A FORTIORI LOGIC
CHAPTER 4 –
APPARENTLY VARIANT FORMS
1. Variations in form and content
2. Logical-epistemic a fortiori
The four copulative and four implicational moods of a
fortiori argument described earlier should be viewed as representative of
this form of argument, but obviously not as limiting its precise possible
contents. They are theoretical models, by way of which we can test whether
cases encountered in practice are ‘true to form’, i.e. valid, or not so. For
in both types we are concerned with a broader range of propositional forms
than may appear at first sight. We shall here describe in some detail some
of the variations on the two theoretical themes that we may encounter in
practice, and then we will enquire as to whether or when mixtures of them
are conceivable[1].
1.
Variations in form and content
In copulative arguments.
I have called the first four moods ‘copulative’ because they involve
categorical relations indicated by the copula ‘is’ (or ‘to be’). But it
should be clear that they could equally well involve other categorical
relations; also, negative polarity may be involved and non-actual modalities
(can, must, and different probabilities in between) of various modes (de
dicto or various types of de re). To give an example: “If this
man can run two miles so fast, he can surely run one mile just as fast”
(positive predicatal) may be counted as a copulative argument; the effective
copula (the relation between terms) here is ‘run’ and the modality
(qualifying the relation) is ‘can’, the terms being ‘this man’ and ‘a
distance of one or two miles in a given lapse of time’. Moreover, past,
present or future tenses may be involved, in various combinations, provided
the major premise justifies it. For example: “If a man is that strong when
old, he was surely as strong or stronger when younger.”
The verb typically used to relate subjects and
predicates in copulative a fortiori arguments is “is” or “to be.” This can
be taken very broadly to refer to any classification. But in practice, most
verbs can be used here: to have (some quality or entity), to do (some action
or go through some process), or whatever, with any object or complement,
provided the statement can credibly be recast in the standard form (this
process is called permutation[2]).
This is true not only of a fortiori argument, but equally of syllogism and
other forms of argument; so it requires no special dispensation. For
example: “she sings Mozart well” can be recast as “she is a [good Mozart
singer].” Sometimes, permutation is formally not possible, or at least not
without careful consideration; for instances, the relations of ‘becoming’
and ‘making’ (or ‘causing’) cannot always be permuted.
One or more of the verbs involved may be of negative
polarity. Be especially careful when one term is negated and another is
posited, for this can confuse. In such cases, i.e. when in doubt, we can
ensure that the argument is true to form (i.e. valid) by obverting the
predicate(s) concerned. Obversion is permutation of the negation, passing it
over from the copula to the predicate. For examples: instead of “is required
not to be P” we would read “is required to be nonP;” or instead of “enough
not to be S,” read “enough to be nonS.” The middle term (R) may likewise be
negative in form, provided it is consistently so throughout the argument.
However, if the major premise is negative, as in “P is not more R than Q” or
“More R is not required to be P than to be Q,” no such obversion of R is
acceptable, although we may be able to convert the comparison involved from
“more” to “less” (though in such case check carefully that the minor premise
and conclusion are true to form).
Natural, temporal or spatial modality may be introduced
in a fortiori argument; i.e. the predications involved may be modal and not
merely actual. For example, in the major premise “More R is required to be
able to be P than to be able to be Q” (and similarly in the minor premise
and conclusion). In such case, I would say that the modality has to be
looked on as part of the effective term. In our example, the effective major
and minor terms are not really P and Q, but “able to be P” and “able to be
Q.” That is, here too permutation of sorts is used to verify that the
inference is true to form. So the natural modality is operative here rather
as in extensional conditional propositions, than as in categoricals[3].
Logical and epistemic modalities, as well as ethical
and legal modalities, are considered separately further on.
In implicational arguments.
Similarly, though I have called the second set of four moods
‘implicational’, the relation ‘implies’ involved in them should not be taken
in a limited sense, with reference only to logical implication. For it is
obvious that, if the moods are valid for that mode, similar moods can be
constructed and validated for other modes of conditioning, such as the
extensional or the natural (to name two examples)[4].
