Handbook of the History of Logic: - Fordham University Faculty

The Development of Supposition Theory in the Later 12 th through 14 th Centuries 169
Sherwood’s Equipollences: 13 Replacing a phrase below by another
on the same line yields a logically equivalent proposition: 14
every A no A not not some A not
no A not some A every A not
some A not no A not every A not
some A not not no A not not every A
An example is the equivalence of the old form ‘No A is B’ with the new form
‘Every A is not B’. It is also assumed that an unquantified predicate term ‘B’ is
equivalent to the particular ‘some B’, so the old form ‘Some A is B’ is equivalent
to ‘Some A is some B’, which is equipollent to ‘Some A is not no B’. Likewise,
‘No A is every B’ is equivalent to ‘Every A is not every B’ andalsoto‘Every A
someBisnot’.
Notice that application of any of these equipollences leaves an affirmative proposition
affirmative, and a negative proposition negative. (Suppose that ‘no’ and
‘not’ are (the only) negative signs. A proposition is affirmative if it contains no
negative signs or an even number of negative signs; otherwise it is negative.) They
thus preserve the principle that affirmatives with empty main terms are false, and
negatives with empty main terms are true.
Suppose that we take Aristotle’s original list of four (non-singular; non-indefinite)
categorical propositions and add the quantifier sign ‘every’ to the predicate. This
gives us a list of eight forms:
13This little chart is from William Sherwood I.19. In applications to categorical propositions
the actual form would have the normal Latin word order ‘not is’ instead of the English order ‘is
not’. It is possible to be precise about what propositions are included in this stock of what I am
calling basic categorical propositions. First, the following rules produce categorical propositions
whose copula occurs at the end of the proposition. (These are meaningful in Latin.):
1. A denoting phrase consists of a quantifier sign (‘every’, ‘some’, ‘a(n)’ or ‘no’) followed by
a common noun.
2. ‘is’ is a partial categorical proposition
3. If φ is a partial categorical proposition with zero or one denoting phrases in it, and if δ is
a denoting phrase, then δ followed by φ is a partial categorical proposition
4. If φ is a partial categorical proposition, so is ‘not φ’.
5. A categorical proposition is any partial categorical proposition with two denoting phrases.
Examples: ‘is’ ⇒ ‘no donkey is’ ⇒ ‘some animal no donkey is’.
‘is’ ⇒ ‘some donkey is’ ⇒ ‘not some donkey is’ ⇒ ‘an animal not some donkey is’
To put the verb in its more natural order in the middle:
If φ is a categorical proposition which ends with ‘is’, if there is a denoting phrase
immediately to its left, then they may be permuted
Example: ‘some animal not no donkey is’ ⇒ ‘some animal not is no donkey’
(For English readers, change the Latin word order ‘not is’ to ‘isn’t’: ⇒ ‘some animal isn’t no
donkey.)
14In Latin the negation naturally precedes the verb. For applying these equipollences the
negation coming before the verb can be treated as if it came after, so that ‘No A not is some
B’ can be treated as ‘No A is-not some B’ ≈ ‘No A is no B’. A similar provision applies to
English, where negation follows the copula ‘is’.