could differentiate a Cartesian dream-state from the state of being awake? How could mindless persons differ from persons with minds? In addition, if Russell’s scenario of a five-year-old earth, replete with fake sciences of age but which are nevertheless duplicates of actual signs of age, be different from a planet that is billions of years old? In the case of the pink elephant dancing on this book in front of me, the sceptic makes it being disproved (and proved!) in that he claims that it ‘has no weight, no colour, smell, or texture’. But we can ask the sceptic:

How does a weightless, colourless, odourless, and texture-less thing differ from nothing – or no thing?

As Wittgenstein put it. A nothing will do as well as a something in such a scenario [1953]. Similarly, how does Russell’s teacup, or the metaphysical realist’s ‘future contingent’, differ from nothing? Neither have truth-conditions. So who cares if they are still either true or false? Future contingents are non-verifiable and have no truth-condition – in principle [see Dummett’s 1982].

If proof is impossible, even in principle, when it comes to these statements about the world, about the future, about other minds, about brains in vats, about pink elephants, about inter-galactic teacups, then we have no need for proof in such contexts. Thus we can erase the word ‘proof’ from these ‘contexts’ [see D. Lewis’s 1996] of empirical and scientific discourse and leave it where it belongs – in logic and mathematics! Indeed not only is proof impossible in the case of these sceptical hypotheses, but the belief in it has proved to be destructive in science and philosophy. Proof, like certainty and ‘ultimate foundations’, are malignant demons when it comes to these disciples. It is far from being a coincidence, then, that not only was Descartes concerned with proof in his philosophy, but he based his philosophical reasonings, or so he claimed or thought, on the deductive systems of maths, geometry and logic [see Descartes’ 1641].

In Hume’s words, ‘matters of fact’ can never be conclusively proved, whereas his ‘relations of ideas’ may well be – if only on Hume’s terms. And as Tractatus Wittgenstein might have put it. We have proof in logic and maths precisely because their propositions are not world-directed – they are not about the experiential or empirical world!

The acknowledgement that proof is impossible or counter-productive outside logic and maths is well captured by C. S. Peirce’s technical term ‘abduction’ [Peirce, 1992 and Hookway, 1985]. When we use abductive logic in science and philosophy what we are essentially looking for is the ‘argument to the best explanation’. We are not looking for an argument which we can prove; and neither are we looking for an argument or theory that has a 100% truth-content and is therefore complete. Instead we are simply looking for the best argument, piece of evidence or theory we have at a particular point in time. So even though our argument or theory is incomplete and not 100% true, it may still be superior to other arguments or theories with less completeness or truth.  In Popper terms [Popper, 1963/72]:

theory1 = 90% truth-content
                10% falsity-content

theory2 = 50% truth-content
                50% falsity-content

(It is acceptable to ask oneself here how it is, exactly, that we can discover such precise percentage levels in scientific and other theories.)

It can be argued that if we somehow know that we only have a 90% truth-content in the case of theory1 above, there must be a sense that we implicitly have 100% knowledge of its truth in that in order to make that distinction between a 90% truth-content and a 10% falsity-content we would need to know why it is the case that 10% of the theory is in fact false. But how could we know that this is the case if we don’t also know what it is that makes 10% of this theory false. Wouldn’t we need to know what are the facts or pieces of evidence which make 10% of this theory false? We surely can only know what is false if we also know what makes it false. And what makes 10% of our theory false may be knowing what makes this slice of the theory false will take us half way to knowing what it is that would make it true. Thus we have a seemingly paradoxical case of holding a theory that is both 10% false and 100% true! However, we could know that 10% is false without having to know what, precisely, makes it false. Similarly, we don’t know the 10% truth-content which makes, as it were, the theory only have a 90% truth-content. We know that it is 10% false, but we don’t know the nature of the truth-content that could or would supersede that 10% negative score. For example, we may know that 848 x 68 = 1,000,000 is false without at the same time knowing what is the correct answer to this multiplication. Perhaps we could know it to be false for purely intuitive reasons – that is, the incorrect answer simply seems too high a number to be the answer to the equation. But we do not know why we think that. We simply have an intuition (whatever that is!) that it is, or that it must be, false!

