From @uga.cc.uga.edu:lojban@cuvmb.bitnet Tue Jun 20 23:40:59 1995 Received: from punt2.demon.co.uk by stryx.demon.co.uk with SMTP id AA3486 ; Tue, 20 Jun 95 23:40:56 BST Received: from punt2.demon.co.uk via puntmail for ia@stryx.demon.co.uk; Tue, 20 Jun 95 16:50:32 GMT Received: from uga.cc.uga.edu by punt2.demon.co.uk id aa22952; 20 Jun 95 17:50 +0100 Received: from UGA.CC.UGA.EDU by uga.cc.uga.edu (IBM VM SMTP V2R2) with BSMTP id 6601; Tue, 20 Jun 95 11:44:39 EDT Received: from UGA.CC.UGA.EDU (NJE origin LISTSERV@UGA) by UGA.CC.UGA.EDU (LMail V1.2a/1.8a) with BSMTP id 7995; Tue, 20 Jun 1995 11:40:49 -0400 Date: Tue, 20 Jun 1995 08:40:28 -0700 Reply-To: "John E. Clifford" Sender: Lojban list From: "John E. Clifford" Subject: pc answers X-To: lojban list To: Iain Alexander Message-ID: <9506201750.aa22952@punt2.demon.co.uk> Status: R Djer's additions about what is covered by the various quantifier prefixes raises an interesting point but has it slightly off. In the first place, z1 is not defined to *be* the set {xyz} but to *range over* that set, i.e., to take members of that set as values in genrating true sentences. And the claim in 2 is just that for at least one (but maybe two or all three) members, the replacement does give a true sentence. For example, it may be that it is true of man x that he touches all three dogs, w,v, and u. And, if not of x, then maybe of y and, if not of y either, then surely of z. Notice that, "for some" ("at least one") includes the possibility of "all," so that djer's list of possible all-dog-touchers actually has seven members. But the members of this list are not the substituends for the quantifier, which takes only individuals as replacements. The subsets come in only in the sense that, if all the members of any one of those seven subsets gives a true sentence, then the original sentence is true (another way to put this is that the original is the disjunction of the corresponding claims about the three members of the set). The universal quantifier, w1 in the case of 2, means that all the instances from the set must give trues for the original to be true (the conjunction of the instances). Expanding 2 out then gives first (w1)xTw1 or (w1)yTw1 or (w1)zTw1 (the disjunction of the z1 instances) and then (xTw and xTv and xTu) or (yTw and yTv and yTu) or (zTw and zTv and zTu) (expanding all of the universal w1's out) If we go back to basics, the nine simple sentences listed above, each claiming one man touched one dog, we can find 512 = 2^9 possible situations. Of these, exactly one, the first in the usual ordering (as for truth tables) makes sentence 1 (all-all) true. For sentence 2 (and 3) (some-all), 169 situations makes each true (with some uncalcuated overlap of cases -- I've forgotten the rule calculating those). (Actually, I calculated the number of ways that this could fail to happen, e.g., where each man touched two or fewer dogs. Since each man can do this in seven different ways -- every way except touching all three -- and they can do these independiently of how the other man act, there are 7^3 = 343 ways. The remaining cases, 169, must be the ones where the sentence holds.) By the same kind of reasoning (seven ways to fulfill for each man -- all but the case where he touches no dog -- independently) 343 cases meet each of 4 and 5. Again, the two sets of cases overlap, significantly more than the 175 cases mathematically required. Finally, sentence 6 is verified in 511 cases, all but the last one in the standard order. The case that satisfies 1 is included in both the 169 sets, the 169 set for 2 is included in the 343 for 5 and similarly for 3 and 4. Indeed, the overlaps of 2 and 3 are included in the overlaps of 4 and 5. And so on. pc>|83