Indeed, we can apply them more broadly still to a wide range causal
propositions (concerning causation, volition or influence, notably). Thus,
all sorts of relational expressions might appear in practice in lieu of
‘implies’ provided such a link is ultimately subsumed.
In implicational a fortiori argument, the items P, Q, R
and S stand for theses instead of terms. It is clear that any categorical
relation may be involved in these clauses – whether the copula ‘is’ or any
other, whether positive or negative, whether actual or modal – and indeed
ultimately any non-categorical relation. The proviso is that the claimed
relations between the clauses and the middle thesis R be indeed applicable
(which is not always the case, of course). The logical (‘de dicto’) relation
of implication is the basic bond in such argument, but this may be replaced
by any ‘de re’ relation that suggests it – such as natural, temporal,
spatial or extensional modes of conditioning, or more broadly by the same
various modes of causation (including logical causation, of course), and
more broadly still (though in such cases the underlying bond becomes more
tenuous, a probability rather than a certainty of sequence) by volition or
influence.
The following is a sample of thoroughly causal a
fortiori (positive antecedental). The important thing to realize here is
that the a fortiori argument per se has nothing to do with causality. It
takes the truth of the premises for granted and merely tells us the
conclusion from them, on the basis of given quantitative relations to the
middle term or thesis. It is an a fortiori argument, and not a causal
argument.
P causes more R than Q does, |
and, Q causes enough R to cause S; |
therefore, P causes enough R to cause S. |
To give an example: “The car’s good looks generate more
sales than its technical features do; and, its technical features generate
enough sales to keep the company afloat; therefore, its good looks generate
enough sales to keep the company afloat.” Here, the causal relations of
generation and maintenance (keeping) replace the logical relation of
implication.
Transformations. We may
also note in this context that often (though not always) the same a fortiori
argument can at will be credibly worded either in copulative form or in
implicational form. If intelligently articulated, such transformations do
not vitiate the argument. Consider for instance the following argument:
A being C implies more E in it than B |
and, B being D implies enough E in it to imply |
therefore, A being C implies enough E in it to |
Here, we have two subjects A and B (which may be the
same subject, in some cases) with four different predicates C, D, E, F,
brought together in truly implicational form. Notice that the middle term is
“E in it” – i.e. it refers the predicate E to a corresponding
subject, and not to just-any subject. In other words, it signifies the
effective middle thesis to be variously “B is E” or “A is E,” as the case
may be. The argument can obviously be restated in truly copulative form, as
follows:
AC is more E than a BD is, |
and, BD is E enough to be F; |
therefore, AC is enough E to be F. |
The terms AC and BD refer respectively to “A when it is
C” and “B when it is D.” This is valid transformation, provided the middle
item E suggests a thesis in the implicational form (as above clarified) and
a term in the copulative form. As we saw earlier, a middle thesis per
se, being a proposition, cannot vary; so that when we say that more or less
of it is implied, we always have in mind something within it that
varies – usually a term (though not always). So, when we transform the
implicational form into a copulative one, we have to identify precisely
which content of the middle thesis to use as our middle term. We could
similarly, of course, transform the copulative argument into the
implicational one, if we proceed carefully.
In practice, it does not matter so much exactly how we
word our argument, in implicational or copulative form, provided it ends up
matching a valid form. The human brain is very clever and able to assimilate
large variations in wording with little difficulty (though it can also, of
course, be misled). So, we should not view the division too rigidly.
However, such transformations are not always possible: we may have
difficulty restructuring the middle item, or at least some information might
be lost or might have to be added in the process. So we are justified in
regarding copulative and implicational species of a fortiori argument as
essentially distinct, even if in some special cases they can be transformed
into each other.
2.
