I’ve received a request from a long-time reader to write a basics post on modal logics. In particular, what is a modal logic, and why did Gödel believe that a proof for the existence of God was more compelling in modal logic than in standard predicate logic.<\/p>\n
The first part is the easy one. Modal logics are logics that assign values to statements that go beyond “This statement is true” or “This statement is false”. Modal logics add the concepts of possibility and necessity. Modal logic allows statements like “It is necessary<\/em> for X to be true”, “It is possible<\/em> for X to be true”, etc.<\/p>\n <\/p>\n The classic presentation of modal logic is basically to take first order predicate logic, and add two modal operators<\/em> to it:<\/p>\n Possibility and necessity are, naturally, related by negation: ¬◊P⇔□¬P (P is not possible if and only if it’s necessary that P is not true), and □P⇔¬◊¬P. (P is not necessary if and only if it’s possible that P is not true.)<\/p>\n Going the modal route allows you to incorporate contingent reasoning<\/em> into an inference process in a very nice way. You can make statements like “If it’s possible that Jane murdered Joe, then it’s necessary to make sure that Jane doesn’t get left alone with other potential victims”. Statements like that are quite easy in a modal logic (◊Murdered(Joe,Jane) ⇒ ∀x:◊Victim(x): □¬LeaveAlone(Jane,x)). <\/p>\n In math circles, the term “modal logic” has also been expanded to include a range of logics that do similar things: things like temporal logics (which add “X is true sometimes”, “X must always be true”, “X can never be true”, “X can be true after Y happens”, etc) are often called modal logics. Intuitionistic logic is also sometimes incorrectly referred to as a modal logic. (Intuitionistic logic still assigns strict truth values to statements; but it also includes the ability to have statements whose truth value is unknown<\/em>.)<\/p>\n The second half of the question is much harder: Why did Gödel find a modal proof to be compelling? The honest answer to that is: I don’t know. My suspicion is that the idea that you could prove that it’s necessary<\/em> for God to exist meant something to him. Even as a religious person, I don’t find the modal proof any more compelling than the non-modal one: at it’s core, it’s really the same old silly argument that in order for human beings to have an idea of truth and beauty, there must be some source of that concept, and that the only possible source of it is some being which is the embodiment of all things good, which must be God.<\/p>\n But Gödel was, sadly, quite seriously mentally ill: he was depressed, paranoid, and extremely obsessive. In the end, he starved himself to death because when his wife became ill and was hospitalized, he refused to eat, because he didn’t trust anyone else – including himself – to prepare food that wasn’t poisoned. Throughout his life, one of his great obsessions was the idea that there was more to the world than just what we can see; he was desperate to find some kind I’ve received a request from a long-time reader to write a basics post on modal logics. In particular, what is a modal logic, and why did Gödel believe that a proof for the existence of God was more compelling in modal logic than in standard predicate logic. The first part is the easy one. Modal […]<\/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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74,33],"tags":[],"class_list":["post-343","post","type-post","status-publish","format-standard","hentry","category-basics","category-logic"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-5x","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/343","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=343"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/343\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=343"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\nof meaning, something that showed we were more that just automatons – but he never found anything
\ncompelling in any of the organized religions – almost every “real world” construct built by human beings fell under the umbrella of his paranoia, and ended being viewed by him as another effort to poison either his body or his mind. The modal proof of God was, perhaps, his answer: an answer in the pureness of mathematics, which he could, in his own mind, verify was untainted by the poison of human hands.<\/p>\n","protected":false},"excerpt":{"rendered":"