by Corry Shores
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[Central Entry Directory]
[Logic & Semantics, Entry Directory]
[Gottlob Frege, Entry Directory]
[Frege, Begriffsschrift, Chapter 1, Entry Directory]
[The following is summary. Bracketed commentary is my own. Please forgive my typos, as proofreading is incomplete.]
Consider a set of nearly identical expressions where one place in the sentence is different in each instance (in other words, it is as if there is a variable term while the rest of the sentence remains the same). (Especially in cases where a certain syntactical unit of the sentence, like the subject, is one of the variables) we can consider the variable part the argument and the invariable part the function. There can be more than one argument per function. For example, consider the sentences: “hydrogen is lighter than carbon dioxide”, “oxygen is lighter than carbon dioxide”, and “nitrogen is lighter than carbon dioxide”. In these cases, hydrogen, oxygen, and nitrogen are the arguments, and “is lighter than carbon dioxide” is the function. However, suppose we add to this list a fourth sentence, “helium is lighter than radon”. Now there would be two arguments per sentence (in the prior case, helium and radon), and the function would be just “...is lighter than...”. The argument or function can be more or less determinate. When it is indeterminate, it is like a variable.
Let us suppose that there is expressed in our formalized language the circumstance of hydrogen’s being lighter than carbon dioxide. In place of the symbol for hydrogen we may insert the symbol for oxygen or nitrogen. This changes the sense in such a way that ‘oxygen’ or ‘nitrogen’ enters into the relations that ‘hydrogen’ stood in before. If an expression is thought of as variable in this way, it is split up into a constant part representing the totality of these relations and a symbol, imagined as replaceable by others, that stands for the object related by the relations. I call the one part a function, the other an argument. This distinction has nothing to do with the conceptual content; it concerns only our way of looking at it. In the manner of treatment just indicated, ‘hydrogen’ was the argument and ‘being lighter than carbon dioxide’ the function; but we can equally look at the same conceptual content in such a way that ‘carbon dioxide’ is the argument and ‘being heavier than hydrogen’ is the function. p. 16] We need in this case merely to imagine ‘carbon dioxide’ as replaceable by other ideas like ‘hydrochloric acid gas’ or ‘ammonia’.
‘The circumstance of carbon dioxide's being heavier than | hydrogen’ and ‘The circumstance of carbon dioxide's being heavier than oxygen’ are the same function with different arguments if we treat ‘hydrogen’ and ‘oxygen’ as arguments; on the other hand, they are different functions of the same argument if we regard ‘carbon dioxide’ as the argument.
Let our example now be: ‘the circumstance that the centre of mass of the solar system has no acceleration provided that none but internal forces act on the solar system.’ Here ‘solar system’ occurs in two places. We may therefore regard this as a function of the argument ‘solar system’ in various ways, according as we imagine ‘solar system’ to be replaceable at its first occurrence or at its second or at both (in the last case, replaceable by the same thing both times). These three functions are all different. The proposition ‘Cato killed Cato’ shows the same thing. If we imagine ‘Cato’ as replaceable at its first occurrence, then ‘killing Cato’ is the function; if we imagine ‘Cato’ as replaceable at its second occurrence, then ‘being killed by Cato’ is the function; finally, if we imagine ‘Cato’ as replaceable at both occurrences, then ‘killing oneself’ is the function.
The matter may now be expressed generally as follows: Suppose that a simple or complex symbol occurs in one or more places in an expression (whose content need not be a possible content of judgment). If we imagine this symbol as replaceable by another (the same one each time) at one or more of its occurrences, then the part of the expression that shows itself invariant under such replacement is called the function; and the replaceable part, the argument of the function. (13, boldface mine, italics his)
By this definition, something may occur in the function both as an argument and also at positions where it is not regarded as replaceable; we must thus distinguish argument-positions in the function from other positions.
p. 17) I should like at this point to give a warning against a fallacy that ordinary language easily leads to. Comparing the two propositions‘the number 20 can be represented as the sum of four squares’and‘every positive integer can be represented as the sum of four squares,’ |it seems possible to regard ‘being representable as the sum of four squares’ as a function whose argument is ‘the number 20’ one time and ‘every positive integer’ the other time. We may see that this view is mistaken if we observe that ‘the number 20’ and ‘every positive integer’ are not concepts of the same rank. What is asserted of the number 20 cannot be asserted in the same sense of [the concept] ‘every positive integer’; of course it may in certain circumstances be assertible of every positive integer. The expression ‘every positive integer’ just by itself, unlike ‘the number 20,’ gives no complete idea; it gets a sense only through the context of the sentence.
We attach no importance to the various ways that the same conceptual content may be regarded as a function of this or that argument, so long as function and argument are completely determinate. But if the argument becomes indeterminate, as in the judgment: ‘whatever arbitrary positive integer you may take as argument for “being representable as the sum of four squares,” the proposition always remains true,’ then the distinction between function and argument becomes significant as regards the content. Conversely, the argument may be determinate and the function indeterminate. In both cases, in view of the contrast determinate-indeterminate or more and less determinate, the whole proposition splits up into function and argument as regards its own content, not just as regards our way of looking at it.
Suppose that a symbol occurring in a function has so far been imagined as not replaceable; if we now imagine it as replaceable at some or all of the positions where it occurs, this way of looking at it p. 18] gives us a function with a further argument besides the previous one. In this way we get functions of two or more arguments. E.g. ‘the circumstance of hydrogen’s being lighter than carbon dioxide’ may be regarded as a function of the arguments ‘hydrogen’ and ‘carbon dioxide.’
The speaker usually intends the subject to be taken as the principal argument; the next in importance often appears as the object. Language has the liberty of arbitrarily presenting one or | another part of the proposition as the principal argument by a choice between inflexions and words, e.g. betweenactive and passive,
‘heavier’ and ‘lighter,’
‘give’ and ‘receive’;but this liberty is restricted by lack of words.