Science
The logic of science
Undeniably true propositions
Proofs
False premises
Theoretical knowledge, factual knowledge
Falsification
Induction and history
Scientific progress
Order
Natural orders
Phenomenal aspects
Phenomena
Inorganic substances: process
Organic, inanimate substances: evolution
Animate substances: conscious behaviour
Persons: self-conscious action
Metaphysical orders
Science
The logic of science
‘Science’ literally means knowledge. “Scientific knowledge” is therefore a pleonasm. By extension, ‘science’ stands for the dedicated search for knowledge and connotes openness to criticism of methods used and claims made. With respect to this connotation, “secret science” is an oxymoron and “scientific knowledge” is not a pleonasm but refers to knowledge obtained by dedicated and public search. The search for knowledge, or truth, is not the search for consensus among experts, eminence in academia, headlines, book sales, research grants, peer recognition by people in search of similar things, or other signs of academic, social, economic or political success.[1] A scientist is someone who searches for truth — he is a scientist only as far as that is what he is doing. Besides, no one is a scientist merely because other people call or believe him to be a scientist, or because he has the legal permission to call himself a scientist.
Here we are going to discuss only some of the main points in the logic of science as knowledge. The logic of research,[2] or search for knowledge, is not on our agenda.
Knowledge implies truth. For every proposition P, if you say that someone knows that P then you imply that P is true. Obviously, something may be true without anybody knowing it to be true, but nobody can logically claim “P is true but nobody (including me) knows it.” However, “He knows that P, but he does not know he knows it” is not necessarily false, although obviously “I know that P but I do not know that I know that P” is a necessarily false statement. Science is knowledge that is known to be knowledge. Because knowledge implies truth, what is false cannot be science; moreover, it can never be that one science contradicts another.
Knowledge is not the same thing as belief. It is not the same as having true belief(s). It makes sense to ask a person why he believes what he believes. He can adequately answer by giving his reasons, even if the questioner finds them unconvincing. In contrast, it does not make sense to ask a person why he knows what he knows. However, we can ask him howhe knows. To answer that question, he must give evidence and arguments that ought to convince the questioners and others of the truth of what he claims to know. The evidence and arguments need not actually convince others, for they may be too dumb to understand them, too lazy to check them or too unconcerned to bother. However, whether the evidence and the arguments justify his claim to know is a question of logic, not of psychology. Every argument implies the claim that it ought to and does respect the laws of logic just as every assertion implies the claim that it is true and ought to be recognized as such. Without those implications, the speaker produces no argument or statement. Hence, as self-respecting persons, who mean to be serious and want to be taken seriously, we ought to respect the laws of logic. Indeed, we cannot argue that we ought not to respect the laws of logic in making arguments. If we did that, we would give ourselves permission to speak without logic — or rather, since speech (as distinct from babbling) implies commitment to logic, permission to pretend to speak without actually doing so. Moreover, and regardless of our conscious intentions, we would give our opponents permission not only to do the same but also to interpret our utterances as incoherent babbling. Similarly, every argument that relies on evidence implies the claim that the evidence is reliable and ought to be recognized as such. Without that implication, the speaker produces no evidence in support of his argument.
A claim to know how to do something can be made good by actually doing it or describing a step-by-step method of doing it in which each step is known or can be shown to contribute unequivocally to the production of the intended outcome. A claim to know that something is true needs to be made good with proof, i.e. a deduction from one or more true or undeniable propositions (the premises of the proof).
Undeniably true propositions
A proposition is undeniable or irrefutable for a particular person if he would not know how to refute it or would not be able to try to do so without involving himself in a contradiction. The contradiction can be a formal one (claiming both P and not-P to be true). It can be a performative contradiction (claiming to do what one is evidently not doing, or not to do what one is evidently doing). It can be a lie (claiming to know that something is so and so while knowing it is not). Or it can be a mistake or a sign of ignorance (claiming truth for something that is known or can be shown to be false). Of course, the fact that some people cannot refute a proposition does not mean that no person can do so; it does not prove that the proposition is undeniable, let alone true.
An undeniable, irrefutable or necessarily true proposition is one that no person can prove false. Any attempt to deny or refute it involves one in a logical or a performative contradiction. Consider, for example, propositions or propositional forms like “A is true only if A is true”, “For any number x, there is a number that is greater than x”, “Eggs come twelve a dozen” or “Minutes before he died, he was still alive.” They are either formal tautologies or else analytical truisms that not even God could deny.
