Logicians discerns different kinds of paradoxes and suggest different remedies. Though not a logician myself, I try to convey a hunch of these and argue that, if a transfer from logic to scientific paradoxes is at all possible, scientists should stick with Quine’s remedy augmented by Priest’s claim for independent evidence.
Broadly defined, “a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it.” (Quine 1976, p. 1)
According to Quine (1976) a paradox can be plain true, plain wrong, or based on inconsistent premises. The following examples illustrate true and false paradoxes.
The statement that a sailor who travelled the seas for 20 years comes home to celebrate his fifths birthdays, if sustained by a well told story, is absurd at first sight. Nevertheless, it turns out to be true once his birthday is identified as a 29th of February.
Conversely, 2 = 1 is false, even if it is sustained by the argument: “Let x = 1, then x2 = x; so x2 – 1 = x – 1. Dividing both sides by x – 1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1” (Quine 1976, p. 3). The fallacy lies in dividing by x – 1, which is zero.
The remaining paradoxes can bring on crises in thought, because “some tacit and trusted pattern of reasoning must be made explicit and avoided or revised” (Quine 1976, p. 5). According to Priest (1979, p. 200), however, mere avoidance is unsatisfactory and a “solution would tell us which premise is false or which step invalid; but moreover it would give us an independent reason for believing the premise or the step to be wrong.” Therefore paradoxes that are open to independent reason can be resolved by searching for contradictions in the premises and corroborating them by independent reasons. In science the latter are usually called evidence.
Unfortunately, some paradoxes are logically closed, like the statement: “This sentence is false.” They allow no independent evidence to impinge on them. Therefore, logicians devised alternative remedies for dealing with this kind of purely logic paradoxes. The orthodox remedy for a closed paradox is a hierarchy of truth values (Tarski 1956). A paradox would be true at one level but false at the adjacent levels and vice verse. That is, “This sentence is false” would be true at level x but false at level x+1 and x-1. Kripke (1975) argued for a gap between true and false. That is, such a paradox is neither true nor false but resides in a gap between these the truth values. Finally, Priest (1979) suggests that semantically closed paradoxes are true contradictions (dialetheia). They do not fall into a gap between true and false, but are both true and false at the same time. The following illustrates the different conceptions of truth values.
The logic where there is a gap between true and false, a region where something is neither true nor false can be applied to the Liar's paradox: "This statement is false." Nevertheless, there is a Liar's revenge paradox: "This statement is either false or neither true nor false." If it's false, then it will be true; and if it falls in the gap between true and false (see figure above) then it will also be true. I don't remember at the moment, whether there is a Liar's revenge on Tarski's solution, but there probably is. Otherwise, why would logicians have gone to creating paraconsistent logics?
Paraconsistent logics are made for the containment of the inference explosion that would otherwise follow. Inference explosion is the phenomenon in logics that, if your premises are inconsistent, you will be able to infer anything, for example, that you are a mink or that the moon is made of cheese. In other words, the world would become utterly nonsensical. Anyway, having no access to independent reason (evidence) one plays around with the truth values, until the contradiction entails no inference explosion. The best online source for an introduction is a new podcast at Rationally Speaking with Graham Priest.
As scientific paradoxes are open to independent reasons called evidence, proposing resolutions that are analogous to Tarski’s, Kripke’s, or Priest’s remedies seem second rate, because they are tailor made for closed paradoxes. It should therefore be possible to address scientific paradoxes with the logic of Quine. That is, ask whether they are plain true or false and if that is not the case search for inconsistencies in the premises (and then for independent evidence).
For example, the so-called invasion paradox states that invasive species have no history of adaptation to their new environment and should therefore be less competitive than the native species. In fact, however, they are invasive. That is, they exclude native species competitively. In introducing the paradox, Sax and Brown (2000) already admitted that it is not really such a bugger, because, invasive species left some enemies behind and usually have a longish history of non-invasive (cryptic) existence in the new habitat where they do eventually get invasive. Hence, the assumption that they have no history of adaptation to the new environment is usually plain false and the one that they should be less competitive is also plain false, because entering an enemy free space they should be more competitive.
Research on the lek paradox, on the other hand, is focussed on scrutinizing premises and finding out what's wrong with them (Kotiaho et al 2008, box 1). That is understandable, because one of the premises of this paradox states that heritable variance in fitness should sink to zero. If true, natural selection could not work and adaptation would grind to a halt. As a consequence, the mate choice seen in these species should degenerate, because investing time and energy into mate choice is useless, if there is no heritable variation in fitness attached to it. Alas, there are no signs of mate choice getting any laxer.
The paradox of sexual reproduction (Williams 1975; Maynard Smith 1978) is the fact that most animals and plants reproduce sexually despite an immediate twofold advantage in reproductive success that should be gained by an asexual (parthenogenetic) mutant female. Given the above arguments, I suspect an inconsistency somewhere in the premises. In my opinion this paradox cannot be plain true or false with such a history of leading evolutionary biologists running their heads against this wall.
P.S.: My attempt at a resolution agreeing with Quine's logic can be found here.
- Kotiaho JS, LeBas NR, Puurtinen M, Tomkins JL (2008) On the resolution of the lek paradox. Trends Ecol. Evol. 23:1-3.
- Kripke S (1975) Outline of a theory of truth. J Philosophy 72:690-716.
- Maynard Smith J (1978) The evolution of sex. Cambridge University Press.
- Priest G (1979) The logic of paradox. J Philosophical Logic 8:219-241.
- Quine WV (1979) The ways of paradox. In: The Ways of Paradox and other Essays. Harvard University Press, pp 1-21 [first published in 1962 in Scientific American 206:84–96].
- Sax DF and Brown JH (2000) The paradox of invasion. Global Ecology & Biogeography 9:363-371.
- Tarski A (1956) The concept of truth in formalized languages. In: Woodger JH (translator) Logic, semantics, mathematics. Oxford University Press, pp 152-278.
- Williams GC (1975) Sex and evolution. Princeton University Press.