Logical-epistemic a fortiori
We need to distinguish between purely ‘ontical’ (or
de re) a fortiori argument and more ‘logical-epistemic’ (or de dicto)
ones. The adjective ‘ontical’[5]
(from Gk. ontos, meaning ‘existence’) applies to the objects of ontology,
the study of being, just as the adjective ‘epistemic’ (from Gk. episteme,
meaning ‘knowledge’) applies to the objects of epistemology, the study of
knowing. Ontical thus characterizes the things we allegedly know, whereas
epistemic characterizes our alleged knowledge of them. Clearly, these terms
are relative, in that something epistemic may be intended ontically.
A logical-epistemic a fortiori argument is one applying
logical and/or epistemic qualifications to some relatively ontical
information. A ‘logical’ qualification logically evaluates the proposed
information in a given context of knowledge: it may logically evaluate a
term as conceivable, significant, clear, precise, well-defined, and so
forth, to various degrees, or the same in negative connotation; or it may
evaluate a proposition through a modality with degrees[6]
like probable, confirmed, evident, consistent, true, or their negations. An
‘epistemic’ qualification concerns the state of belief, opinion or knowledge
of the speaker rather than the content spoken of or its purely logical
evaluation; this refers to characterizations like credible, reliable,
believable, understandable, to varying degrees, and their negative
equivalents.
The distinction can be tested as follows: E.g. for
‘credible’ when we ask ‘to whom?’, we can answer ‘to this person’, or ‘to
most people’, or ‘to everyone’, signifying that the issue is relatively
subjective; whereas for ‘probable’, we would refer to a more objective
issue, such as how often similar subjects have the same predicate.
A logical or epistemic thesis, then, is one which
predicates such a logical or epistemic term to an ontical term or thesis.
E.g. ‘term X is vague’, ‘thesis X is probable’ are logical propositions,
‘term X is generally understood’, ‘thesis X is widely believed’ are
epistemic propositions. Of course, logical and epistemic propositions are in
a sense themselves ontical; but they are always relative to information
which is more ontical.
The following are examples of purely copulative
logical-epistemic a fortiori argument, the first being subjectal and the
second predicatal:
Term P is ‘better defined’ (R) than term Q is, |
and, term Q is well defined (R) enough to be |
all the more, term P is well defined (R) enough |
‘Better definition’ (R) is required of a term |
and, term S is well defined (R) enough to |
all the more, term S is well defined (R) enough |
Note that both samples involve only terms (i.e. they
are not hybrid) and both have as their middle term R the logical
qualification ‘well defined’. In the subjectal example, R characterizes the
terms P and Q, whereas in the predicatal example it characterizes the term
S. In the subjectal example, the subsidiary term S is ‘comprehensible’, an
epistemic qualification suitably related to R, and its major and minor terms
P and Q are ontical (at least, relative to the two other terms). In the
predicatal example, the major and minor terms P and Q are logical (‘pinpoint
its instances’) or epistemic (‘comprehensible’) qualifications suitably
related to R, while the subsidiary term S is (at least relatively) ontical.
The following are examples of purely implicational
logical-epistemic a fortiori argument, the first being antecedental and the
second consequental:
Thesis P implies more ‘correct predictions’ (R) |
and, Q implies correct predictions (R) enough |
all the more, P implies correct predictions (R) |
More ‘correct predictions’ (R) are required to |
and, thesis S implies correct predictions (R) |
all the more, S implies correct predictions (R) |
Note that both samples involve only theses (i.e. they
are not hybrid) and both have as their middle thesis R the logical
proposition that ‘many of its predictions are correct’[7].
In the antecedental example, R characterizes the theses P and Q, whereas in
the consequental example it characterizes the thesis S. In the antecedental
example, the subsidiary thesis S is the logical proposition, suitably
related to R, that ‘thesis A is probably true’, while the theses P and Q are
(at least relatively) ontical. Here, the probability of Q due to correct
prediction is declared in the minor premise high enough to imply A probable;
therefore, given the major premise, the same can be concluded with regard to
P and A. In the consequental example, the theses P and Q are the logical
propositions, suitably related to R, that ‘thesis A is probably true’ and
‘thesis B is probably true’, respectively, while the subsidiary thesis S is
(at least relatively) ontical. Here, the probability of S due to correct
prediction is declared in the minor premise high enough to imply A probable;
therefore, given the major premise, the same can be concluded with regard to
S and B.
