tag:blogger.com,1999:blog-8027844571839885250.post5418112103049956556..comments2022-03-25T07:20:12.468-04:00Comments on Matters of Substance: A paradox concerning propositionsGabriele Contessahttp://www.blogger.com/profile/13607158011908969169noreply@blogger.comBlogger37125tag:blogger.com,1999:blog-8027844571839885250.post-77542098837330667512011-10-26T13:27:41.197-04:002011-10-26T13:27:41.197-04:00Thanks Valdi for the link. I read it with interest...Thanks Valdi for the link. I read it with interest, and yes I'm well aware of the issues and agree with you. I think independent arguments would need to be given (such as <a href="http://nd.academia.edu/JoshuaRasmussen/Papers/637687/What_Propositions_Correspond_To_and_How_They_Do_It_A_Dissertation_Submitted_to_the_Graduate_School_of_the_University_of_Notre_Dame" rel="nofollow">here</a>, pp 62-78).Joshua Rasmussenhttps://www.blogger.com/profile/03271147200091927898noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-39798336324317973502011-09-14T04:02:42.047-04:002011-09-14T04:02:42.047-04:00Joshua, you do recognise that when you say " ...Joshua, you do recognise that when you say " if there is [the property -Q] then surely there is the proposition that -Q exists", commits you to Platonism about propositions, i.e. to the idea that there is a realm of reality outside space and time where 'ideas' roam. I know a lot of philosophers do this routinely, mistakenly believing that it is either intuitively right, or simply convenient and innocent. In 'Truthmakers without Truth' I argue that it is neither intuitively right or innocent (it might be convenient, but if false it is devastating). The paper is available here: http://durham.academia.edu/RögnvaldurIngthorsson/Papers/308786/Truthmakers_Without_TruthValdihttp://durham.academia.edu/RögnvaldurIngthorsson/Aboutnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-57371697385359665922011-09-12T07:26:52.067-04:002011-09-12T07:26:52.067-04:00where in your dichotomy you include the following ...<i>where in your dichotomy you include the following solution: The stipulation "let L = the proposition that the property of ~Q exists" fails</i><br /><br />Perhaps under "there is no such property as ~Q" (because if there is, then surely there is the proposition that ~Q exists). But your points about the meaning of "~Q" are helpful.<br /><br /><i>But surely the domain of quantification in the definition of 'Loves(x,x)' doesn't need to have any language in it.</i><br /><br />Right, but I was dealing with the <i>metalinguistic</i> definition of 'About(x,x)' which explicitly quantifies over predicates... I agree, though, that expansion is deserved.Joshua Rasmussenhttps://www.blogger.com/profile/03271147200091927898noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-91161821797129904522011-09-11T09:58:43.545-04:002011-09-11T09:58:43.545-04:00Josh:
"The problem is that we cannot even me...Josh:<br /><br />"The problem is that we cannot even meaningfully ask if About(P,P) unless the domain of quantification in the definition of 'About(x,x)' includes the pieces of language necessary to express P".<br /><br />This needs some expansion. Consider this parallel: "The problem is that we cannot even meaningfully ask if Loves(Romeo,Romeo) unless the domain of quantification in the definition of 'Loves(x,x)' includes the pieces of language necessary to express Romeo". But surely the domain of quantification in the definition of 'Loves(x,x)' doesn't need to have any language in it.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-29146975857519174382011-09-11T09:53:37.819-04:002011-09-11T09:53:37.819-04:00Josh:
You say "let L = the proposition that ...Josh:<br /><br />You say "let L = the proposition that the property of ~Q exists". Where in your disjunctive taxonomy of ways out ("either L doesn't say anything, or ~Q doesn't exist") is the suggestion that the stipulation fails?<br /><br />There are three ways a stipulation of the form "Let L = the F" can fail:<br /> i. There is more than one F.<br /> ii. There are no Fs.<br /> iii. The expression "the F" is nonsense.<br /><br />In this case, (i) is not a big deal since you can just do existential-elimination, and let L be any one of the Fs.<br /><br />But (ii) and (iii) are real issues. Are they distinct from your option "L doesn't <em>say</em> anything"? I think so, since if the stipulation of L fails, then it's dubious that it's correct to affirm anything "of L", including that it doesn't say anything.