Some statements are analytically true (e.g., “A bachelor is not married”, “Until he died, he was still alive”). Other statements are analytically false (“An even number of eggs cannot be distributed among two persons so that the one gets as many eggs as the other”, “A hectolitre is 10 litres”). Their truth-value can be assessed a priori, without investigation of the things mentioned in them. We need only know, understand and analyse the meanings of the words used in making them and be able to make deductions from them. The relevant meanings should be looked up in dictionaries or in specialized vocabularies or terminological lists. However, these sources may not always agree, which means that the same sentence can express an analytical proposition under one linguistic convention but not under another. Some dictionaries may even include in their definitions elements that should be reserved for encyclopaedias. Compare “Whale: common name for big animals that spend their entire life in the sea” with “Whale: common name for big mammals that spend their entire life in the sea.” Under the first definition, “A whale is not a mammal” is false, but not analytically false. Under the second definition, it is still false but also analytically false — but that definition includes some knowledge about whales, not just about the common use of the word ‘whale’. A person may be able to use the word ‘whale’ correctly (e.g., he is infallibly able to pick out the pictures of whales from a miscellaneous collection of photographs), yet know zilch about what distinguishes a mammal from a fish.
Some statements are such that their truth-value cannot be assessed a priori. “This morning, a whale was found on the beach near Dunkirk” is an example. To assess its truth-value, we need to find out about the things mentioned in it (not just about the words used to make it). Statements of this kind are synthetic (not analytic). Their truth-value can be found only a posteriori, after an investigation of those things.
Not all synthetic statements are a posteriori, however. The truth-value of some synthetic statements can be known a priori. E.g., “All human beings are mortal.” This is undeniable, irrefutable and, in that sense at least, true (although there is no positive, constructive proof of it). It is not just that all the evidence we have confirms its truth. No matter how long a man lives, his living that long is not proof that he will live forever. Methuselah died eventually, but even his still being alive today would not prove his immortality. In fact, there will never be a moment when we have proof that some man is immortal. Hence, it is useless to say, “But the concept of immortal man is not contradictory in the way the concept of square circles is. It is not logically impossible that there will be at least one immortal man in the future.” Indeed, it is logically possible that at least one human being is immortal — which is why “All human beings are mortal” is not a formal tautology or an analytic proposition.
“Man is mortal” is an example of a synthetic statement that is true a priori: synthetic because “being mortal” is not part of the lexicographical definition of “man”, and a priori because mere thought can establish that no event in space or time can possibly prove it false. For the same reason, “No compound or complex material thing lasts forever” is undeniable. Again, there is ample evidence that supports its truth, and there could not ever be evidence that it is not true.
From the examples about men and physical objects, we cannot inductively infer that nothing lasts forever. Metaphysical things, such as numbers have no physical existence. There is no direct physical evidence of their existence,[3] although there may well be physical evidence of the effect of their being. For example, a machine may be proof that its maker had access to mathematical knowledge, even if there is no physical evidence of its maker. No empirical evidence of metaphysical things going out of existence is possible — and of course, there can be no physical evidence to the effect that they will never cease to exist. Whether they can pass from being to not being (nothingness) is a question we cannot answer by referring to physical evidence. Indeed, as long as one of us is around to raise the question, such metaphysical things will have to be there, or the question will not be meaningful. For metaphysical things, to be is to be eternally or beyond time. This too is an undeniable, irrefutable truth. Mathematical being has no history, no development, no evolution; the only history of mathematics is the history of human discoveries of mathematical truths.
“Every existing physical thing can be the outcome of a coincidence of blind, unintentional or purposeless events or processes” is a synthetic a priori proposition. The coincidence may be extremely unlikely; but given the vast expanse of the universe both in its spatial and temporal dimensions, it cannot be excluded a priori. The famous “parallel roads” of Glen Roy, Scotland, which certainly look as if they were designed by man, are now deemed effects of natural forces (water and ice). Even a computer is nothing more than a particular physical arrangement. It is easy enough to prove that the computers we actually use are intentional products of intelligent design and purposeful manufacture; however, that does not prove that nothing that is physically identical or similar to a computer could come into existence by natural, unplanned, unintelligent coincidences. In fact, there can be no proof to the effect that such coincidences are impossible. This we know a priori, because what was put together on purpose was put together; it would have come about just the same if the physical process of putting it together had happened accidentally.
