‘The only truths which may be known a priori are trivial and conventional truths of meaning.’
The distinction between a priori and a posteriori truths is a distinction concerning how truths are known, as opposed to a distinction concerning what truths are. Such a distinction arises from our two different ways of acquiring knowledge. On one hand, we acquire knowledge through our natural faculties of sense-experience - by sights, smell, taste, sound, touch and so on. The truths arrived are by such a mechanism are called a posteriori or empirical. On the other hand, we also seem to be able to arrive at truths through mere reflection - by applying our ‘faculty of reason’, without any recourse to experience.* Such truths are called a priori, which is to say they are arrived at in an a priori way. The investigation called for here is into the nature of truth revealed through the latter channel; specifically, the charge is that a priori truths are trivial and true by virtue of definition or meaning.
Let us look at the kind of truths encompassed by the landscape of a priori truths. All mathematical knowledge and laws of logic, for example, are said to be true independent of sense-experience. We do not need our senses to check whether 7+5 =12, whether the angles of a triangle sum to 180 degrees, and whether the law of excluded middle is true. If indeed, someone claims that the angles of a triangle to not sum to 180 degrees, then at least on of the terms is being used in a different sense that it normally is - maybe the triangle is non-euclidean or angles defined and measured in a different way. The statement ‘Either I am going to submit the essay or not going to submit’ holds true without any reliance on what happens in the natural world. One might argue that the statement might not be true. For example, one might say that if I submit half the essay, then I would have both submitted the essay and not submitted the essay. Any such attempt would, however, be inevitably be based on vagueness of the terms in the statement. But as long as the terms are well defined, no matter what the definition is, the statement holds true.
The above attacks on the validity of a priori truths suggest that definitions and meaning of terms (or symbols, as in the case of 5+7 = 12 example) play a crucial role. If all we need to do to defend priori propositions is play around with definitions, then is their truth the result of a mere language game?
Let us look at this argument in detail. It follows the following line of reasoning, based on the analytic-synthetic distinction:
Any self-respecting mathematician would recoil at the conclusion of that argument, and I sympathise with them. For mathematical knowledge is anything but trivial.
In putting the role of definitions as the arbiter of meaning, the argument puts the cart before the horse. Meaning exists prior to definition, and the same definiendum (object of a definition) can mean different things in different contexts*. For example, while the number 9 and the number of planets in the solar system point to the same abstract entity, they mean different things. If I were to employ the argument in this context, it would posit an equivalence between the proposition like “The number of planets in the solar system is 9” and the proposition “9 is 9”. While the former is an empirical proposition whose truth is established by experience, the latter is a tautological relation whose truth is known a priori. To say, that 7+5 is the same entity as 12 or 7+5 = 12 is thus in a similar category, where the meaning of the left hand side is different from the meaning of the right hand side, although they point towards the same abstract entity as their referent.
Statements like “All bachelors are unmarried” and “7+5 =12”, thus, are more like the statements “The number of planets in the solar system is 9” or “Morning Star = Evening Star” than the statements “12=12” and “All men who are not married are unmarried”. The analytic-synthetic distinction, as outlined in (1) above, is not clear-cut, and the premise of the argument stands on shaky grounds. This criticism has been fully developed by Quine, who further argued that analyticity hinges on a notion of synonymity (interchangeability salva variate) which itself depends on analyticity for conceptual support.
Second, there is also a problem with (3) and (5). While a priory statements such as mathematical theorems might be tautologies in an absolute sense, it does not follow that they are trivial. It might be trivial to a being with infinite vision and potency, but to such a being, what is non-trivial? Is there any truth which is non-trivial from an objective point of view? What is so special about empirical truths that we accord them non-triviality? That the venus, the evening star and the morning star are one and the same thing is an empirical truth, but then the reason this is non trivial is only because we are at a certain standpoint in space time where it is not immediately apparent to us . To an omniscient being this truth would be trivial. As another example, that the earth follows the gravitational equation might be an empirical truth, but this truth might be trivial to an omniscient being. Indeed, an omniscient being might not even see an categorization or distinction between earth and vacuum but see the whole of reality as one inseparable whole. The categorization of earth and planets and vacuum hold meaning for us because we have a certain view of reality. Similarly, different mathematical propositions hold different meaning for us because our minds have certain cognitive capacities of interpreation. When a relation between two such propositions, existing by virtue of their very definitions or language, is revealed through a priori reasoning, then the truth we get to know is as trivial as the truth of being revealed that morning star = evening star = venus.