<br /><br />Here is a technicality. What does "~Q" mean? There are two readings of "~Q". One reading is that it's a name for the property that negates Q. The other reading it's a definite description like "the property that negates Q". If it's a name, then if there is no property that negates Q, "Let L = the F" may fail in way (iii) by containing a non-referring name (this depends on what you think about non-referring names). In this case, your solution that ~Q doesn't exist and (iii) end up both holding.<br /><br />Suppose "~Q" is a definite description. Then we need to Russelize the definition of L as:<br /> "The proposition that there is a unique property that negates Q and for every property that negates Q, that property exists." <br />And then your argument that if L lacks Q, then L has Q fails, since L does not say of any property that it exists. (Unless you think that the second conjunct does. But we can also say that the second conjunct says of every property that it exists or doesn't negate Q.) <br /><br />So I think you need to take "~Q" as a name. So, suggestion for presentation in the future. Instead of using "~Q", introduce a new letter:<br /> Let N be the property that negates Q.<br />(Or if you're worried--one should be--about uniqueness, do an existential-elimination subproof, and let N be "any one" of the properties that negate Q.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-35428388357416136202011-09-11T09:21:21.093-04:002011-09-11T09:21:21.093-04:00Josh:
"So, to get out of the contradiction, ...Josh:<br /><br />"So, to get out of the contradiction, either L doesn't say anything, or ~Q doesn't exist."<br /><br />Just to make sure we're on the same wavelength, let me query where in your dichotomy you include the following solution: The stipulation "let L = the proposition that the property of ~Q exists" fails.<br /><br />A stipulation of the form "let L = the F" fails when (and only when?) there is more than one F or there are no Fs. In this case, if there is more than one F, that's perhaps less of a problem. We should be able to let L be any one of the Fs, and in effect proceed inside an existential-elimination subproof. But where in your taxonomy is the suggestion that there is no F--no proposition that says that ~Q exists?Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-47010264922626886612011-09-09T18:13:42.787-04:002011-09-09T18:13:42.787-04:00To be clear: the post above that starts with "...To be clear: the post above that starts with "this one is trickier" is dealing with the definition "About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy') or True('~Gy'))".Joshua Rasmussenhttps://www.blogger.com/profile/03271147200091927898noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-91089484115878705012011-09-09T18:10:53.462-04:002011-09-09T18:10:53.462-04:00(Of course, there may be other Liar sentences, but...(Of course, there may be other Liar sentences, but the question is whether any of them call into question my solution to the aboutness paradox.)Joshua Rasmussenhttps://www.blogger.com/profile/03271147200091927898noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-57755322660765600372011-09-09T17:56:44.992-04:002011-09-09T17:56:44.992-04:00Now to meet your necessary condition:
1. Strengthe...Now to meet your necessary condition:<br />1. Strengthened Liar: What I said about semantic circularity applies. "this statement is false" is semantically circular because the meaning of "this" is dependent upon the meaning of "this statement is false" which ultimately implies that the meaning of "this" is dependent on the meaning of "this". So, strengthened liar fails to express anything. (Or it expresses a necessary falsehood--because the circularity problem occurs at the level of referents, not "meanings"; that may make better sense of coordination cases...)<br /><br />2. Truth-teller: Same as above.<br /><br />3. Coordination cases. Same as above. :)<br /><br />I had planned to bring in my theory of propositions (as arrangements of properties) and aboutness (in terms of the exemplifiable parts of a proposition), but it seems that wasn't necessary for a fully satisfying solution. :)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-60355636446196645482011-09-09T17:54:21.110-04:002011-09-09T17:54:21.110-04:00This one is trickier. Since it is metalinguistic, ...This one is trickier. Since it is metalinguistic, it won't suffice to deny the existence of the property of ~About(x,x). The paradox just needs the existence of the predicate '~About(x,x)'. And that exists: just look back a few words! <br /><br />However, there is a semantic circularity problem (which comes up for certain Liar sentences, btw). The problem is that we cannot even meaningfully ask if About(<i>P</i>,<i>P</i>) unless the domain of quantification in the definition of 'About(x,x)' includes the pieces of language necessary to express <i>P</i>, yet one piece of language that's required to express <i>P</i> is 'About(x,x)'. In other words, 'About(x,x)' will not have an extension that allows us to interpret 'About(<i>P</i>,<i>P</i>)', unless 'About(x,x)' already enjoys the semantic power to express <i>P</i>. Put simply: 'About(x,x)' must already have meaning before the definition of 'About(x,x)' can help us interpret 'About(P,P)'. And that, my friends, is viciously circular. <br /><br />Thus, the solution is to recognize that if we define 'About(x,x)' meta-linguistically, then asking whether <i>P</i> is about itself (in that same sense) is meaningless. And no contradiction can come from a meaningless question.<br /><br />So, to review, we've seen that the commonsense notion of aboutness is compatible with supposing that <i>P</i> is about itself. And upon scrutiny, we saw why certain stipulated definitions also fail to generate a contradiction.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-5319244429979724822011-09-09T17:50:30.087-04:002011-09-09T17:50:30.087-04:00About(p,y) iff there are properties F and G, such ...About(p,y) iff there are properties F and G, such that p says that everything that has F has G, and y has F.<br /><br />This leads to a paradox if there is such a property as ~About(x,x). (To see it, just ask yourself if <i>P</i> is about itself on this definition). <br /><br />But I will now argue that there is no such property as ~About(x,x) if it is true that propositions can <i>say</i> anything.<br /><br />I begin with a definition:<br />‘x has Q’ =df ‘there is a property f, such that x says that f exists, and x has f’<br /><br />Now let L = the proposition that the property of ~Q exists<br /><br />If it makes sense to suppose that a proposition can say something, then L surely says that the property of ~Q exists. But ask yourself: does L have Q, or not? Either answer leads to contradiction. Suppose L has Q. Then by the definition of ‘x has Q’, it follows that L has ~Q, which is inconsistent with its having Q. Suppose, on the other hand, L lacks Q. Then it follows that L doesn’t have ~Q (else it would satisfying the definition for having Q), which is inconsistent with its lacking Q. Either way we have a contradiction. <br /><br />So, to get out of the contradiction, either L doesn't <i>say</i> anything, or ~Q doesn't exist. I see no other serious options. (Notice that self-reference isn't even in play here!) But if ~Q doesn't exist, then by parity of reasoning, it seems that neither does ~About(x,x): for there seems to be no relevant difference between the respective paradoxes that would explain why ~Q can't exist while ~About(x,x) can. So, we have an independent reason to think that if propositions say anything, then there's no such property as ~About(x,x). And if there's no such property as ~About(x,x), then the paradox dissolves: <i>P</i> is simply not about itself (because it doesn't say anything about ~About(x,x)).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-90673163704842929092011-09-09T17:47:49.459-04:002011-09-09T17:47:49.459-04:00my necessary condition for a fully satisfying solu...<i>my necessary condition for a fully satisfying solution: it needs to handle, strengthened liar sentences, truth-teller sentences, contingent strengthened liar sentences, contingent truth-teller sentences, and coordination cases (e.g., n propositions arranged in a circle, each saying of the next that it's false). :-)</i><br /><br />Let's see what we can do. I will begin by giving a solution to the problem that results on an intuitive, undefined understanding of 'about'. I'll then tackle the problem for various stipulated definitions of 'about'. I'll conclude by meeting your necessary conditions. :)<br /><br />To solve the problem at an intuitive level, it suffices to show how our intuitions about aboutness are compatible with thinking that <i>P</i> is about itself. To do that, I note with Valdi that <i>P</i> is plausibly about every proposition if it's about any. For it begins "every proposition that..." Now since <i>P</i> is itself a proposition, then <i>P</i> is about itself. Thus, it is simply false that <i>P</i> is about <i>only</i> those propositions that aren't about themselves. It's compatible with commonsense that <i>P</i> is <i>also</i> about propositions that are about themselves, for it is compatible with commonsense that <i>P</i> is about every proposition, including itself.<br /><br />This solution doesn't go deep enough, of course. For we can stipulate other nearby senses of "about" such that <i>P</i> would seem to <i>only</i> be about those propositions that aren't about themselves.<br /><br />I'll begin with what I take to be the most intuitive definition:Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-67423907869558179002011-09-05T12:44:34.412-04:002011-09-05T12:44:34.412-04:00Josh:
