H. P. Grice, "The 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. 

J.L. 4 



50