Another deflationary and fallibilistic reality of scientific schemes is that there may be many competing theories that deal with the same evidence and data. Scientists happily accept this lively competition [see Quine’s 1953 and 1969]. They think that it is a good thing. What this is called, in technical terms, is the well-know thesis of the ‘underdetermination of theory by the evidence’. After all, perhaps the favourite theory has less truth-content than its neglected rivals; even if that is not always a case of comparing truth-content with truth-content. For example, even if theory2 has only a 70% truth-content to the 80% of theory1, the former may be preferred because of its simplicity, explanatory power or even because of its aesthetic appeal. So scientists as a matter of course accept rival or even complementary explanations and theories even if they don’t themselves utilise them. They simply acknowledge the well-accepted fact that they ‘are often presented with more than one possible explanation for an event or a state of affairs with no conclusive way of knowing which one is correct’. If that is indeed the case, then the scientist may simply accept the ‘best explanation’ or theory even if that has less truth-content that some or its rivals. It may, for example, be simply that other theories are less revolutionary - or more revolutionary [see Quine’s 1953 and 1969].

In terms of explanatory power’, instead of proof, we cannot prove that other people have minds or that the earth is not five years old, but we can safely assume that they have minds and that the earth is billions of years old. We can do so because believing that other people have minds ‘explains why it is they talk like they do, act like they do, have the same physiology as us’ [Baggini, 2002]. Similarly, considering the not inconsiderable amount of evidence for the earth’s old age and the absurdity of Russell’s hypothesis (even if non-refutable), it is quite simply rational to accept the scientists and laugh at the sceptics who truly believe the sceptical scenario or just its possibility. In other words, the hypothesis or theory that other people have minds and that the earth is ancient have more explanatory power than the theories and hypotheses which deny them. Again, in terms of Peircian abduction, theories are ‘arguments to the best explanation’. They are superior to all the arguments and theories that would deny – if not entertain – the fact (?) that other people have minds and that the earth is not the age of a toddler. However, we still do not have proof that other people have minds and that the earth is old. We may not even need proof.

Take the interesting case of conspiracy theories [see Baggini, 2002]. Such things generally have less explanatory power than their generally-accepted rivals. We can offer a list of factors and reasons which should make us - even more - suspicious of the majority of such conspiracy theories:

i)                    Their ‘facts’ are not firmly established - or established at all.
ii)                  Conspiracy theorists hypothesise huge amounts of supposedly ‘suppressed information’.
iii)                Conspiracy theories do not have available the requisite evidence or data. (This, ironically, is often seen, by conspiracy theorists, as evidence for the conspiracy theory!)
iv)                Conspiracy theories are often not simple but they are complex instead (almost by definition).
       v)         The complexity of conspiracy theories often make them more exciting    and  pleasing to conspiracy theorists (admittedly this is a psychological point).

All the above give us good reasons to reject most conspiracy theories.


References and Further Reading

Baggini, J – (2002) Making Sense, BCA
Descartes, R – (1641) Meditations on First Philosophy, various editions
Dummett, M – (1982) ‘Realism’, in Synthese 52, pp. 55-112
Frege, G – (1884/1950) Foundations of Arithmetic, trans. J. L. Austin, Oxford
Hookway, C – (1985) Peirce, London
Hume, D – (1739) A Treatise of Human Nature, various editions
Husserl, E – (1900/1970) Logical Investigations, tr. London
Lewis, D – (1996) ‘Elusive Knowledge’, in Australasian Journal of Philosophy 74, 4, pp. 549-67
Moore, G. E – (
Peirce, C. S – (1992) Reasoning and the Logic of Things, Cambridge Mass.
Popper, K – (1974) ‘Darwinism as a Metaphysical Research Programme’, in his Unended Quest, Fontana
                  - (1963/1972) Conjectures and Refutations, Routledge and Kegan Paul

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