I'd love to see your fully satisfying s...Josh:<br /><br />I'd love to see your fully satisfying solution. Just let me give ahead of time my necessary condition for a fully satisfying solution: it needs to handle, strengthened liar sentences, truth-teller sentences, contingent strengthened liar sentences, contingent truth-teller sentences, and coordination cases (e.g., n propositions arranged in a circle, each saying of the next that it's false). :-)<br /><br />I've played with the idea that we need to distinguish between the logic of sentences and the logic of propositions. The formation rules for the logic of sentences in natural languages are unsurprisingly messy, and I don't expect a good account. In formalized languages they can be neater, but it is not surprising if a formalized language fails to do the work we need it to do, say handle both semantic and syntactic stuff together.<br /><br />Now, the best way to get self-reference in a language (you can consider the collection of propositions as a trivial language, where each proposition counts as expressing itself) is to have the language have enough machinery to allow it to parse itself. Once we can self-parse, we can form liar sentences by a diagonal lemma or by a constructive method like <a href="http://alexanderpruss.blogspot.com/2011/09/easy-constructive-proof-of-version.html" rel="nofollow">here</a>.<br /><br />But I am not sure propositions have the right kind of parseability. You invoked the idea of a proposition p saying that everything that has F has G. That's a parseability assumption. To keep it at the level of propositions (and properties--after all, propositions are just nullary properties), we need a ternary property like: lambda_x lambda_y lambda_z (x says that everything that has y has z). <br /><br />I don't know whether this ternary property exists. An initial worry is that it might turn out that <all Fs are Gs> is the same proposition as <all ~Gs are ~Fs>. This particular worry doesn't come up in your case: for your purposes, we can use the ternary property lambda_x lambda_y lambda_z (x says that everything that has y has z or that everything that has ~z has ~y). But there may be further worries along these lines. <br /><br />In particular, I have a general worry about quantifying into contexts like "x has F". The worry comes from Kripke-type puzzles like <a href="http://alexanderpruss.blogspot.com/2011/08/puzzle-about-stipulation.html" rel="nofollow">this one</a>.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-19595439091868642672011-09-05T11:13:50.596-04:002011-09-05T11:13:50.596-04:00We can stipulate that <(x)(~Fx or Gx)> is ab...<i>We can stipulate that <(x)(~Fx or Gx)> is about all and only those xs that satisfy F</i><br /><br />Or: we can stipulate that it's about all and only those xs that satisfy either F or ~G. Then we won't need to assume that <~Fx or Gx> is a different proposition from . So we have:<br /><br />About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy') or True('~Gy')).<br /><br />And then we form the proposition Paradoxical = <(p)(~About(p,p)→ About(p,p))>. And we get paradox when we ask if About(Paradoxical, Paradoxical) or not. (Note: Paradoxical is necessarily false, but the paradox arises from its mere existence.)<br /><br />It should be no surprise that this paradox is connected with others (like Liar). After all, it's conclusion is certainly false. You've got to be right, Alex, that to get out of it, one must make restrictions in the formation rules of logic. But part of the puzzle is in seeing precisely how and why to make those restrictions in non-arbitrary ways.<br /><br />Also, I'm not sure we need a meta-linguistic formulation. Here's a formulation that has intuitive appeal: <br /><br />About(p,y) iff there are properties F and G, such that p says that everything that has F has G, and p has F.