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<br><!--StartFragment--><p class="MsoNormal"><span style="font-family:Chalkboard">JL Speranza’s latest note, partly in response to an earlier note of mine, contains some observations on Quine’s and other logicians treatments of “and.”<span style="mso-spacerun: yes"> </span>I agree with many of JL’s observations, but I want to call attention to the fact that knowledgeable logicians are well aware of the fact that the truth-functional conjunction, often represented by an upside-down 'v'</span><span style="font-family:Chalkboard"> but sometimes by ‘&’, is a sentence or clause connective, not a true counterpart to the English ‘and.’ The latter, unlike the former, can properly connect expressions from many categories other than clauses.<span style="mso-spacerun: yes"> </span>It can connect noun phrases, as in ‘Tom’s dog and Mary’s cat,’ verb phrases, as in ‘slipping and falling,’ adjectival phrases, as in ‘powerful and threatening,’ and adverbial phrases, such as ‘cautiously and prudently.’<span style="mso-spacerun: yes"> </span>(I speak of phrases here, instead of simply nouns, verbs, adjectives, and adverbs, because longer verbal, adverbial, etc. units can also be joined by ’and,’ ‘or,’ ‘neither,’ ‘nor,’ and other vernacular conjunctions.)<span style="mso-spacerun: yes"> </span>To say this is not to criticize the logician’s familiar symbols.<span style="mso-spacerun: yes"> </span>Sometimes ‘and’ and so forth are used as clause connectives for which operations such as commutation holds, as in ‘2 + 2 = 4 and 3 + 2 = 5.’<span style="mso-spacerun: yes"> </span>Of course, as J.L. observed, commutation sometimes fails for the truth-functional ‘and,’ as in ‘Tom sat down and started to eat,’ which might be written more perspicuously as ‘Tom sat<span style="mso-spacerun: yes"> </span>down <i style="mso-bidi-font-style: normal">and then</i> he started to eat.’<span style="mso-spacerun: yes"> </span>G.H. Von Wright once had a little calculus featuring an ‘and then’ connective. A moral of my observations here is that, to avoid error, we have to be very careful when putting vernacular inferences into symbolic notation. If an 'or' isn't used as a truth-functional clause-connective, it should not be represented by a logician's '&'.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-family:Chalkboard">J.L. also comments on Excluded Middle, saying he thinks it is really a law about the tilde rather than ‘or’ (or ‘v’).<span style="mso-spacerun: yes"> </span>I think it is “about” both symbols if it is about either.<span style="mso-spacerun: yes"> </span>Actually, it <i style="mso-bidi-font-style:normal">uses</i> both and <i style="mso-bidi-font-style:normal">mentions</i> neither.<span style="mso-spacerun: yes"> </span>But Excluded Middle holds only for sentences (or clauses) that have a determinate meaning and satisfy the principle of Bivalence: when they are either true or false but not both.<span style="mso-spacerun: yes"> </span>Sentences containing vague predicates such as ‘fat’ don’t (without regimentation) satisfy bivalence and so provide counter instances to Excluded Middle. Jack Sprat is clearly thin and his wife is clearly fat, but if Jack’s brother is a borderline case, neither clearly fat nor clearly thin, then the sentence ‘Jack’s brother is fat’ is (without regimentation) neither true nor false, and ‘Jack’s brother is fat v ~( Jack’s brother is fat)’ is not true and so is an exception to Excluded Middle. <o:p></o:p></span></p><p class="MsoNormal"><span style="font-family:Chalkboard">For various reasons, Carnap thought that the meaning of some predicates, including vague ones, can usefully be clarified incompletely by ‘A-Postulates.’<span style="mso-spacerun: yes"> </span>If I wish to use the predicate ‘fat’ in a discussion where I want my meaning to be relatively clear, I might offer a partial clarification of its meaning by offering two A-postulates:<o:p></o:p></span></p><p class="MsoListParagraphCxSpFirst" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><span style="font-family:Chalkboard;mso-fareast-font-family:Chalkboard;mso-bidi-font-family: Chalkboard"><span style="mso-list:Ignore">1)<span style="font:7.0pt "Times New Roman""> </span></span></span><span style="font-family:Chalkboard">(x)(x is fat -> ~(x is thin))<o:p></o:p></span></p><p class="MsoListParagraphCxSpLast" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><span style="font-family:Chalkboard;mso-fareast-font-family:Chalkboard;mso-bidi-font-family: Chalkboard"><span style="mso-list:Ignore">2)<span style="font:7.0pt "Times New Roman""> </span></span></span><span style="font-family:Chalkboard">(x)(x is obese -> x is fat).<o:p></o:p></span></p><p class="MsoNormal"><span style="font-family:Chalkboard">Carnap regarded A-postulates as semantical rules, so the two formulas I have just given could be considered true by virtue of the semantical rules of a certain system.<span style="mso-spacerun: yes"> </span>As such they would count as analytic truths of that system. I defend Carnap on this matter in chapter 3 of my recent book; I can think of no enable objection to his procedure.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-family:Chalkboard">My thanks to JL for the comments.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-family:Chalkboard"><o:p> </o:p></span></p> <span style="font-size:11.0pt;mso-bidi-font-size:12.0pt;font-family:Chalkboard; mso-fareast-font-family:Cambria;mso-fareast-theme-font:minor-latin;mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:minor-bidi;mso-ansi-language:EN-US; mso-fareast-language:EN-US">Bruce</span><!--EndFragment--> </body></html><tt></tt>