Consider for example the Darwinian “idea of evolution by natural, accidental variation and natural selection”. Darwin's idea was essentially an extrapolation from the intentional, artificial selection of animals and plants by humans. If farmers and breeders were able to produce new varieties of species in the short span of human history and on the limited area of land fit for human habitation and husbandry, think what natural variation and selection could accomplish over the full reach of time since the solar system came into existence! Thus, the Darwinian idea of evolution and the proposition that specimens of every kind of physical thing might have come about unintentionally by a coincidence of unintelligent factors are both synthetic a posteriori truths. However, although they are undeniable, there is no positive, constructive proof for them. In contrast, the Darwinian theory of evolution, which states that Darwin's idea actually explains the existence of every animate or inanimate organism, is not a synthetic a priori proposition. To appreciate the difference, compare “Accidental variation and selection might have produced X” (Darwin's idea) with “Accidental variation and selection did produce X — and did so without the assistance of any other factor” (the Darwinian theory).[4]
Other synthetic a priori truths are: “Human persons, such as you and I, can act”, “Being capable of acting, we have notions of past, present and future events or situations; of causal connections, regularities, means and ends, making choices; of better and worse, rights and obligations, being honest and pretending”. It is useless to try to demonstrate that they are not true. Every attempt to do so requires you to act, to consider what you hope to achieve by that action or how doing it will produce the intended effect, and to explain why others ought to accept that you have successfully refuted one of the above propositions. Yet, none of this you could do without presupposing their truth throughout your argument, which is itself essentially an intentional, purposive, claim-stating action.
“If you can choose, you must choose”, “Every choice has a cost” or “Choosing one thing entails giving up another” — these are synthetic a priori truths. So is “The utility of an additional unit of a good is lower than the utility of an already acquired unit of that good.” Yet another example “A person uses the means available to him to try to achieve his most highly valued ends for which he thinks them suitable, while postponing attempts to achieve ends he values less until additional means are available.” The undeniability of these propositions becomes clear as soon as we consider the plight of a person who would set out to argue that they do not apply to him.
It is of course possible to transform synthetic propositions (whether a priori or not) into analytical propositions by referring to a specialised list of definitions. This happens frequently in many sciences, when attempts at formalisation are made. Then so-called technical definitions are supplied not only to remove the ambiguities of natural languages but also to “pack” terms with essential knowledge. Such technical terms consequently refer not to a general dictionary based on common speech habits but to an encyclopaedia providing essential knowledge of a particular field. Recall the two definitions of the word ‘whale’ that were mentioned earlier. In a formal presentation of zoology (such as a textbook), “A whale is a mammal” will be an analytic proposition, if the text refers to a definition of ‘whale’ that already includes basic knowledge of whales, in particular, the knowledge that they are mammals. Still, that particular item of knowledge in the vocabulary of the text is synthetic. If the proposition about whales being mammals is analytic in the text itself then that is merely an artefact of the formalisation — not a qualification of that item of knowledge.
Note that the distinction between a priori and a posteriori is relative to the assessor of the truth or falsity of a proposition. On the one hand, we, humans, know a priori that the proposition that all men are mortal is undeniably true. On the other hand, gods, who are beyond time, might know that some humans are (or will be) immortal. They might even know that humans do not really exist or that they are actually incapable of acting or thinking. However, we could not know those things, since to assess their truth-values, we would have to engage in acting and, in particular, in reasoning, experimenting or otherwise marshalling the required evidence. In other words, our attempts to assess the truth-values of these supposed items of divine knowledge already presuppose our existence and our capacity of acting, thinking and speaking. If, in the eyes of the gods, we are merely deluding ourselves when we believe that we exist or can act, we can nevertheless never discover the illusionary nature of those beliefs. Thus, some propositions that are not necessarily true in themselves are undeniable for us. Because we are interested here in human science, not in scientia divina, we should accept that such propositions represent human knowledge.
Proofs
A proof can be direct, i.e. constructive: it leads by logically valid steps from true or undeniable premises to the proposition to be proven. The general form of such a proof is as follows:
P is true
Proof
Q is undeniably true
QP [the actual construction]
Therefore, [arguing in modus ponendo ponens[5]]
P is true
Q.E.D.
Alternatively, a proof can be indirect or non-constructive: without being constructive, it nevertheless shows that the proposition in question is undeniable or irrefutable, for example by a reductio ad absurdum of its negation. This kind of proof takes the following form:
P is undeniable
(i.e. ¬¬P)
Proof