From nobody@digitalkingdom.org Fri Jul 14 09:07:36 2006 Received: with ECARTIS (v1.0.0; list lojban-list); Fri, 14 Jul 2006 09:07:37 -0700 (PDT) Received: from nobody by chain.digitalkingdom.org with local (Exim 4.62) (envelope-from ) id 1G1QCD-0000by-Sg for lojban-list-real@lojban.org; Fri, 14 Jul 2006 09:07:17 -0700 Received: from web81310.mail.mud.yahoo.com ([68.142.199.126]) by chain.digitalkingdom.org with smtp (Exim 4.62) (envelope-from ) id 1G1QCC-0000br-2G for lojban-list@lojban.org; Fri, 14 Jul 2006 09:07:17 -0700 Received: (qmail 51033 invoked by uid 60001); 14 Jul 2006 16:07:14 -0000 DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=s1024; d=sbcglobal.net; h=Message-ID:Received:Date:From:Subject:To:In-Reply-To:MIME-Version:Content-Type:Content-Transfer-Encoding; b=gS38DwswtUqn+QFUjOixgxkD3QadoSs+ezQNdA+wHtavDGUY5i8c8JEiy1ZXB6msLOKJxgYgVF4psgJZg6mZSb9eaR5O8dluQuFJoI3BxtpXrG+XkVP1mymcbpgEqeN2Xtf9dBNSSLrmDe+YQFbUKkvay3WBxwEdXbhQiTra+hc= ; Message-ID: <20060714160714.51031.qmail@web81310.mail.mud.yahoo.com> Received: from [70.230.169.179] by web81310.mail.mud.yahoo.com via HTTP; Fri, 14 Jul 2006 09:07:14 PDT Date: Fri, 14 Jul 2006 09:07:14 -0700 (PDT) From: John E Clifford Subject: [lojban] Re: A (rather long) discussion of {all} To: lojban-list@lojban.org In-Reply-To: <20060713211125.79798.qmail@web81315.mail.mud.yahoo.com> MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Spam-Score: -0.6 (/) X-archive-position: 12197 X-ecartis-version: Ecartis v1.0.0 Sender: lojban-list-bounce@lojban.org Errors-to: lojban-list-bounce@lojban.org X-original-sender: clifford-j@sbcglobal.net Precedence: bulk Reply-to: lojban-list@lojban.org X-list: lojban-list First round of corrections, thanks to xorxes. --- John E Clifford wrote: > I send this along for corrections and questions before using it (in its revised form) to answer > Maxim's questions. > > Singular v. Plural Semantics > > Language: > > Variables: > Names: > Predicates: Relation: Y > Sentential connectives: ~, & (others by usual definitions) > Quantifiers: E > Descriptor: t > > Terms: a variable is a term, a name is a term, if F is a formula containing free variable > x, then txF is a term. > Formula: A predicate followed by a term is a formula, A followed by two terms is a > formula, a formula preceded by ~ is a formula, two formulas preceded by & is a > formula, a formula preceded by a variable preceded by E is a formula > A formula contains a free variable x just in case there is an occurrence of x in that formula > which is not in any subformula which begins Ex nor in a term which begins tx > > A sentence is a formula which contains no free variables. > > A singularist model: > > Domain D: a non-empty set > Masses M: Power D – 0. the set of all non-empty subsets of D > Concepts: > > Interpretation: a function, I that assigns to: Each concept an object from M, with at least one concept for each singleton in M > Each name a concept > Each predicate a function from concepts into {0, 1} > I(Y) is the function from pairs of concepts such that I(A)(c1,c2) = 1 iff I(c1) is included in > I(c2) > > A is an assignment iff A is a function from variables to concepts > A(c/x) is an assignment just like A except that it assigns the concept c to variable x instead > of A(x). > If a is a term, R(a) = I(a) if a is a name, R(a) = A(a) if a is a variable, R(a) is a concept c such that F is true for I and A(c/x), if a = txF > > i is an individual just in case i is in M and is a subset of each of its subsets (is identical > with each of its subsets, has only one member, i is a singleton). > > Where P is a predicate and a a term, Pa is d-true for I and A iff for every individual i > included in I(R(a)) and for every concept c s.t. I(c) = i, I(P)(c) = 1 > > Where P is a predicate and a a term, Pa is c-true for I and A iff I(P)(R(a)) = 1 > > A Pluralist model > > Domain: Some things > Concepts > C is a relation between concepts and items in D, such that for every d in D, there is at least once c such that c is related by C only to d [We designate a selected such concept C/d, for each d] > > > An interpretation I is a function which assigns > To each name a concept > To each predicate a function from concepts into {0,1} To Y the function from pairs of concepts into {0,1} such that I(A)(c1,c2)) = 1 iff for every thing d such that c1Cd holds, c2Cd holds > > An assignment A is a function from variable to concepts > A(c/x) is an assignment just like A except for assigning c to x in place of A(x). > For term a, R(a) = I(a) if a is a name, R(a) = A(a) if a is a variable, is a concept c such that F is true for C,I and A(c/x) if a = txF > Pa is d-true for C,I and A iff for every d such that R(a)Cd, I(P)(C/d) = 1 Pa is c-true for C,I and A iff I(P)(R(a)) = 1 > > In either case, > > A formula F is true for [C,]I and A > > If it is Pa, for some predicate P and some term a and either Pa is d-true for [C,]I and A or Pa > > is c-true for [C,]I and A > > If it is Yab and I(Y)(R(a) R(b)) =1 > > If it is ~S for some formula S and S is not true for [C,]I and A > > It is &GH for some formulae G and H and both G and H are true for [C,] I and A > > It is ExG for some variable x and some formula G and for some concept c, G is true for [C,] I and > A(c/x) > > Otherwise not. > > > > > > > To unsubscribe from this list, send mail to lojban-list-request@lojban.org > with the subject unsubscribe, or go to http://www.lojban.org/lsg2/, or if > you're really stuck, send mail to secretary@lojban.org for help. > > To unsubscribe from this list, send mail to lojban-list-request@lojban.org with the subject unsubscribe, or go to http://www.lojban.org/lsg2/, or if you're really stuck, send mail to secretary@lojban.org for help.