{"id":1591,"date":"2023-05-26T23:30:00","date_gmt":"2023-05-27T03:30:00","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=1591"},"modified":"2023-05-26T21:29:34","modified_gmt":"2023-05-27T01:29:34","slug":"is-true-of-part-2-linguistic-analog","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=1591","title":{"rendered":"Is True Of – Part 2 Linguistic Analog"},"content":{"rendered":"\n

This month\u2019s post is part 2 of 2 of the exploration of the YouTube video entitled Russell’s Paradox – a simple explanation of a profound problem<\/a> by Jeffery Kaplan.  What we discussed last month was the summary of Russell\u2019s paradox in which we found that the set of all ordinary sets defined as<\/p>\n\n\n\n

\\[ {\\mathcal Q} = \\{ x | x \\mathrm{\\;is\\;a\\;set\\;that\\;does\\;not\\;contain\\;itself} \\} \\; \\]<\/div><\/p>\n\n\n\n

creates a paradox.  If we assume ${\\mathcal Q}$ does not contain itself (i.e., it is ordinary) then the membership comprehension \u2018is a set that does not contain itself\u2019 instructs us that ${\\mathcal Q}$ in fact does contain itself.  Alternatively, if we assume that ${\\mathcal Q}$ does contain itself (i.e., it is extraordinary) then membership comprehension instructs us that it doesn\u2019t. <\/p>\n\n\n\n

This type of self-referential paradox mirrors other well-known paradoxes that arise in linguistics such as the liar’s paradox<\/a> or the concept of the self-refuting idea<\/a>.  What makes Kaplan\u2019s analysis interesting (whether or not it originates with him) is the very strong formal analogy that he draws between the common act of predication that we all engage in nearly continuously, and the more abstruse structure of Russell\u2019s paradox that few among us know or care about.<\/p>\n\n\n\n

The heart of Kaplan\u2019s analogy is the explicit mapping of the \u2018contains\u2019 idea from set theory \u2013 that sets contain members or elements, some of which are sets, including themselves \u2013 with the \u2018is true of\u2019 idea of predication. <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

To understand what Kaplan means by the \u2018is true of\u2019, we will again follow the structure of his examples with some minor verbal modifications to better suit my own taste.<\/p>\n\n\n\n

Predication is the act of saying something about the subject of a sentence by giving a condition or attribute belonging to the subject.  In the following sentence<\/p>\n\n\n\n

Frodo is a brave hobbit.<\/div>\n\n\n\n

The subject of the sentence is \u201cFrodo\u201d and the predicate is \u201cis a brave hobbit\u201d.  The predicate \u201cis a brave hobbit\u201d is true of Frodo, as anyone who’s read Lord of the Rings can attest.  Kaplan then points out that the first basic rule of na\u00efve set theory, which he states as<\/p>\n\n\n\n

Rule #1 of sets: there is a set for any imaginable collection of a thing or of things\u201d<\/div>\n\n\n\n

has, as its formal analogy in predication, the following:<\/p>\n\n\n\n

Rule #1 of predication: there is a predicate for any imaginable characteristic of a thing.<\/div>\n\n\n\n

The two key rules of set theory that lead to Russell\u2019s paradox have their analogs in predication as well.   <\/p>\n\n\n\n

Rule #10 of sets, which allows us to have sets of sets, is mirrored by Rule #10 of predication that tells us we can predicate things about predicates.  As an example of this, consider the following sentence:<\/p>\n\n\n\n

“Is a Nazgul” is a terrifying thing to hear said of someone.<\/div>\n\n\n\n

The predicate \u201cIs a Nazgul\u201d is the subject of that sentence and \u201cis a terrifying thing to hear said of someone\u201d is the predicate.<\/p>\n\n\n\n

Rule #11 of sets, which allows sets to contain themselves (i.e., self-reference), finds its analog in Rule #11 of predication that tells us that predicates can be true of themselves.<\/p>\n\n\n\n

Here we must proceed a bit more carefully.  Let\u2019s start with a simple counterexample:<\/p>\n\n\n\n

“Is a hobbit” is a hobbit.<\/div>\n\n\n\n

This sentence is clearly false as the subject, the predicate \u201cIs a hobbit\u201d, is clearly not a hobbit itself, it is a predicate.  But now consider the following sentence, which Kaplan offers:<\/p>\n\n\n\n

“Is a predicate” is a predicate<\/div>\n\n\n\n

This sentence is clearly true as the subject, the predicate \u201cIs a predicate\u201d, is clearly a predicate.  And, so, Rule #11 of predication works.<\/p>\n\n\n\n

Kaplan then constructs a table similar to the following (again only minor verbal tweaks are done for the predicates that are not true of themselves to suit my own taste)<\/p>\n\n\n\n

Predicates not true of themselves<\/th>Predicates true of themselves<\/th><\/tr>
\u201cis a brave hobbit\u201d<\/td>\u201cis a predicate\u201d<\/td><\/tr>
\u201cis a Nazgul\u201d<\/td>\u201cis a string of words\u201d<\/td><\/tr>
\u201ckeeps his oaths\u201d<\/td>\u201ctypically comes at the end of a sentence\u201d<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Note that the predicate \u201cis true of itself\u201d is a predicate that is true of all the predicates that are true of themselves, that is to say, of all the predicates that can be placed in the right column of the table above. The next step is then to ask what is the predicate of all the predicates that can be placed in the left column of the above table.  A little reflection should satisfy oneself that the predicate \u201cis not true of itself\u201d fits the bill. <\/p>\n\n\n\n

The final step is to ask in which of the two column does \u201cis not true of itself\u201d fall, or, in other words,<\/p>\n\n\n\n

is \u201cis not true of itself\u201d true of itself?<\/div>\n\n\n\n

If we assume that it is true of itself then the content found between the quotes tells us that it is not true of itself.  Equally vexing, if we assume that it is not true of itself, that assumption matching the content found between the quotes tells us that it is true of itself.  In summary: if it is then it isn\u2019t and if it isn\u2019t then it is.  And we\u2019ve generated the predicate analogy to Russell\u2019s paradox.<\/p>\n\n\n\n

Of course, this is just a form of the well-known Liar\u2019s Paradox, so we might be willing to just shrug it off as a quirk of language, but I think Kaplan is making a deeper point that is worth deeply considering.  At the root of his analysis is the realization that there are objective rules (or truths), that these rules generate self-referential paradoxes, and, so, one is forced to recognize that paradoxes are an essential ingredient in not just all of language but of thought itself.  And no amount of patching, such as was done to na\u00efve set theory, can rescue us from this situation.  This observation, in turn, has the profound philosophical implication that there is only so far that logic can take us.<\/p>\n","protected":false},"excerpt":{"rendered":"

This month\u2019s post is part 2 of 2 of the exploration of the YouTube video entitled Russell’s Paradox – a simple explanation of a profound problem by Jeffery Kaplan.  What… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1591","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1591","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1591"}],"version-history":[{"count":3,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1591\/revisions"}],"predecessor-version":[{"id":1635,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1591\/revisions\/1635"}],"wp:attachment":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}