# Seeking clarification of how an argument from Aristotle is found fallacious using Frege's quantification tools

G. E. M. Anscombe writes in An Introduction to Wittgenstein's Tractatus (page 15-16):

Again, the following fallacious piece of reasoning is found in Aristotle: 'All chains of means to ends must terminate in a final end. This final end will be the supreme good.' The first statement is reasonable; the second assumes that the first has shewn that there is some one end, the same for all chains of means to ends, in which they all terminate: the fallacy is immediately avoided by writing:

For all x, if x is a chain of means to ends, there is a y such that y is a final end and x terminates in y,

is a final end and x terminates in y,

which is quite different from:

There is a y such that y is a final end, and for all x, if x is a chain of means to ends, x terminates in y.

• Is there a reference to Aristotle justifying Anscombe's claim?
• Could someone symbolize further the two statements (which I put in italics) showing they are not the same? Part of my problem with symbolizing this is I do not know what the domains of "x" and "y" are. Specifying the domains would be important for the answer.
• Regarding your second question, you don't need to know the domain in order to symbolize statements in first-order logic. Could you clarify how specifying the domain would help? – Eliran Nov 23 '18 at 16:47
• @Eliran Anscombe appears to be claiming that the statements challenge supposed views of Aristotle as fallacies claiming that quantification techniques help to expose the fallacies. One quantifies over variables which range over members of domains. Those domains are not empty. To see if Anscombe is right I need to know explicitly what the domains are. – Frank Hubeny Nov 23 '18 at 19:04

The reasonable general principle is :

For all x (if x is a Chain-means-to-ends, then there is an y such that y is a Final-end and y Terminates x).

The debatable conclusion :

"This final end will be the supreme good",

implicitly assumes that there is exactly one Final-end common to all Chain-means-to-ends, and thus we can call it the Supreme-good.

A possible source for Anscombe's example (page 16) may be : Nic.Eth, Bk I, 1094a :

Every art and every investigation, and likewise every practical pursuit or undertaking, seems to aim at some good: hence it has been well said that the Good is That at which all things aim. [...] If therefore among the ends at which our actions aim there be one which we will for its own sake, while we will the others only for the sake of this, and if we do not choose everything for the sake of something else （which would obviously result in a process ad infinitum, so that all desire would be futile and vain）, it is clear that this one ultimate End must be the Good, and indeed the Supreme Good.

Maybe useful : The Importance of telos in Aristotle.