<br /><br />What we would like is a solution that get's underneath the intuitions (however vague they might be) that generates the paradox. Finding faults with particular rules of logic formulation doesn't seem to me to do that. That's not fully satisfying.<br /><br />(I'll give my "fully satisfying" solution eventually; I first want people to hunger for it. :) )Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-1419076869979450752011-08-30T00:32:58.876-04:002011-08-30T00:32:58.876-04:00Actually, I would prefer to replace "True(...Actually, I would prefer to replace "True('Fy')" with "Satisfies(y,F)". But of course if you can form a liar sentence with "True", you can form a liar-like sentence with "Satisfies".Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-35592648853192811632011-08-30T00:31:19.496-04:002011-08-30T00:31:19.496-04:00I remember seeing the "about" thing, eit...I remember seeing the "about" thing, either in an article by John Post or in Grim's book.<br /><br />Anyway, start with a fairly minor issue. Read "All Fs are Gs" in the FOLish way as a quantified material conditional. Then for your argument to work, you need to take <(x)(Fx → Gx)> to be about all and only the xs that satisfy F. But (x)(Fx → Gx) is equivalent to, and perhaps even defined as, (x)(~Fx or Gx). And why think this quantified disjunction is about all and only the xs that satisfy F? Surely it's about <em>all</em> xs.<br /><br />Now, we may be able to get out of this by stipulation. We can stipulate that <(x)(~Fx or Gx)> is about all and only those xs that satisfy F. This would require, however, that <~Fx or Gx> be a different proposition from <Gx or ~Fx>, and that is far from clear.<br /><br />Anyway, let's allow that we get to stipulate that <(x)(Fx→Gx)> is about all and only those things that satisfy F. <br /><br />So, we have an About predicate: About(p,y) iff there are predicates F and G such that p = <(x)(Fx→Gx)> and Fy. This requires substitutional quantification. But we can eliminate the substitutional quantification in favor of metalinguistic objectual quantification:<br />About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy')).<br /><br />Then we say that Mundane(p) iff ~About(p,p). And then we form the proposition Paradoxical = <(p)(~About(p,p)→Mundane(p))>. And we get paradox when we ask if Mundane(Paradoxical) or not. <br /><br />But now notice that the paradox uses significant metalinguistic machinery, both syntactic and semantic. On the syntactic side, we need concatenation and some quoted strings (both are implicit in "'(x)(Fx→Gx)'", which we should probably logically represent as something like Concat(Concat(Concat(Concat('(x)(', F), 'x→'),G),'x)'). On the semantic side we have both "expresses" and "True".<br /><br />Once we have concatenation, a sufficient number of quoted strings, "True" and standard first order logical stuff with equality, we can form a strengthened liar sentence. So it's not particularly surprising that with the ingredients at hand one can get paradoxical sentences.<br /><br />Nor is the paradox really that far from the liar in substance, but that's just an intuition here.<br /><br />How to get out of it? I think one has to either deny a classical rule of inference, or get rid of an unrestricted truth predicate or deny unrestricted compositional formation rules in logic. (I go for the last of these.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-89715273801089539102011-08-24T21:00:03.122-04:002011-08-24T21:00:03.122-04:00"Is your main objection that if if a proposit..."Is your main objection that if if a proposition is an ordered series of objects and the relation holding between them it will not be possible to truly deny that something exists?"<br /><br />That's the heart of it. (To get 5 from 3 and 4 substitute "P is true" in 3 with "it is true that Scott Soames no longer exists", keeping in mind that P is the proposition that Scott Soames no longer exists. 6, like 4, is a schema. Treat it as a rule for substituting expressions--as modeled by what I said here about 4.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-43929186902755874212011-08-24T18:14:05.817-04:002011-08-24T18:14:05.817-04:00Josh, can you please spell out how you are derivi...Josh, can you please spell out how you are deriving 5 from 3 and 4?<br /><br />Further, what do you mean by 6? Do you mean --<br /><br />It is necessary that (it is not true that something is true if and only if that thing is true).<br /><br />Now if it is not true that something is true, I suppose that thing is not true. So then it seems you are saying:<br /><br />It is necessary that (something is not true if and only if it is true).