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We shall deal with the validation of all these arguments further on. Meanwhile, the following
clarifications should also be kept in mind:
 The expression “all the more,” and others like it (such as “a fortiori,” “how much more,” and so
on), are often used in practice to signal an intention of a fortiori argument. This is useful
specifically when the argument is only partly explicit; but when the argument is fully explicit, as
shown above, such expression is in fact redundant, and (as we shall see) can even be misleading
(suggestive of ‘proportionality’). When the argument is stated in full, it is sufficient to say
“therefore” to signal the conclusion; nothing is added by saying “all the more.”
Incidentally, in practice people sometimes reserve “all the more” for argument that goes from
minor to major and “all the less” for argument that goes from major to minor; but it is also true
that the expression “all the more” (and others like it) is also often used indiscriminately, and this is
the way we usually intend it here.
 The four arguments function just as well if the major term is greater (in respect of the middle term)
than the minor term, or if they are equal. Whence, I have inserted in brackets in each mood: an “as
much as” alternative clause to “more than” in the major premise, and an “equally” alternative to
the traditional expression “all the more” in the conclusion. So though we have four figures, we
may say that they contain two moods each, a ‘superior’ and an ‘egalitarian’ one, making a total of
eight moods. Egalitarian a fortiori argument is also sometimes called ‘a pari’.
 Note that for subjectal moods, I have specified the major premise as “P is more R than Q (is R)” –
this is done to avoid confusion with a proposition of the form “P is more R than (P is) Q.” If we
try using the latter with “P is Q enough to be S” to conclude “P is R enough to be S,” we would
have an argument vaguely resembling a fortiori but which is in fact invalid . In the valid form, Rp
> Rq; whereas in the fake form Rp > Qp. Watch out for occurrences of this fallacy in common
The major premise of predicatal argument, i.e. “More R is required to be P than to be Q,” does not
have the same potential for ambiguity. Note, however, that it could alternatively be formulated as
“To be P requires more R than to be Q (requires R)” – in which form it might be confused with the
major premise of subjectal argument, viz. “What is P is more R than what is Q (is R).”
 The major premise may occasionally in practice be converted – i.e. it may be stated, in subjectal
argument, as “Q is less R than P” instead of as “P is more R than Q;” and in predicatal argument,
as “Less R is required to be Q than to be P” instead of as “More R is required to be P than to be
Q.” The validity of the argument in such cases is not affected, provided the minor premise and
conclusion remain the same. Note this proviso well. Very often, such conversion of the major
premise confuses people and they erroneously transpose the minor premise and conclusion .
Arguments involving such converted major premises, which may be labeled ‘inferior’, should not
be counted as distinct moods.
 In practice, the major premise is very often simply left out. The proponent of a given argument
may have it explicitly or tacitly in mind. But he may also be quite unaware of it, in which case it is
only we logicians who tell him it is logically present in the background and playing an active role
in the inference. This is not something peculiar to a fortiori argument, but is likewise often
encountered in syllogism and other forms of argument. It is called enthymemic argument (a mere
technical term); you can call it abridged or abbreviated argument, if you like.
 Concerning the minor premise and conclusion, the phrase “R enough to be” is often left out in
practice. This may occur with the major premise absent, so that the middle term (R) is completely
unstated (though of course still logically implicit); or it may occur with the major premise present,
in which case the mention of the middle term in it is deemed sufficient for the whole argument.

4 For example, “This screw is longer than it is wide; and it is wide enough to fit into this hole; therefore it
is long enough to do so.” Clearly, this would be fallacious reasoning; the conclusion does not follow from the
We might also put the major premise of subjectal argument in the form: “More R is found in P than in
Q.” However, the most natural form for the subjectal major premise is active and that for the predicatal major
premise is passive.
6 To avoid confusion always simply reflect on the question: which term ‘is more R’ or ‘requires more R’
than the other?
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