William M. Briggs

Statistician to the Stars!

Good Ways Of Speaking About Truth

Quid est veritas?” Pilate asked. Famously, his interlocutor did not answer, perhaps because Pilate didn’t give Him the chance. Then Pilate may have been (understandably) addled because the Answer was standing there.

Anyway, Aristotle, under less pressure, had a go at a definition (one that Pilate almost certainly would have known). He said, “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.”

That is lovely, understandable, and complete. It—the definition—is called realism, a pleasant and accurate label. Actually, it is called Aristotelian or ‘moderate’ realism to distinguish it between the hyper and over-literal realism of his pal, Plato. That difference makes no difference to us today.

There are other ideas of truth, all of them wrong, which follow two main roads: idealism (everything exists in your head, therefore your head doesn’t exist) and nominalism (what’s in your head doesn’t exist, therefore there is nobody there to think up and fret over nominalism). But we’ll pass these by today, too.

Yesterday, we agreed it was true that ‘all men are mortal’ and that ’2 + 3 = 5′. It is the nature of men to die and for integers to behave undeviatingly according to certain rules. These are universal truths. There also exist contingent truths, which are propositions that accord with Aristotle’s definition conditionally. Unfortunately, there is only word in English for both, which means universal and contingent truths are often confused—which leads to hurt feelings.

To explain. Universal truths are those which begin with indisputable axioms and lead inexorably and necessarily to certain truths. For example, once we accept, without proof and based on no evidence save introspection, that “For all natural numbers x and y, if x = y, then y = x” and a couple of other similar sounding axioms, it is necessarily true that ’2 + 3 = 5′. Because we don’t know why or how the axioms can be true—we just know that they are—we don’t know why or how ’2 + 3 = 5′ is true, except in the weak sense that we say the equation is true because the axioms and intermediate theorems are. But we cannot say why it didn’t turn out that ’2 + 3 = 7′ (don’t even think of arguing over the symbols).

Contingent truths are those propositions which follow from premises that might themselves not be universally true. For example, if we accept “All cats speak French & Whiskers is a cat” then it follows, i.e. it is contingently true, that “Whiskers speaks French”. Yet nobody but a cat lady would run into the street and claim Whiskers’s linguistic ability were universally true. That’s because the first premise is, according to other well known premises, false. Therefore, on that evidence, the conclusion, while contingently true, is universally false. True and false simultaneously, at least speaking loosely, and therefore something to fight over.

The “Laws” of science are all contingently true. Any one or even all of these Laws may be universally true, but we don’t (possibly yet) know it. If they were universally true, then they would all be in the same epistemic boat as mathematics and logic. We would start with introspection, decide what follows from beliefs we just know are true, and then build theorem upon theorem until we reached the Law of Gravity.

That’s almost how it works, but not quite. Inside the Law of Gravity are several fudge factors, “constants” of the universe which are derived via observation, i.e. which are not deduced from first principles. And (we read) there are one or two other dicey premises which are not entirely convincing. The Law of Gravity, which nobody doubts in practice, cannot be said to be universally true (no pun; nay, not even from me), even though it contingently is.

Because the Laws of science are only, or at least, contingently true the premises which accompany them may be argued over. It is not unscientific to do so. It is prudent. When physicists argue over gravitation, it is clear to everybody that the conclusion is accepted because it is observed to hold in most places, and so discussion centers on the premises which would make the Law hold in all places.

The situation is different in climate science, for example, where the conclusion itself is in doubt (rampant global warming will kill half the population by 2009—oops, I meant 2017), and where the premises are so beloved that they are Not To Be Questioned. The (suitably modified to keep current) doom conclusion is contingently true, but that does not make it truly true, i.e. universally true. Failing to understand that distinction is what leads the weak to shout “Denier!”


13 Comments

  1. “Universal truths are those which begin with indisputable axioms”. Only humans dispute (so far as I know) so if any human disputes any axiom then there are no indisputable axioms. I’m told (it was a long time ago) that some mathematicians have rejected proof by contradiction and built another mathematics on what’s left so some heretofore indisputable axioms have actually been disputed. So if we can’t truly identify what is indisputable then we cannot identify universal truths. The converse isn’t true: what nobody disputes isn’t necessarily a universal truth.

