{"id":1916,"date":"2009-11-28T18:11:11","date_gmt":"2009-11-28T17:11:11","guid":{"rendered":"http:\/\/emilkirkegaard.dk\/en\/?p=1916"},"modified":"2013-01-05T10:44:59","modified_gmt":"2013-01-05T09:44:59","slug":"infallible-knowledge-the-modal-fallacy-and-modal-collapse","status":"publish","type":"post","link":"https:\/\/emilkirkegaard.dk\/en\/2009\/11\/infallible-knowledge-the-modal-fallacy-and-modal-collapse\/","title":{"rendered":"Infallible knowledge, the modal fallacy and modal collapse"},"content":{"rendered":"

The much mentioned the modal fallacy<\/a> is not a fallacy (that is, is a valid inference rule) if one accepts an exotic view about modalities and necessities that is logically implied by a particular understanding of infallible knowledge and a knower.<\/p>\n

Infallible knowledge<\/h3>\n

Some people seem to think that some known things are false and thus the need for a term like infallible knowledge, for that kind of knowledge that cannot be of false things. However that term \u201cinfallible knowledge\u201d (and it’s under-term \u201cinfallible foreknowledge\u201d) is subject to some interpretation. Is it best understood as:<\/p>\n

A. If something is known, then it is necessarily true.<\/p>\n

Or?:<\/p>\n

B. Necessarily, if something is known, then it is true.<\/p>\n

Or equivalently, in terms of \u201ccannot\u201d instead of \u201cnecessarily\u201d:<\/p>\n

A. If something is known, then it cannot be false.<\/p>\n

B. It cannot be false that, if something is known, then it is true.<\/a>1<\/sup><\/p>\n

I contend that the second interpretation, (B), is the best. However suppose that one accepts the first, (A).<\/p>\n

The assumption of the existence of a foreknower<\/h3>\n

Now let’s assume that there is someone that knows everything (which is the case), the knower. He posses infallible knowledge \u00e1 la (A). Now we can work out the implications.<\/p>\n

The foreknower exists and knows everything (that is the case):<\/p>\n

1. There exists at least one person and that for all propositions, that a proposition is the case logically implies that that person knows that proposition.<\/p>\n

(\u2203x)(\u2200P)(P\u21d2Kx(P))<\/p>\n

Whatever is known is necessarily the case (A):<\/p>\n

2. For all propositions and for all persons, that a person knows a proposition logically implies that that proposition is necessarily true.<\/p>\n

(\u2200P)(\u2200x)(Kx(P)\u21d2\u25a1P)<\/p>\n

Thus, every proposition that is the case is necessarily the case:<\/p>\n

Thus, 3. For all propositions, that a proposition is the case logically implies that it is necessarily the case.<\/p>\n

\u22a2 (\u2200P)(P\u21d2\u25a1P) [from 1, 2, HS]<\/p>\n

Thus, everything that is logically possible is the case:<\/p>\n

Thus, 4. For all propositions, that a proposition is logically possible logically implies that it is the case.<\/p>\n

\u22a2 (\u2200P)(\u25caP\u21d2P) [from 3, others]<\/a>2<\/sup><\/p>\n

Thus, everything that is logically possible is necessarily the case:<\/p>\n

Thus, 5. For all propositions, that a proposition is logically possible logically implies that it is necessarily the case.<\/p>\n

\u22a2 (\u2200P)(\u25caP\u21d2\u25a1P) [from 3, 4, HS]<\/p>\n

This is called modal collapse. The acceptance of that all possibilities are necessarily the case.<\/p>\n

Thus, the modal fallacy is no longer a fallacy:<\/p>\n

Thus, 6. For all propositions (P) and for all propositions (Q), that a proposition (P) is the case, and that that proposition (P) logically implies a proposition (Q), logically implies that that proposition (Q) is necessarily the case.<\/p>\n

\u22a2 (\u2200P)(\u2200Q)(P\u2227(P\u21d2Q)\u21d2\u25a1Q) [from 3]<\/a>3<\/sup><\/p>\n

And so we can validly infer from a proposition being the case and that that proposition logically implies some other proposition to that that other proposition is necessarily the case.<\/p>\n

Notes<\/h3>\n
\n

<\/a>1Or \u201ccannot be not-true\u201d to avoid relying on monoalethism (and the principle of bivalence<\/a>) which means that truth bearers only have a single truth value.<\/p>\n<\/div>\n

\n

<\/a>2This follows like this: I. \u25a1P\u21d4\u00ac\u25ca\u00acP (definition of \u25ca). II. Thus, P\u21d2\u00ac\u25ca\u00acP. [I, 3, Equi., HS] Thus, \u25ca\u00acP\u21d2\u00acP. [II, CS, DN] Thus, III. \u25caP\u21d2P. [II, Substitution of \u00acP for P]<\/p>\n<\/div>\n

\n

<\/a>3This follow like this: P\u2227(P\u21d2Q)\u21d2Q is just MP, and from 3 it follows that any proposition that is the case is necessarily the case.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

The much mentioned the modal fallacy is not a fallacy (that is, is a valid inference rule) if one accepts an exotic view about modalities and necessities that is logically implied by a particular understanding of infallible knowledge and a knower. Infallible knowledge Some people seem to think that some known things are false and […]<\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[1482,14,1481,1275],"class_list":["post-1916","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-infallible-knowledge","tag-modal","tag-modal-collapse","tag-the-modal-fallacy","entry"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/1916","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/comments?post=1916"}],"version-history":[{"count":3,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/1916\/revisions"}],"predecessor-version":[{"id":3524,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/1916\/revisions\/3524"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=1916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=1916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=1916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}