{"id":1859,"date":"2023-07-12T14:14:51","date_gmt":"2023-07-12T18:14:51","guid":{"rendered":"https:\/\/www.fecundity.com\/nfw\/?p=1859"},"modified":"2023-07-12T15:42:47","modified_gmt":"2023-07-12T19:42:47","slug":"all-some-and-a-rhetorical-misstep","status":"publish","type":"post","link":"https:\/\/www.fecundity.com\/nfw\/2023\/07\/12\/all-some-and-a-rhetorical-misstep\/","title":{"rendered":"All-some and a rhetorical misstep"},"content":{"rendered":"\n

John Norton breezes through an example of a deductive inference so as to characterize induction by contrast. His example of a valid deductive inference form is: “All As are B. Therefore, some As are B.” He even dubs this the all-some<\/em> schema.<\/span>1<\/sup><\/a><\/span><\/p>\n\n\n\n

It is a perplexing example. In old-school Aristotelean logic, the all-some<\/em> schema is valid. In modern first-order logic, however, A may be an empty predicate. There being no As makes \u2200x(Ax\u2192Bx) true and \u2203x(Ax&Bx) false, showing that the schema is invalid.<\/p>\n\n\n\n

This got me thinking about whether the modern reading of the schema is really better than the classical one. I think it is.<\/p>\n\n\n\n\n\n\n\n

Start with a few things that seem intuitively true:<\/p>\n\n\n\n

    \n
  1. Some As are Bs<\/em> is false if there are no As.<\/li>\n\n\n\n
  2. All As are Bs<\/em> is equivalent to It is not the case that some As are non-Bs<\/em>, where not the case<\/em> can be understood as truth-functional negation.<\/li>\n\n\n\n
  3. The class of As is the class of non-non-As.<\/li>\n<\/ol>\n\n\n\n

    These together entail that All As are Bs<\/em> is true when there are no As. The third might even be unnecessary. So holding that all-some<\/em> is valid requires either giving up one of these, which doesn’t seem like an appealing move. One could reject 1 and 2 on principle by holding that empty predicates are logically impossible\u2014 but I can’t muster even an iota of enthusiasm for such a move.<\/p>\n\n\n\n

    Of course, Norton not is concerned with any of that. He just wants a quick and easy example of a deductive schema. As I reader, I understand his point. Since his example distracted me enough to think through this mishegas, though, it hasn’t served his rhetorical purposes very well.<\/p>\n\n\n\n

    Tangentially related postscript<\/h2>\n\n\n\n

    Last year, I posted about wanting a better name<\/a> than the material theory of induction<\/em> for the fact that scientific inference always relies on domain-specific background knowledge. Still looking for one. Suggestions welcome.<\/p>\n","protected":false},"excerpt":{"rendered":"

    John Norton breezes through an example of a deductive inference so as to characterize induction by contrast. His example of a valid deductive inference form is: “All As are B. Therefore, some As are B.” He even dubs this the all-some schema. It is a perplexing example. In old-school Aristotelean logic, the all-some schema is … Continue reading “All-some and a rhetorical misstep”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[9],"tags":[68,13],"class_list":["post-1859","post","type-post","status-publish","format-standard","hentry","category-philosophy","tag-logic","tag-teaching"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7PjAo-tZ","jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/posts\/1859","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/comments?post=1859"}],"version-history":[{"count":3,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/posts\/1859\/revisions"}],"predecessor-version":[{"id":1863,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/posts\/1859\/revisions\/1863"}],"wp:attachment":[{"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/media?parent=1859"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/categories?post=1859"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fecundity.com\/nfw\/wp-json\/wp\/v2\/tags?post=1859"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}