  2. This statement is FALSE: “There are other ideas of truth, all of them wrong, which follow two main roads: idealism (everything exists in your head, therefore your head doesn’t exist) and nominalism (what’s in your head doesn’t exist, therefore there is nobody there to think up and fret over nominalism).”
    “Truth” is a function of how one defines “truth” AND relative to the yardstick used to measure “truthfulness.” The above, like so much of philosophical assertions, is founded in ambiguous jargon the meaning of which is left obscure. Using eugenics—the philosophical rationale used to support because it is curiously the same as that used to support Global Warming Alarmism, Michael Crichton illustrates how vague terminology helped conceal what was really going on.
    Briggs—you really need to incorporate clear definitions of the terms you use in conjunction with broad, noble-seeming concepts…doing otherwise is a basic logical flaw sure to lead to confusion, erroneous conclusions, etc.
    At any rate, the M. Crichton essay, “Why Politicized Science is Dangerous” (included at the end of his book, State of Fear) is readily found on-line: http://www.crichton-official.com/essay-stateoffear-whypoliticizedscienceisdangerous.html

    Back to the point about shades of truth relative to the yardsticks used—there being many “truths” depending on the measures used, and as such they are not wrong –an interesting example is made in the movie, A Man for All Seasons, where Sir Thomas More (a historical figure renowned both for his judicial expertise and religious piety, and, in hindsight, a significant bit of religious hypocrisy also founded on his strict application of philosophy) was on trial for not endorsing King Henry VIII’s latest divorce & remarriage. S. T. More thought he’d get away with that by being silent on the matter; as we know, that didn’t work. The movie provides some interesting dialogue in which “truth” as defined by law is battled against “truth” as indicated by what everybody knows the facts to be – and these are very different “truths” that, relative to the appropriate measuring stick are both correct though diametrically opposed. As this was a court trial, More’s rationale based on legal principles (as opposed to “common sense” interpretation) prevailed (so they applied perjured testimony to ultimately convict him). Here’s the dialogue from the movie illustrating “truth” relative to law vs. “truth” relative to what is common knowledge and known fact (as much or all of the actual testimony was preserved, there’s a good chance this is close to what was actually said, though for this example that doesn’t matter):
    Cromwell: Now, Sir Thomas, you stand on your silence.
    Sir Thomas More: I do.
    Cromwell: But, gentlemen of the jury, there are many kinds of silence. Consider first the silence of a man who is dead. Let us suppose we go into the room where he is laid out, and we listen: what do we hear? Silence. What does it betoken, this silence? Nothing; this is silence pure and simple. But let us take another case. Suppose I were to take a dagger from my sleeve and make to kill the prisoner with it; and my lordships there, instead of crying out for me to stop, maintained their silence. That would betoken! It would betoken a willingness that I should do it, and under the law, they will be guilty with me. So silence can, according to the circumstances, speak! Let us consider now the circumstances of the prisoner’s silence. The oath was put to loyal subjects up and down the country, and they all declared His Grace’s title to be just and good. But when it came to the prisoner, he refused! He calls this silence. Yet is there a man in this court – is there a man in this country! – who does not know Sir Thomas More’s opinion of this title?
    Crowd in court gallery: No!
    Cromwell: Yet how can this be? Because this silence betokened, nay, this silence was, not silence at all, but most eloquent denial!
    Sir Thomas More: Not so. Not so, Master Secretary. The maxim is “Qui tacet consentire”: the maxim of the law is “Silence gives consent”. If therefore you wish to construe what my silence betokened, you must construe that I consented, not that I denied.
    Cromwell: Is that in fact what the world construes from it? Do you pretend that is what you wish the world to construe from it?
    Sir Thomas More: The world must construe according to its wits; this court must construe according to the law.

  3. Ye Olde Statistician

    9 April 2013 at 11:17 am

    The two basic forms of truth — correspondence and coherence — roughly match contingent and universal. Scientific propositions must be “true to the facts,” that is, faithful to the known data, i.e., corresponds with empirical reality. Mathematical propositions must be “true to the postulates,” i.e., coherent in being logically derived from them. Hence, much discussion about the data, like the Airy disks that turned out not to be actual diameters at all; and mathematical explorations of reduced postulate sets.

    cf. http://plato.stanford.edu/entries/truth/

    A favorite example: given the axioms of ZF set theory, including the Axiom of Choice (AC), you cannot prove the Continuum Hypothesis (CH), full stop. However, if you drop AC and replace it with CH, you can prove AC. This is taken to indicate that CH is “true, but unprovable.”

  4. Sander van der Wal

    9 April 2013 at 11:29 am

    @Rich

    This mathematician would be Brouwer: http://en.wikipedia.org/wiki/Luitzen_Egbertus_Jan_Brouwer.

    Back to the subject. A premise is not necessarily a statement about the world as it is, it can also be about the world as it could be, i.e. a theory.

