Further Logical Proof!
World wide goatist research continues apace. John Berry in the USA has created the following marvellous proof:
Inductive proof: Everything is a goat.
1) Base case: Consider a set of elements consisting of one goat. That goat is a goat, so all elements in this set are goats.
2) Inductive hypothesis: If any set of k elements are all goats, then any set of k+1 elements are all goats.
Proof of inductive hypothesis: Let S be any set of k+1 goats. Number them 1 to k+1, by any selection method you like.
Now remove goat #1. The remaining set of goats 2 to k+1 is a set of k goats. So by assumption, they are all goats.
Now put back goat #1 and remove goat #k+1. Again, the remaining set of k goats (#1 to #k) are goats.
Now since there was an overlap between the two sets (for example, goat #2 was in both sets), and in each set all the goats are goats, then all the goats in the larger set of k+1 must also be goats.
We have proven by induction that any set of things, of whatever size, consists of goats. So everything is a goat.
Further great work from the United States in this great proof from James Dreier:
Consider the sentence (G):
(G) If G is true, then everything is a goat.
We can demonstrate that (G) is in fact true, by Conditional Proof:
1 SUPPOSE that G is true.
2 Therefore, "If G is true, then everything is a goat" is true. (Substitution)
3 Therefore, if G is true, then everything is a goat. (Disquotation)
4 Therefore, everything is a goat. (From 1 and 3)
On the supposition that G is true, we have proved that everything is a goat. So we have proved that *if* G is true, then everything is a goat, QED.
This completes the proof of the important lemma, that if G is true, then everything is a goat. We now proceed to the desired conclusion.
5 If G is true, then everything is a goat. (proved above by conditional proof)
6 "If G is true, then everything is a goat" is true (by Quotation)
7 G is true. (substitution)
8 Everything is a goat. (From 5 and 7)
Some will find the conclusion obvious, but it is nice to have an absolute proof.
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