Though all the above examples are positive, we can
easily construct similar arguments in negative form. In all of them, the
logical-epistemic middle item (R) may be viewed as the basis of the
deduction, and the suitably related logical-epistemic subsidiary item (S) or
major and minor items (P and Q) may be viewed as the goal of the deduction;
the remaining item(s) usually have ontical content, though they may in
special cases (when that is what is discussed) be logical or epistemic too.[8]
These four samples make clear that logical-epistemic a
fortiori arguments function like purely ontical a fortiori argument; there
is nothing special about them, other than the logical-epistemic nature of
some of the items involved. Nevertheless, such arguments seem rare; or at
least, I find it difficult to formulate many examples of them. The matter
gets more complicated when we, further on, look into ‘hybrid’ a fortiori
arguments, which seem to involve mixtures of terms and theses.
3.
Ethical-legal a fortiori
In my book Judaic Logic[9],
I showed that, although the eight moods of a fortiori argument listed
earlier are formulated very generically, they can be adapted to ethical or
legal a fortiori argumentation. Generally, the middle item R may be any
quantitative factor shared in some way by the other three items. In ethical
or legal argument, this common thread will be specifically an ethical/legal
characterization, or a proposition involving such characterization, by which
I mean expressions like desirable, advantageous, useful, valuable, good,
moral, ethical, legal, obligatory, demanding, important, stringent, and so
on – and their negative versions – all of which, note well, have degrees.
Coupled with that, either the subsidiary item or the major and minor items
must refer to a physical, mental or spiritual action or event related
to the ethical-legal qualification; for examples, as something desirable is
sought after, or something good is preferred. The remaining
item(s) are not ethical-legal in content.
A fortiori arguments involving such ethical or legal
expressions must be examined and evaluated carefully, because these
characterizations are rather vague and complex. One can easily err using
them if one does not take pains to clarify just what they are intended to
mean in each case. Consider, for instance, the following subjectal argument:
P is more valuable (R) than Q, |
and, Q is valuable (R) enough to make A |
therefore, all the more, P is valuable (R) |
This argument can be interpreted and rewritten as
follows:
·
Major premise means: ‘P does more to produce some value R than
Q does’,
which in turn means:
‘P produces R to degree R_{P}’,
and ‘Q produces R to degree R_{Q}’, and
‘R_{P} is greater
than R_{Q}’ – whence, ‘if R_{P} then R_{Q}’.
·
Minor premise means: ‘Q produces R to degree R_{Q}’,
and
‘if R_{Q} then S (=
the term ‘makes A is imperative’)’.
·
Conclusion means: ‘P produces R to degree R_{P}’
(given), and
‘if R_{P} then S’
(since R_{P} implies R_{Q}, and R_{Q} implies S).
Alternatively, it might be read and rendered
negatively, as follows:
·
Major premise means: ‘nonP does more to inhibit some value R
than nonQ does’,
which in turn means:
‘nonP inhibits R to degree
nonR_{nonP}’, and ‘nonQ inhibits R to degree nonR_{nonQ}’ ,
and
‘nonR_{nonP} is
greater than nonR_{nonQ}’ – whence, ‘if nonR_{nonP} then
nonR_{nonQ}’.
·
Minor premise means: ‘nonQ inhibits R to degree nonR_{nonQ}’,
and
‘if nonR_{nonQ} then
S (= the term ‘makes A is imperative’)’.
·
Conclusion means: ‘nonP inhibits R to degree nonR_{nonP}’
(given), and
‘if nonR_{nonP} then
S’ (since nonR_{P} implies nonR_{Q}, and nonR_{Q}
implies S).
Sometimes, both these interpretations are intended
together. P and Q are two values; and S is some trait or behavior that is
being recommended, say. The important factor here is of course the middle
term R, which is implicit in the expression ‘valuable’. What does it mean to
be more or less valuable, or valuable enough? This has to refer to some
causal concept – namely, the positive concept of production and/or the
negative concept of inhibition. Where did R come from? It is implicit in the
concept of value that something is valuable relative to some standard of
value – call it R. So ‘valuable’ means valuable in the pursuit of (say) R.
