{"id":3972,"date":"2019-02-05T05:37:19","date_gmt":"2019-02-05T05:37:19","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2019\/02\/05\/excluded-middle-and-bivalence-again-1\/"},"modified":"2019-02-05T05:37:19","modified_gmt":"2019-02-05T05:37:19","slug":"excluded-middle-and-bivalence-again-1","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2019\/02\/05\/excluded-middle-and-bivalence-again-1\/","title":{"rendered":"Excluded Middle, Bivalence, and Disquotation"},"content":{"rendered":"
\n

LEM: For every  p, p v ~p.<\/span><\/p>\n

BV: Every proposition is either true or false.<\/span><\/p>\n<\/blockquote>\n

These principles are obviously not identical<\/em>.  Excluded Middle is syntactic principle, a law of logic, whereas Bivalence is a semantic principle. The first says nothing about truth or falsity. The second does. (See Michael Dummett, Truth and Other Enigmas<\/em>, Harvard UP, 2nd ed. , 1980, p. xix; Paul Horwich, Truth<\/em>, Oxford UP, 2nd ed., 1998, p. 79) Though not identical they might nonetheless be logically equivalent.  Two propositions are logically equivalent iff each entails the other.  Entailment is the necessitation of material implication. Can it be shown that (LEM) and (BV)  entail each other? Let's see.
<\/span><\/p>\n

The logical equivalence of the two principles can be demonstrated if we assume the disquotational schema:<\/span><\/p>\n

\n

DS: p<\/em> is true iff p. <\/span><\/p>\n<\/blockquote>\n

For example, snow is white<\/em> is true iff snow is white. Or, if you insist, 'snow is white' is true iff snow is white. In the latter forrmulation, which does not involve reference to propositions, the truth predicate  — 'is true' — is merely a device of disquotation or of semantic descent. On either formulation, 'is true' adds no sentential\/propositional content:  the sentential\/propositional content is the same on both sides of the biconditional.  The content of my assertion is exactly the same whether I assert that snow is white or I assert that snow is white<\/em> is true.  But if (DS) is granted, then so is:
<\/span><\/p>\n

\n

DS-F: p<\/em> is false iff ~p.<\/span><\/p>\n<\/blockquote>\n

For example, snow is white<\/em> is false iff ~(snow is white).     <\/span><\/p>\n

Now if the disquotational schemata exhaust what it is to be true and what it is to be false, then (LEM) and (BV) are logically equivalent. <\/span><\/p>\n

Given (DS) and (DS-F), we can rewrite (LEM) as <\/span><\/p>\n

\n

LEM-T: For every p, p <\/em> is true v p<\/em> is false.
<\/span><\/p>\n<\/blockquote>\n

Now (LEM-T) is simply a restatement of (BV). The principles are therefore logically equivalent given the disquotational schemata.  <\/span><\/p>\n

But this works only if falsehood can be adequately explained in terms of the merely logical operation of negation.  This will NOT work if negation can only be explained in terms of falsehood.  For then we would enter  an explanatory circle of embarrassingly short diameter.  <\/span><\/p>\n

Ask yourself: when is one proposition the negation of another? The negation of p<\/em> is the proposition that is true<\/em> iff p is false<\/em> and false<\/em> iff p is true<\/em>.  To explain the logico-syntactic notion of negation we have to reach for the semantic notions of truth and falsehood.  But then falsehood cannot be exhaustively understood or reduced to negation.<\/span><\/p>\n

It is telling that to explain negation and the other logical connectives we use TRUTH tables.  Such explanation is satisfactory.  But it would not be if the redundancy or disappearance or disquotational schemata gave the whole meaning of 'true' and 'false.'  (The point is made by M. Dummett, Truth and Other Enigmas<\/em>, p. 7)<\/span><\/p>\n

I take this explanatory circle to show that there is more to truth and falsehood than is captured in the above disquotational schemata.<\/span><\/p>\n

Conclusion: if one's reason for accepting the logical equivalence of (LEM) and (BV) is (DS) then that is a bad reason.<\/span><\/p>\n

Are there counterexamples to (DS)? <\/span> It seems to fail right-to-left if 'Sherlock Holmes is a detective' is plugged in for 'p' on the RHS of (DS).  Arguably, Holmes is a detective, but it is not true<\/em> that Holmes is a detective.  For it to be true that Holmes is a detective, 'Holmes' would have to refer to something that exists.  But this requirement is not satisfied in the case of purely fictional items.  I am assuming that veritas sequitur esse<\/em>, that truth 'follows' or supervenes upon being (existence):
<\/span><\/p>\n

\n

VSE:  There are no true predications about what does not exist.<\/span><\/p>\n<\/blockquote>\n

Since Holmes does not exist, 'Holmes is a detective' appears to express a proposition that is neither true nor false. Likewise for its negation, 'Holmes is not a detective.'  (LEM) is not violated since either Holmes is a detective or Holmes is not a detective. But (BV) is violated since the two Holmes propositions are neither true nor false.
<\/span><\/p>\n

It is worth noting that from 'Only propositions have truth-values' one cannot validly infer 'All propositions have truth-values.'   <\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

LEM: For every  p, p v ~p. BV: Every proposition is either true or false. These principles are obviously not identical.  Excluded Middle is syntactic principle, a law of logic, whereas Bivalence is a semantic principle. The first says nothing about truth or falsity. The second does. (See Michael Dummett, Truth and Other Enigmas, Harvard … Continue reading “Excluded Middle, Bivalence, and Disquotation”<\/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":[408,108,228],"tags":[],"class_list":["post-3972","post","type-post","status-publish","format-standard","hentry","category-language-philosophy-of","category-logica-docens","category-truth"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/3972","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=3972"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/3972\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=3972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=3972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=3972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}