{"id":12805,"date":"2009-02-20T16:06:03","date_gmt":"2009-02-20T16:06:03","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2009\/02\/20\/further-modal-concepts-consistency-inconsistency-contradictoriness-and-entailment\/"},"modified":"2009-02-20T16:06:03","modified_gmt":"2009-02-20T16:06:03","slug":"further-modal-concepts-consistency-inconsistency-contradictoriness-and-entailment","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2009\/02\/20\/further-modal-concepts-consistency-inconsistency-contradictoriness-and-entailment\/","title":{"rendered":"Further Modal Concepts: Consistency, Inconsistency, Contradictoriness, and Entailment"},"content":{"rendered":"

I <\/font>argued earlier<\/font><\/a> that the validity of argument forms is a modal concept.  But the same goes for consistency, inconsistency, contradictoriness, and entailment.  Here are some definitions. 'Poss' abbreviates 'It is broadly-logically possible that ___.' 'Nec' abbreviates 'It is broadly-logically necessary that ___.' '~' and '&' are the familiar truth-functional connectives. 'BL' abbreviates 'broadly logically.'<\/font><\/p>\n

D1. A pair of propositions p, q is BL-consistent =df<\/sub> Poss(p & q).<\/font><\/strong><\/p>\n

Clearly, any two true propositions are consistent. (By 'consistent' I mean consistent with each other.  If I mean self-consistent, I'll say that.)   But there is more to consistency that this.  It is a modal notion.  Consistency cannot be defined in terms of what is actually the case.  One must also consider what could have been the case.  As long as p, q are contingent, they are consistent regardless of their truth-values. If both are true, they are consistent.  If both are false, they are consistent.  If one is true and the other false, or vice versa, they are consistent.<\/font><\/p>\n

D2. A pair of propositions, p, q, are BL-inconsistent =df<\/sub> ~Poss(p & q).<\/font><\/strong><\/p>\n

D3. A pair of propositions p, q are BL-contradictory =df<\/sub> ~Poss(p & q) & ~Poss (~p & ~q).<\/font><\/strong><\/p>\n

Note the difference between inconsistency and the stronger notion of contradictoriness.  If two propositions are inconsistent, then they logically cannot both be true.  If two propositions are contradictory, then they are inconsistent but also such that their negations logically cannot be true.<\/font><\/p>\n

Example. All men are rich<\/em> and No men are rich<\/em> are inconsistent in that they cannot both be true.  But they are not contradictory since their negations (Some men are not rich<\/em>, Some men are rich<\/em>) are both true.  All men are rich<\/em> and Some men are not rich<\/em> are contradictory.  Some men are rich<\/em>, Some men are not rich<\/em> are neither inconsistent nor contradictory.<\/font><\/p>\n

D4. P entails q =df <\/sub>~Poss(p & ~q).<\/font><\/strong><\/p>\n

Entailment, also called strict implication,  is the necessitation of material implication.  If '–>' stands for the material conditional, then the right hand side of (D4) can be put as follows: Nec (p –> q).<\/font><\/p>\n

(Alethic) modal logic's task is to provide criteria for the evaluation of arguments whose validity or lack thereof depends crucially on such words as 'possibly' and 'necessarily.'  But if I am right, many indispensable concepts of nonmodal<\/em> logic (e.g., standard first-order predicate logic with identity) are modal concepts.  <\/font><\/p>\n

<\/font> <\/p>\n

<\/font> <\/p>\n","protected":false},"excerpt":{"rendered":"

I argued earlier that the validity of argument forms is a modal concept.  But the same goes for consistency, inconsistency, contradictoriness, and entailment.  Here are some definitions. 'Poss' abbreviates 'It is broadly-logically possible that ___.' 'Nec' abbreviates 'It is broadly-logically necessary that ___.' '~' and '&' are the familiar truth-functional connectives. 'BL' abbreviates 'broadly logically.' … Continue reading “Further Modal Concepts: Consistency, Inconsistency, Contradictoriness, and Entailment”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[108,235],"tags":[],"class_list":["post-12805","post","type-post","status-publish","format-standard","hentry","category-logica-docens","category-modal-matters"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/12805","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=12805"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/12805\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=12805"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=12805"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=12805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}