What does ‘makes A is imperative’ (S) mean? It means
that A is absolutely necessary for some unstated goal – or more probably for
the ultimate goal here sought, namely R. However, note well, the necessity
of A here referred to does not play any part in the actual a fortiori
inference. The subsidiary item here is really not just A but the whole
clause S (i.e. ‘makes A is imperative’). Another such term like ‘makes A
allowed’ or even ‘makes A not imperative’ or ‘makes A forbidden’ could
equally well have occurred in that position without affecting the argument
as a whole. Clearly, then, the conclusion can be formally inferred from the
given premises, so the a fortiori argument as a whole is valid.
Of course, many questions can be asked about how we
come to know the premises in the first place. The hierarchy of values P and
Q proposed in the major premise has to be justified; and why the minor value
Q implies the imperativeness (or whatever) of ‘A’ is not here explained (but
taken for granted at the outset). The scale of values on which P and Q are
measured could be a merely subjective scale, or one based on biological
considerations, or again one based on spiritual ones. ‘A’ might for instance
be a cause of Q, P and/or R, though need not be. But these issues stand
outside the a fortiori reasoning as such. The a fortiori argument as such
does not need more information than the said premises give to draw the said
conclusion – provided that the message of each premise and of the conclusion
are well understood.
Let’s look at another sample, for instance the
predicatal argument:
More ‘virtue’ (R) is required to be (or have or |
and, S is virtuous enough to be (or have or do) |
therefore, all the more, S is virtuous enough |
In this case, S refers to a person supposedly, and P
and Q to character traits, or maybe behavior patterns, which require
different degrees of ‘virtue’ (by S) to achieve. Here, the middle term
‘virtue’ has to be understood in a sufficiently uniform manner that the
inference becomes possible. Obviously, if it means something different in
each proposition – say, courage in one and perseverance in another – we
cannot logically draw the conclusion from the premises. Here again, then,
caution is called for.
Apart from these words of warning, much the same can be
said for ethical-legal a fortiori as was said regarding logical-epistemic a
fortiori, so I won’t repeat myself here.
4.
There are no really hybrid forms
I have already shown that my inventory of copulative
and implicational a fortiori arguments is in principle exhaustive[10],
i.e. that ‘hybrid’ arguments are formally non-existent even if we often in
everyday discourse seem to make use of them. The main reason given was that
a standalone term cannot imply or be implied by a whole proposition. Terms
can only be subjects or predicates; only theses can be antecedents or
consequents.
This is true notwithstanding the fact, which we
admitted, that since a thesis as such cannot have degrees like a term, the
middle thesis of implicational arguments must be examined carefully, to
determine what it is in it that is variable (i.e. more, equal or
less, or sufficient or insufficient). The variable factor may be a subject
or a predicate or a quantity or a modality, or a compound of such elements.
Thus, we can safely say that, formally speaking, there
are no hybrid a fortiori argument. There are in principle no partly
copulative and partly implicational a fortiori arguments. The four items P,
Q, R, S of such arguments are necessarily either all terms (i.e. the main
constituents of propositions) or all theses (i.e. propositions of whatever
form, constituted by terms). Even if in everyday speech we often give the
impression that terms and theses can be mixed indiscriminately, there is
always some unspoken intent that explains the illusion. Some commentators
have nevertheless tried, wittingly or unwittingly, to propose hybrid forms
like the following:
P is more R than Q is, |
and, Q is R enough to imply S; |
therefore, P is R enough to imply S. |
In the above ‘mostly subjectal’ example, S seems to be
a consequent of Q and P, although they seem to be subjects of predicate R.
The solution may be that S is in fact a term, and what is thought of as
implied is the thesis ‘it (i.e. the subject Q or P, as appropriate) is S’.
