H. P. Grice, "Paradoxes of Entailment"

The paradox of implication assumes many forms,  some of which are not easily recognised as involving  mere varieties of the same fundamental principle. But     COMPOUND PROPOSITIONS 47   I believe that they can all be resolved by the consider-  ation that we cannot ivithotd qjialification apply a com-  posite and (in particular) an implicative proposition to  the further process of inference. Such application is  possible only when the composite has been reached  irrespectively of any assertion of the truth or falsity of  its components. In other words, it is a necessary con-  dition for further inference that the components of a  composite should really have been entertained hypo-  thetically when asserting that composite.   § 9. The theory of compound propositions leads to  a special development when in the conjunctives the  components are taken — not, as hitherto, assertorically —  but hypothetically as in the composites. The conjunc-  tives will now be naturally expressed by such words as  possible or compatible, while the composite forms which  respectively contradict the conjunctives will be expressed  by such words as necessary or impossible. If we select  the negative form for these conjunctives, we should write  as contradictory pairs :   Conjunctives {possible) Composites {fiecessary)     a. p does not imply q   1, p is not implied by q   c. p is not co-disjunct to q   d. p is not co-alternate to q     a, p implies q   b, p is implied by q   c, p is co-disjunct to q   d, p is co-alternate to q     Or Otherwise, using the term 'possible' throughout,  the four conjunctives will assume the form that the several  conjunctions — pq^pq, pq ^-nd pq — are respectively /^i*-  sidle. Here the word possible is equivalent to being  merely hypothetically entertained, so that the several  conjunctives are now qualified in the same way as are  the simple components themselves. Similarly the four     48 CHAPTER HI   corresponding composites may be expressed negatively  by using the term 'impossible,' and will assume the  form that the ^^;yunctions pq^ pq, pq and pq are re-  spectively impossible, or (which means the same) that  the ^zVjunctions/^, ^^, pq Rnd pq are necessary. Now  just as 'possible* here means merely 'hypothetically  entertained/ so 'impossible' and 'necessary' mean re-  spectively 'assertorically denied' and 'assertorically  affirmed/   The above scheme leads to the consideration of the  determinate relations that could subsist of p to q when  these eight propositions (conjunctives and composites)  are combined in everypossibleway without contradiction.  Prima facie there are i6 such combinations obtained by  selecting a or ay b or 3, c or c, d or J for one of the four  constituent terms. Out of these i6 combinations, how-  ever, some will involve a conjunction of supplementaries  (see tables on pp. 37, 38), which would entail the as-  sertorical affirmation or denial of one of the components  / or q, and consequently would not exhibit a relation of  p to q. The combinations that, on this ground, must be  disallowed are the following nine :   cihcd, abed, abed, abed] abed, bacd, cabd, dabc\ abed.   The combinations that remain to be admitted are  therefore the followino- seven :   abld, cdab\ abed, bald, cdab^ dcab\ abed.   In fact, under the imposed restriction, since a or b  cannot be conjoined with c or d, it follows that we must  always conjoin a with c and d\ b with e and d\ c with  a and b\ ^with a and b. This being understood, the     COMPOUND PROPOSITIONS 49   seven permissible combinations that remain are properly  to be expressed in the more simple forms:   ab, cd\ ab, ba, cd, dc\ and abed   These will be represented (but re-arranged for purposes  of symmetry) in the following table giving all the  possible relations of any proposition/ to any proposition  q. The technical names which 1 propose to adopt for  the several relations are printed in the second column  of the table.   Table of possible relations of propositio7i p to proposition q.     1. {a,b)\ p implies and is implied by q   2. (a, b) : p implies but is not implied by q,   3. {b^d): p is implied by but does not imply q,   4. {djb^'c^d): p is neither implicans nor impli   cate nor co-disjunct nor co-alternate to g.   5. {dy c)\ /is co-alternate but not co-disjunct to $r,   6. {Cyd): /isco-disjunctbutnotco-alternateto$^.   7. {Cjd)'. p is co-disjunct and co-alternate to q,     p is co-implicant to q  p is super-implicant to q.  p is sub-implicant to q.   p is independent of q     p is sub-opponent to q  p is super-opponent to q,  p is co-opponent to q,   Here the symmetry indicated by the prefixes, co-,  super-, sub-, is brought out by reading downwards and  upwards to the middle line representing independence.  In this order the propositional forms range from the  supreme degree of consistency to the supreme degree  of opponency, as regards the relation of/ to ^. In tradi-  tional logic the seven forms of relation are known respec-  tively by the names equipollent, superaltern, subaltern,  independent, sub-contrary, contrary, contradictory. This  latter terminology, however, is properly used to express  the formal relations of implication and opposition,  whereas the terminology which I have adopted will apply  indifferently both for formal and for material relations.