For the logical positivist, there are only two types of statements that are literally significant. A statement is factually significant if it is either true solely in virtue of the meaning of its constituent symbols (if it is analytic) or if it is found true upon empirical verification. In order for mathematics and logic to have factual significance, the positivist must show that it exhibits one of these two qualities. The second of the two options is completely undesirable, because it lacks all consistency. As Hume stated, “The fact that a law has been substantiated in n-1 cases affords no logical guarantee that it will be substantiated in the nth case also, no matter how large we take n to be” (Miller 93). So, if the logical positivists took this stance, they would have to admit that some day two plus two could equal five. The fact that it has equaled four thus far only makes four at best the most probable answer. Therefore, the logical positivist must argue that mathematics and logic are analytic or say that they are not literally significant and are senseless. The latter of the two would have disastrous effects upon the world, so the argument must be made for the analytic.
In attempting to prove that mathematics and logic are analytic, Ayer decides to tackle the two central and problematic features of the two: necessity and utility. In the case of necessity, the problem is reconciling the idea that our experience cannot prove logical or mathematical claims wrong with the belief that we possess knowledge of mathematical and logical truths. Ayer finds the answer to this problem linguistically. For Ayer, mathematical truths simply “record our determination to use words in a certain fashion” (Miller 95). Thus, “we cannot deny them without infringing the conventions which are presupposed by our very denial” and “Our knowledge that no observation can ever confute the proposition ‘7+5=12’ depends simple on the fact that the symbolic expression ‘7+5’ is synonymous with ‘12’” (Miller 95). For Ayer this is the same as saying that the symbol bachelor is synonymous with the symbol unmarried man. This also holds for all other a priori truths. Thus, the truths of mathematics and logic are necessary, because they “arise from conventional connections between the symbolic expressions of our language.” The way in which language is used produces logic.
Because logic comes from the connections between expressions in our language and not from the language itself, it can not be said to say anything about objects, or rather, “The certainty and universal validity, or better, the irrefutability of a proposition of logic derives just from the fact that it says nothing about objects of any kind” (Miller 95). This is where the next problematic feature of mathematics and logic arises, namely utility. If mathematics and logic say nothing about objects, then how can it be said that there are discoveries in the field, and how can these discoveries be useful at all? The answer to this lies in the finitude of man. According to Ayer, “A being whose intellect was infinitely powerful would take no interest in logic and mathematics. For he would be able to see at a glance everything that his definitions implied, and, accordingly, could never learn anything from logical inference which he was not fully conscious of already” (Miller 96). Because man is not all knowing, he does not see immediately and fully the logical truth asserted by his statements. To express this logic, it must be deduced from his statements. Miller provides an example to show this. He knows two things: Jones is not wearing a rose in this buttonhole, and either Jones is wearing a rose through his buttonhole or is wearing a carnation in his buttonhole. Even if Miller does not understand what this implies, he can figure it out by using the logical knowledge of the truth of (¬P ^ (P v Q))→Q. In doing this he has utilized his knowledge of logical truth and extended his knowledge via discovery. There is also a sense in which this does not add to his knowledge. If Miller knows the facts above about Jones, he knows implicitly already that Jones has a carnation in his buttonhole. However, when logic is applied, the fact becomes explicit. In this way logic does not add to knowledge but converts it from implicit to explicit. This does not take away at all from the utility of mathematics and logic. At this point logical positivism has made a claim for the a priori status of mathematics and logic by accounting for their necessity and utility without resorting to metaphysics, but Miller seeks to show its implausibility with an argument from Quine.
Quine’s argument is against the use of general linguistic conventions as the determination of the a priori status of mathematics and logic. The way the argument is formulated is by using a logical truth, such as If (Bob is a mammal and (If Bob is a mammal then Bob has hair)) then Bob has hair. Because this is a logical truth, it is true in virtue of linguistic convention, but Quine is left with the question of from which convention it proceeds. If a specific convention is made for this specific case, then a specific convention must be made for each individual logical truth that exists. Since, this number is infinite, there would be an infinite number of conventions. This is not at all desirable in the least, so the convention must not be general but specific. Miller offers the convention If (P and (If P then Q)) then Q. Therefore, if the first statement results from the second due to uniform substitution of P and Q, then it is held true no matter what. With this and the fact that the first statement results from the second due to uniform substitution of P and Q, then it is possible to derive that what is claimed to be the convention governing the first statement, which is that it should be held true no matter what. This is a problem in the view of Quine, because it sets up an infinite series of logic and convention due to the fact that logic is relied to show the general convention. As Quine himself put it,
In the adoption of the very conventions whereby logic itself is set up, however, a general difficulty remains to be faced. Each of these conventions is general, announcing the truth of every one of an infinity of statements conforming to a certain description; derivation of the truth of any specific statement from the general convention this requires a logical inference, and this involves us in an infinite regress. (Miller 98-99)The attempt to derive logic from a general convention leads to the use of a logical presupposition in order to derive the logic that is supposed to proceed a priori from the convention. Because of this argument from Quine, it appears that the a priori status of mathematics and logic as the logical positivist would have it is impossible.
It is not clear that Quine’s argument does anything to hurt the logical positivist position on the a priori. There are a few different questions that need to be answered about the argument. Is the infinite regression shown in the example a really problem for the logical positivist argument? Generally any infinite regression is treated as a bad thing (such as the problem of infinite regression in the Network and Background). However, when seeking to discover the general conventions that govern language, the discovery has to be made within the realm of language. Whatever the discovery is found to be, it will also be a piece of language. That piece of language will also have a general convention from which logic comes governing it. It is impossible for Quine to not presuppose logic in order to prove that logic comes from the conventions, simply because in order to construct any type of argument or course of discovery, language must be used. This language that is used has underlying conventions and thus, according logical positivism, logic a priori. Also, is it necessary to have an infinite regression at all? It seems that it is possible to stop once it is found that the original statement is true no matter what. Any attempt to use the information from the proof after the convention has been shown is not a regression but the start of a new proof. The fact that this new proof produces the same convention by utilizing the same structure as the first proof is not a troublesome issue but a benefit. It seems to simply illustrate the way that the language of the first proof produces logic just as the positivist claims. Miller’s account of Quine’s argument has not produced a problematic infinite regression, but has simply illustrated the common linguistic convention to several different proofs (having the same structure), and has shown the manner in which the language necessary to his proofs produces logic.
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