Alternatively, if S is in fact a thesis, it contains ‘it’ (which refers to Q
or P, as appropriate) as subject and some additional term (here tacit) as
predicate.
More R is required to be P than to be Q, |
and, S implies R enough to be P; |
therefore, S implies R enough to be Q. |
In the above ‘mostly predicatal’ example, S is both
antecedent and subject, since it both implies R and is P and Q. Here, the
solution may be that R is in fact a term, and by ‘S implies R’ is meant
simply ‘S is R’. Alternatively, if R is in fact a thesis, the thought may be
that some proposition of which S is the subject (and whose predicate is here
tacit) implies R.
P implies more R than Q (implies R), |
and, Q implies R enough to be S; |
therefore, P implies R enough to be S. |
In the above ‘mostly antecedental’ example, P and Q
seem to be both antecedents and subjects, since they both imply R and are S.
The solution here may be that P, Q and R are indeed theses, and ‘to be S’ is
intended to mean ‘to imply it (i.e. the subject, here tacit, of thesis Q or
P, as appropriate) to be S’.
More R is required to imply P than to imply Q, |
and, S is R enough to imply P; |
therefore, S is R enough to imply Q. |
In the above ‘mostly consequental’ example, S is both
subject and antecedent, since it both is R and implies P and Q. Here, the
solution may be that R is in fact a thesis, and by ‘S is R’ is meant ‘S
implies R’; or maybe, ‘the subject (here tacit) of S has the predicate given
(here tacitly) in R’. Alternatively, if R is in fact a term, ‘S is R’ might
signify ‘the subject (here tacit) of S is R’.
On the surface, the above four examples may seem
conceivable, because we are dealing in symbols. But if we examine
them more closely we find that appearance misleading. For it is a rule of
logic that the same item cannot at once be a term and a thesis, as occurs in
all of the above proposed moods. So these hybrids are not valid forms,
strictly speaking. In each of them, some intent has been left tacit or some
verbal or conceptual confusion occurred in the formulation. Nevertheless, it
should be kept in mind that in practice we often do so word our sentences as
to give the impression that we are mixing copulative and implicational
clauses. This is occasionally confusing, but not always.
Let us analyze some more
specific cases where confusion or doubt might occur in practice. These are
mostly logical-epistemic or ethical-legal arguments that look partly
implicational but are in fact wholly copulative. The reason such
hybrid-looking arguments arise is that in them a thesis may actually
function as (a) a subject-term or (b) a predicate-term.
(a) In the propositions “X is probable” or “X is
desirable,” where ‘X’ is a thesis, say ‘that A is B’, and ‘probable’ or
‘desirable’ is a predicate, thesis ‘X’ may be said to function effectively
as a term (a subject), because it is taken as a unitary whole rather than as
composed of parts.
For example, consider the a fortiori argument “Given
that ‘A is B’ is more probable than that ‘C is D’, it follows that if ‘C is
D’ is probable enough to be relied on, then ‘A is B’ is probable enough to
be relied on.” We might here think that since ‘A is B’ and ‘C is D’ are
theses (the major and minor, respectively), the argument is implicational.
On the other hand, since ‘probable’ and ‘relied on’ are terms (the middle
and subsidiary, respectively), the argument seems copulative. The solution
is not that the argument is hybrid, but that the major and minor theses are
in this context intended as terms – i.e. they are the subjects for which the
middle and subsidiary terms are predicates. Thus, the form of the argument
is really subjectal, and not antecedental or hybrid.
The following is an example of predicatal form with
similar effect. “More satisfaction of inductive criteria (R) is needed to
adopt a thesis (P) than to merely conceive it possible (Q); and, thesis S
satisfies inductive criteria (R) enough to be adopted (P); therefore, thesis
S satisfies inductive criteria (R) enough to be conceivable (Q).” Here,
although S is a thesis (say, ‘that A is B’), it functions in the present
context as a term (a subject), for which R, Q and P are indeed predicates.
So, the form of the argument is really predicatal, and not consequental or
hybrid.