<br /><br />How could that be?<br /><br />***<br /><br />Is your main objection that if if a proposition is an ordered series of objects and the relation holding between them it will not be possible to truly deny that something exists?firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-72765447272115383292011-08-24T17:29:20.112-04:002011-08-24T17:29:20.112-04:00Samuel, I also have a lesser worry concerning your...Samuel, I also have a lesser worry concerning your proposal, namely that it seems to me to be false. Here's one of my reasons. Let S be the proposition that Scott Soames no longer exists. It seems to me that S can be true. Here’s an argument for that:<br />1. Suppose for reductio that S cannot be true.<br />2. Then: it is not possible that S is true.<br />3. Thus: it is necessary that it is not true that S is true. (by the inter-definability of modal operators)<br />4. Necessarily, If the proposition that P is true, then it is true that P. (premise schema)<br />5. Thus: it is necessary that it is not true that it is true that Scott Soames no longer exists. (3,4)<br />6. Necessarily, it is not true that it is true that P iff P. (premise schema)<br />7. Thus: it is necessary that Scott Soames no longer exists. (5,6) <br />But (7) is not a happy result—moreover, it’s false. So, it seems to me that S really is possibly true. But that’s hard to square with the hypothesis that S has Scott Soames as a constituent—assuming constituents are essential. (There are truth-in truth-at issues here, which I'm prepared to discuss.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-44894018441069845922011-08-24T14:21:42.312-04:002011-08-24T14:21:42.312-04:00I think I see what you mean about the proposition ...I think I see what you mean about the proposition only needing to exist. Any proposition which contains the term "propositions not about themselves" is about itself (that is, contains a term referring to itself) if and only if it is not about itself, regardless of whether or not it is true.firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-37412914463875988332011-08-24T12:57:59.171-04:002011-08-24T12:57:59.171-04:00In fact, I think you need the negation to be term ...In fact, I think you need the negation to be term negation, otherwise this proposition is about propositions about themselves.<br /><br />Why do you say it only requires the proposition exist? The derivations of the paradox we've seen so far all assume it is true -- or what am I missing?firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-24488289257070043972011-08-24T12:55:23.230-04:002011-08-24T12:55:23.230-04:00Samuel, forget about aboutness. Just ask yourself ...Samuel, forget about aboutness. Just ask yourself whether P stands in relation R to itself, where R is a relation a proposition bears to an x iff it quantifies over instances of a kind of which x is an instance. Is your complaint that 'R' is not a well-defined relation? <br /><br />firezdog, the paradox doesn't require the premise that P could be true--only that P exists.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-25332442604609308332011-08-24T12:43:57.780-04:002011-08-24T12:43:57.780-04:00But negation is it seems very deeply integrated in...But negation is it seems very deeply integrated into your term, Josh -- the "not" is not sentential negation, is it? It seems it must be term negation.firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-1532977188885606132011-08-24T12:41:28.744-04:002011-08-24T12:41:28.744-04:00In fact it seems no proposition which contains the...In fact it seems no proposition which contains the term "propositions not about themselves" would be true, because from all such propositions a paradox can be derived. (Even the proposition: "Propositions not about themselves are not about themselves" would be false, along with the proposition, "Propositions not about themselves are about themselves.". Perhaps this in itself, a seeming violation of the law of noncontradiction, is enough to invalidate my solution.)firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.comtag:blogger.com,1999:blog-8027844571839885250.post-39951749086606804572011-08-24T12:31:55.235-04:002011-08-24T12:31:55.235-04:00Josh, I suggest we reject the proposition: proposi...Josh, I suggest we reject the proposition: propositions that are not about themselves are mundane. What is wrong with that?firezdoghttps://www.blogger.com/profile/11473050286104950159noreply@blogger.com