{"id":97,"date":"2025-07-06T15:45:37","date_gmt":"2025-07-06T15:45:37","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2025\/07\/06\/russells-paradox-explained-1\/"},"modified":"2025-07-06T15:45:37","modified_gmt":"2025-07-06T15:45:37","slug":"russells-paradox-explained-1","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2025\/07\/06\/russells-paradox-explained-1\/","title":{"rendered":"Russell’s Paradox Explained"},"content":{"rendered":"
\n
\n
A reader asked for an explanation of Russell's Paradox. My pleasure.<\/span><\/p>\n

1. From a contradiction, anything follows. Ex contradictione quod libet<\/em>. Another way of putting it would be to say that every argument having contradictory premises is valid. 'Valid' is a technical term. An argument A is valid =df<\/sub> no argument of A's form has true premises and a false conclusion. Now if A has two premises and they contradict each other, then the Law of Non-Contradiction (LNC) assures us that one of the premises must be false. It follows that no argument of A's form can have true premises and a false conclusion.<\/span><\/p>\n

As long as 'valid' is understood as a technical term having all and only the meaning it is defined as having, then there should not be any trouble understanding how every argument with contradictory premises is valid.<\/span><\/p>\n

Now consider this derivation:<\/span><\/p>\n

a. Al is fat and Al is not fat.<\/span><\/p>\n

b. Al is fat. (From (a) by Simplification)<\/span><\/p>\n

c. Al is fat or Bush is blind (From (b) by Addition)<\/span><\/p>\n

d. Al is not fat. (From (a) by Commutation and Simplification)<\/span><\/p>\n

e. Bush is blind. (From (c) and (d) by Disjunctive Syllogism)<\/span><\/p>\n

This illustrates how any proposition follows from a contradiction.<\/span><\/p>\n

2. Now if Russell's Paradox is a contradiction, then set theory harbors a contradiction. And if anything follows from a contradiction, this is a serious problem for the logicist program of reducing all of mathematics to set theory.<\/span><\/p>\n

3. Unrestricted Comprehension<\/em> is the intuitively attractive idea that for any condition, or open sentence, or propositional function, there is a corresponding set. Thus, corresponding to the condition 'x<\/em> is a cat' there is the set {x<\/em>: x<\/em> is a cat}, in plain English, the set of all cats. Intuitively, it seems that no matter how strange or complex the condition, there ought to be a set of things that satisfy the condition. Thus, corresponding to 'x is either an apple or a sparkplug' there is the set of all x such that x is either an apple or a sparkplug. Unrestricted Comprehension<\/em> appears to be self-evident.<\/span><\/p>\n

4. Now consider 'x<\/em> is not in my pocket.' That condition picks out the set S of all things not in my pocket. Thus my wife and the Eiffel Tower are members of S. But so is S! The set of all things not in my pocket is not in my pocket. Thus S is a member of S. S is a self-membered set. But other sets are non-self-membered. The set of philosophers, for example, is not a member of itself. No set is a philosopher.<\/span><\/p>\n

5. Now consider R, the set of all non-self-membered sets. Is R self-membered or not? Clearly, R is self-membered if and only if R is not self-membered — which is a contradiction.<\/span><\/p>\n

This contradiction is known in the trade as Russell's Paradox. The name, I'd say, is a misnomer. It ought to be called Russell's Antinomy since a paradox need not issue in a contradiction.<\/span><\/p>\n

6. It is easy to see that the antinomy cannot arise without the Unrestricted Comprehension axiom which implies that, corresponding to the condition 'x is the set of all non-self-membered sets' there corresponds the set R. So one solution to the antinomy is via  rejection of Unrestricted Comprehension<\/em>.  <\/span><\/p>\n

Why not reject LNC instead?<\/span><\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

A reader asked for an explanation of Russell's Paradox. My pleasure. 1. From a contradiction, anything follows. Ex contradictione quod libet. Another way of putting it would be to say that every argument having contradictory premises is valid. 'Valid' is a technical term. An argument A is valid =df no argument of A's form has … Continue reading “Russell’s Paradox Explained”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[108,80,124],"tags":[],"class_list":["post-97","post","type-post","status-publish","format-standard","hentry","category-logica-docens","category-paradoxes","category-russell"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/comments?post=97"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/97\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=97"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}