Common Knowledge > Proof of Proposition 2.11 (Stanford Encyclopedia of Philosophy/Spring 2024 Edition)

Processing math: 100%

Supplement to Common Knowledge

Proof of Proposition 2.11

Proposition 2.11.
If $A$ holds, and if $A$ is a common reflexive indicator in the population $P$ that $x$ , then there is common reason to believe in $P$ that $x$ .

Proof. (Cubitt and Sugden 2003)

1.	$\mathbf{R}_i A$	(from RCI and the assumption that $A$ holds)
2.	$A \indi \mathbf{R}_j A$	(RCI2)
3.	$A \indi x$	(RCI3)
4.	$\mathbf{R}_i x$	(from 1 and 3, using CS1)
5.	$\mathbf{R}_i (A \indj x)$	(from 3, using RCI4)
6.	$A \indi \mathbf{R}_j x$	(from 2 and 5, using CS5)
7.	$\mathbf{R}_i \mathbf{R}_j x$	(from 1 and 6, using CS1)
8.	$\mathbf{R}_i(A \indj \mathbf{R}_k x)$	(from 6, using RCI4)
9.	$A \indi \mathbf{R}_j (\mathbf{R}_k x)$	(from 2 and 8, using CS5)
10.	$\mathbf{R}_i (\mathbf{R}_j (\mathbf{R}_k x))$	(from 1 and 9, using A1)
11.	$\mathbf{R}_i (A \indj \mathbf{R}_k (\mathbf{R}_l x))$	(from 9, using RCI4)

And so on, for all $i, j, k, l$ etc. in $P$ . Lines $4, 7, 10, 3n+1 (n \gt 3)$ establish the theorem.

Return to Common Knowledge

This is a file in the archives of the Stanford Encyclopedia of Philosophy.
Please note that some links may no longer be functional.

Supplement to Common Knowledge

Proof of Proposition 2.11

Browse

About

Support SEP

Mirror Sites