    The first premise “All cats speak French” is a theory (a toy theory in this case). “Whiskers is a cat” is a statement about the world as it is. If Whiskers does not speak French, also a statement about the world as it is, then the theory is false, it is not a theory about the world as it is. And if Whiskers happens to speak French, then the theory is still not proven to be true, unless Whiskers is the only cat in the world.

    The second premise, “Whiskers is a cat” doesn’t have to be a statement about the world as it is either. “All Elves are great shots with bow and arrow” is a premise about a fictional human-like species in a famous book. “Legolas is an Elf” is known to be true, as everybody who has read the book and/or seen the movie knows. And “Legolas is a great shot with bow and arrow” is true too, it is in the book, and the movie.

  5. Person of Choler

    9 April 2013 at 12:26 pm

    “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.” ~Aristotle

    “That depends on what the meaning of ‘is’ is”. ~Bill Clinton

  6. @Rich:

    I’m told (it was a long time ago) that some mathematicians have rejected proof by contradiction and built another mathematics on what’s left so some heretofore indisputable axioms have actually been disputed. So if we can’t truly identify what is indisputable then we cannot identify universal truths.

    This is wrong. The issue is more complicated (and more interesting) than you paint it, but just to stick to one of the problems, intuitionists like Brower only dispute the validity of a couple of axioms and deductive rules, not of *all* axioms or *all* deductive rules, so that intuitionistic mathematics is a proper subset of classical mathematics. Every intuitionistic proof is classically valid (since it uses less axioms and deductive rules) so that it is perfectly possible to follow and understand papers by intuitionists and communicate with them because there is common ground, or to use the terminology used by our host, undisputed axioms.

    Such undisputed axioms would include the law of identity, of non-contradiction, etc. as every deductive argument relies on them. As such there can be no deductive argument against them, for that would be self-refuting, which is precisely what your argument is: self-refuting. Employing reason to undermine reason is not to be tolerated in rational discourse.

  7. G. Rodrigues: it’s not necessary for intuitionists to dispute *all* axioms, just one of them, to make my point that what has been held to be indisputable may later be disputed. And I can’t see anything wrong at all with, “if we can’t truly identify what is indisputable then we cannot identify universal truths”. It’s only necessary to demonstrate that what is indisputable can be identified to show that there universal truths.

    My point, such as it is, is no more than that we cannot escape our human limitations but sometimes we gloss over them as when we talk about “proof” as though it somehow existed independently of us.

  8. @Rich:

    it’s not necessary for intuitionists to dispute *all* axioms, just one of them, to make my point that what has been held to be indisputable may later be disputed.

    I suggest you read again what I wrote, because you have not addressed it.

    And while I am at it, and to reinforce the point I made, intuitionists, and constructivists in general, do not discard all proofs by contradiction. They deem as invalid the rule

    (A) not-P => F |- P

    but not the deductive rule (sometimes called proof by negation)

    (B) P => F |- not-P

    (A) is deemed invalid because it is equivalent to the law of excluded middle (LEM for short), which is constructively invalid. But the interesting point, is why a constructivist deems the LEM invalid. The reason in short, is that a constructivist interprets the usual logical apparatus in a different way than a classical mathematician, so that when a constructivist denies the LEM he is not saying the same thing a classical mathematician is saying if he would per absurdum deny the LEM.

  9. All men are mortal…
    If we examine all birth and death records, we will see a subtantial fraction of those that have been born have not died.

    2+3=5 — didn’t it take Betrand Russel 200 pages to prove something similar?

  10. Briggs

    10 April 2013 at 1:53 pm

    Doug,

    Well, about 100 billion have come before us, all who have handed in their dinner pails. We infer from this the essence of man is to cease upon the eventual morrow.

    But I recall a Larry King interview with this group which claimed you didn’t have to die. How? Just believe you wouldn’t die. Who knew it would be so simple?

  11. @Briggs:

    I intend to live forever!

    So far I’m off to a great start. ;)

  12. Sanderr van der Wal

    11 April 2013 at 12:11 am

    @Briggs

    That all men are mortal is a theory, so far without an observation falsifing it.

  13. Briggs

    11 April 2013 at 6:03 am

    Sanderr,

    This is a good point, but wrong. The proposition is a recognition of the essence of what it means to be a man, an essence we have deduced from, inter alia, noticing that all men thus far (except those wandering about now) have died, from seeing that we are biomechanical creatures which wear out, etc.

    If we wanted to turn it into an induction, we could say, “All the men observed thus far are mortal.” This strips the essence away and leaves us with more of a dice role situation.

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