Frege (§9) Begriffsschrift, Chapter 1 (Geach transl.), “The Function", summary

by Corry Shores
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[The following is summary. Bracketed commentary is my own. Please forgive my typos, as proofreading is incomplete.]

Summary of

Gottlob Frege

Begriffsschrift, Chapter 1
(Geach transl.)

§9 The Function

Brief Summary: 
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.

Summary

[For Frege’s first idea, consider if we have this inequality: 4<6. Rather than 4, we could have 2 or 1. So we can instead use a variable to mean that a variety of values can be put there: x<6. In this case, we might consider “being less than 6” as the function, and the values less than 6, (like 2, 1, –5, etc.) as the arguments. (Note: in light of later considerations, it might be inappropriate here to think of the argument as a variable. See the discussion later on indeterminateness. For now, we merely think of the arguments being the specified variable terms. This note holds for the following comments as well).]

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’.
(12)

[So consider again our instances of 4<6 and 2<6, which we expressed more generally as x<6. Here the function is “<6” and the arguments are 2 and 4. But consider instead that we have the first one, 4<6, and with it as well 4<8 and 4<12; and also, we separately have along with 2<6 as well 2<3 and 2<4. In the first case, we could have 4<x rather than x<6, and in the second case we could have 2<x, rather than x<6.]

‘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.
(13)

[Consider the formulation: 5≤5. And consider three possible sets it can belong to:

1)  5≤5, 4≤5, –1≤5. (here: x≤5)

2) 5≤5, 5≤8, 5≤90. (here: 5≤x)

3) 5≤5, 4≤12, -1≤10. (here: x≤y)

In the first case, the function is “being less than or equal to 5”; in the second case, “5 is less than or equal to...”; and in the third, “... is less than or equal to ...”. As it seems in these cases at least, the argument is the variable term, and the function is the constant term plus its relation to the variable term. (Were there just one term, the argument, then the function I suppose is merely the relation to that variable term). In the following, I am still not sure about the importance of one term being repeated twice in the same sentence. For, whether a term happens once or twice, in both cases the variable and function need to be determined by knowing what can vary and what cannot.]

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.
(13)

[Frege then gives a formal definition for a function and an argument. Again, the part that can vary is the argument, and the part that cannot 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)

[I am not sure I understand the next part. He says that an argument can be found in a function. But if the function is what is invariable, then how can something variable be a part of it? Perhaps the idea is that you can have two-place predicates that when articulated in a sentence, the second argument is found syntactically in the sentence predicate and thus seems like it is in the function, but were it written symbolically, it would be shown to not be within the function. For example, x is taller than y. Let me quote:]

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.
(13)

[I am very uncertain what the idea is in the next paragraph. I will quote it first and discuss it.]

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.
(13-14)

[My best guess is that he is saying the following. The “arguments” from each sentence are not comparable, because the first is a constant term and the second is a variable. I am really not sure, however.]

[The next idea I cannot discern. Again let me quote first.]

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.
(14)

[My best guess is that, suppose you have “20 can be represented as the sum of four squares”. Here, since we have already determined that 20 is an argument for “can be represented as the sum of four squares”, then we do not need to call into question its relation to the function. We have already affirmed that it properly fulfills that function. However, if we have “x can be represented as the sum of four squares” (where x is a positive integer), then in order to know what x can be, we need to think more about the meaning of the functional expression. In this case, the function is determinate and the argument is indeterminate. However, suppose we have “20 ‘fulfills some arbitrary condition’”. Here, the argument is determinate but the function is indeterminate. In order to know what kinds of functions could belong in this sentence, we need to think about the conceptual content of 20. I am not sure if that is right, because I seem to be confusing replaceability with indeterminateness. But I am not sure what else it means.]

[Functions can have more than one argument. We illustrated this with with the x≤y examples, I think.]

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.’
(14)

[The next idea I think is the following. The function relation is similar to the subject-predication relation, with the argument being the subject and the function being the predication. For this reason, grammatically speaking, when we express a functional relation in a sentence of everyday language, we often put the argument in the subject position and the function in the predicate position. However, we can place the argument in the predicate position of a sentence if we use the passive voice or if we use an inverted formulation for the function. So consider “x gave something to Jane”. Using the passive voice, we can make that, “Jane was given something by x.” Or we can say, “Jane received something from x.” And “x is less than 10” can become “10 is greater than x”.]

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. between

active and passive,
‘heavier’ and ‘lighter,’
‘give’ and ‘receive’;

but this liberty is restricted by lack of words.
(14-15)

Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

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