“Not so fast!” One might say. Biological Conventions do not entail Logical Conventionalism. In fact, we do not even have a story on how possibly Biological Conventions (i.e., biological rules for the assignment of meanings to logical idioms) can make Logical Conventionalism (i.e. a thesis about the truth-makers of logical sentences) true.
I acknowledge this difficulty. I think the only way Biological Conventions can make Logical Conventionalism true is if (1) they are understood in a teleosemantic way, and (2) the formal system they implement is Intuitionistic Logic.
A word of caution is in place. For, to my knowledge, neither a teleosemantic account of logic has been fully developed, nor has Intuitionistic Logic regarding non-mathematical…show more content… On Millikan’s account, logical axioms, inference rules and typing rules are sometimes equivalent, and can be substituted for one another. For instance, identity between a and b (a=b) is the typing rule that these should be considered as token symbols of the same type. Laws of commutativity are given by grouping symbol tokens into types. For another example, “we read “p” as a symbol of the same type as “b” except that it has been turned upside down. Then we use turning upside down for the negation transformation. We negate propositional constants and variables by turning them upside down; we negate strings by turning the whole string upside down. Double nega- tion elimination now no longer appears as an axiom, theorem, or rule. It can’t be stated, or can’t be differentiated from p implies p” (Millikan 2000,…show more content… Up to now, I addressed the first point; now I should address the second. How could Logical Conventionalism be true, if our biological conventions implemented the formal system of Intuitionistic Logic?
According to Intuitionism, knowing that statement S is true means having a proof of it, i.e., having derived it from some other formulas according to some rule of inference. Intuitionistic logic differs from classical logic in that the traditional concept of “truth” has been replaced by that of constructive probability. The existence of an object is demonstrated by providing a method for creating an object. For Intuitionism, there is no fact of the matter on whether
“there is a y greater than x such that both y and y+2 are prime numbers” (Moschovakis 2015)
is true or false until we have some direct evidence for the case. As Logical Conventionalism claims, there is no fact of the matter on whether logical sentence S is true before it was proved to be so. Intuitionists reject the principle of excluded middle (P or ~P), and they advanced by a quarter of a century Godel’s incompleteness