Third, What the argument also ignores is the unreasonable effectiveness of mathematics in the real world. While the earth follow a trajectory governed by a mathematical equation is an empirical fact which might not hold tomorrow, it is still true that given that it follows the trajectory, we can predict with great accuracy a great deal about everything - our seasonal variations, dates of the solar eclipse. One might say that the statement “Earth follows the gravitational equation” is equivalent to saying “A solar eclipse will happen on this day and date” or “Country X will be have length of day given by Y”, but these revelations about relations between propositions, supported by a priori truths are not trivial. That mathematical knowledge can be reduced to knowledge about certain relations between meanings (in this case the relation between “Earth follows the gravitational equation” and statements of the type “Earth is going to do this at this time of year”) doesn’t imply that it doesn’t have any real world import. Indeed, the unreasonable effectiveness of mathematics in the real world strongly suggests that mathematical truths or propositions capture some extra-linguistic truth about the real world.
Consider the example of the Monty Hall problem: Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
The solution to the puzzle, arrived at mathematically (a priori reasoning), is that switching doors doubles your advantage. Under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their choice have only a 1/3 chance. The problem is a veridical paradox, because the correct result (you should switch doors) is so counterintuitive it can seem absurd, although it is demonstrably true. If someone doesn’t consider the fact that switching doors is beneficial as a matter of fact, as a law, then it could cost him his life (or money). If mathematical truths are trivial tautologies, mere relations of symbols, then there needs to be an account for why these relations of symbols have such real world import. And do other empirical truths, which presumably represent non-trivial knowledge, hold the same power? Ayer, in his argument that a priori knowledge, in virtue of their nature as tautologies don’t present any new knowledge says, ‘‘It is easy to see that the danger of error in logical reasoning can be minimized by the introduction of symbolic devices, which enable us to express highly complex tautologies in a conveniently simple form, And this gives us an opportunity for the exercise of invention in the pursuit of logical inquiries. For a well-chosen definition will call our attention to analytic truths, which would otherwise have escaped us. And the framing of definitions which are useful and fruitful may well be regarded as a creative act’’ The question is, if analytical truths are trivial and mere symbolic, with no bearing on the real world, why is their a danger (of error in logical reasoning) in the first place?
Fourth, while a priori truths might follow from meaning constrained by laws of logic , the validity of the laws of logic themselves has not been accounted for. In other words, while the enterprise of mathematics might be akin to following a rule-based language game, and hence mathematical truths be the conventional truths of meaning, it is still not clear how the rules themselves can be conventional truths of meaning or follow from definitions. To illustrate,
I might hold a rule that :
Whenever If P then Q is true and P is true, then Q is true. (Rule A)
And on the basis of the above rule, observer that If P then Q is true and P is true, and deduce that Q is true. While this might account for how Q is true, the validity of Rule A is still not acccounted for.
If I add another rule to say that:
Whenever Rule A is true and P then Q is true and P is true, then Q is true. (Rule B)
and from this arrive at the truth of Q, I still haven’t accounted for the validity of Rule B.
And so on.
This is an infinite regress problem: all truths of logic cannot be true relative to other truths; there have to be grounded on something. We can’t ground logical truths in definitions, because all definitions do is give the meaning of one term in terms of another term. “A definition, strictly, is a convention of notational abbreviation. . . . Functionally a definition is not a premise to a theory, but a license for rewriting theory by putting definiens for definiendum or vice versa. By allowing such replacements definition transmits truth: it allows true statements to be translated into new statements which are true by the same token.”
Thus, to represent a priori truths as trivial and conventional rules of meaning is a bit of hyperbole. First, there is no clear cut analytic synthetic distinction, and even analytic truths can have empirical relevance. Second, the charge of triviality comes from the point of view of an omniscient being, not in relation to us. Third, the argument gives no account for t the unreasonable effective of mathematics in the natural world. Fourth, in assuming that logic too depends on convention, it runs into a regress problem.
05 February 2015