Is function a Deontic Modal Word? Michael Beebe Draft of June 2017 First Draft January 2017, available at Academia.edu 1.0 Introduction In this paper I develop a theory of function and function as a deontic modal word and phenomenon. Kratzer’s account of the semantics for the deontic modals is invoked and using her approach a formal schema for the semantics of function-sentences is proposed. My account of function is a modalized and extended version of Cummins’ systems-type account of function. In the biological and physical sciences, on this account, function is a complex empirical deontic modal property. It is built on the property of X’s doing Y well enough to enable Z, which is implicitly deontic because of the evaluative but nonetheless empirical in its biological and physical applications. This account of function resolves many of the traditional puzzles about biological and physical function, and extends naturally to include the other types of function. With a variant treatment of the semantics, this account is argued also to apply to mathematical function, where it is shown to be interestingly related to Frege’s account of function. 2.0 Function and function 2.1 The peculiarities of function and function To see some of the puzzles about function, consider the function-sentences in 1a) – g) below, all of which are commonly used in ordinary speech and are consistent with our naive understanding of function-talk. They can’t all be true, we understand that, so the endeavor is to find the account which preserves as much truth and good sense as possible, and also provides the best fit with other, related, linguistic phenomena. 1a) The function of the heart to pump the blood b) The heart is for pumping the blood c) The purpose of the heart is to pump the blood d) The heart is supposed to pump the blood
g) the function of a mordant is to fix the dye h) the function in steel-making of trace metals, e.g., molybdenum, is to make the metal stronger (tougher, less brittle, stainless, etc.) Next, and more unusually but I think relevantly, these last two instances. The first is in physics where the system invoked is fundamental physics, a yes/no system of laws of physical necessity rather than a system of more/less, better/worse relations of the sort found in complex and derivative causal systems such as the biological. The second is the system of roles of mathematical objects in mathematical systems, which is also a yes/no system because mathematical roles and relations are a matter of mathematical necessity. i) mass in Newtonian physics functions to generate the gravitational field And then in mathematics, consider the function of squaring. Think first about my calculator. One of its functions is squaring: j) my calculator functions to square numbers Then think about the abstract mathematical object the function squaring: k) the function squaring functions to square numbers This sounds clumsy, but parse it carefully and you will see that it makes sense: the function squaring is the thing, and it has the function of squaring, and it functions to square numbers. I will devote considerable effort to getting all this laid out in clear and distinct ways. But back to the gravitational force function l) F = g(m1)( m2)/r2 If we look just at the function with its variables removed we get: m) f( ) = g(…)(…)/(…)2 of which we can say f( ) is an inverse square function and inverse square functions are important in field theories. I will return to this question at the very end of Section 2.5 where I attempt to articulate the conceptual structures of functions both empirical and mathematical. I will also argue that Frege’s conception of function (Frege 1952, 107-116; “What is a Function?”) fits my account of function. His conception of function is as a type of abstract object that has the capacity to “map”, that is, to take one object as input and yield another object as output. Input-output talk is metaphorical, but it can be made literal and I will show how. 2.3 The problematic features ‘for’ and ‘supposed to’
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