(b) Again, looking the propositions “X makes Y
probable” or “X makes Y desirable,” where ‘X’ is a term, and ‘Y’ is a term
or a thesis, say ‘that A is B’, and ‘probable’ or ‘desirable’ is a
predicate, we are tempted to view the relation ‘makes’ as equivalent to an
implication (which it indeed implies) and the combination ‘Y is probable’ or
‘Y is desirable’ as an implied thesis, in which case the given proposition
as a whole seems to be implicational. However, because X is a subject-term
(noun), we have to look upon ‘makes’ as a mere copula (verb) and upon the
thesis made, i.e. ‘Y is probable’ or ‘Y is desirable’, as a predicate-term
(object).
An example of this would be the following argument:
“Term P is more well-defined (R) than term Q; and, term Q is well-defined
(R) enough to ‘make term or thesis A conceivable or credible’ (S);
therefore, term P is well-defined (R) enough to ‘make term or thesis A
conceivable or credible’ (S).” This argument might be interpreted as partly
copulative (since P, Q, and R are terms) and partly implicational (since S
seems to refer to an implication, i.e. a thesis). But in fact it is wholly
copulative, because S is a term, i.e. the clause ‘makes term or thesis A
conceivable or credible’ must be taken as a unit and not be cut up. This
example is thus subjectal.
A similar predicatal example would be the following:
“More precision of definition (R) is required to ‘make term or thesis A
comprehensible’ (P) than to ‘make term or thesis B comprehensible’ (Q); and,
term S is precisely defined (R) enough to make A comprehensible (P);
therefore, term S is precisely defined (R) enough to make B comprehensible
(Q).” Here, the argument might be interpreted as partly copulative (since S
and R are terms) and partly implicational (since P and Q seem to refer to
implications, i.e. theses). But in fact it is wholly copulative, because P
and Q are terms, i.e. the clauses ‘make term or thesis A/B comprehensible’
must be taken as units and not be cut up.
All the above examples involve logical-epistemic
qualifications. We can similarly construct hybrid-looking arguments with
ethical-legal qualifications. E.g. “That ‘A be B’ (P) is more desirable (R)
than that ‘C be D’ (Q); and, Q is desirable (R) enough to be pursued
regularly (S); therefore, P is desirable (R) enough to be pursued regularly
(S).”
In conclusion, hybrid a fortiori argument do not really
exist: when they do seem to occur, as they often enough do in
logical-epistemic or ethical-legal contexts, it is due to some thesis being
taken as a whole, i.e. as effectively a term.
5.
Probable inferences
Very often in practice, though the given argument
somehow seems to be an a fortiori, it is really not one at all. We may upon
closer scrutiny decide that it is more precisely a hypothetical syllogism or
an apodosis. Very often we are misled by expressions like ‘all the more’
indicative of a fortiori argument being inappropriately used in other forms
of argument. Inversely, an argument may on the surface not look like
an a fortiori at all, but really be one deeper down. Caution is always
called for in interpreting arguments. We have to ask what form the
underlying reasoning takes, irrespective of the wording used. In some cases,
of course, no reasoning is at all intended; yet some people might assume an
a fortiori argument to be intended, because a comparison or a threshold is
mentioned. We have to always ask how the speaker intends his statement to be
taken.
As just stated, some arguments do not immediately
appear to be in standard a fortiori format, although one senses that there
is an a fortiori ‘flavor’ to them. Consider the following arguments: Are
these a fortiori in nature or something else? How are they to be validated?
Copulative form (X, Y, Z are terms):
X more often occurs in Y than in Z; therefore: |
If X is found in Z, it is probably also in Y |
If X is not found in Y, it is probably also not |
Implicational form (X, Y, Z are theses):
X more often occurs in conjunction with Y than |
If X is found in conjunction with Z, it is |
If X is not found in conjunction with Y, it is |
These closely resemble a fortiori argumentation. There
are copulative and implicational forms (four in all), the former involving
terms and the latter theses. In each case, the first proposition is the
major premise, and the if–then propositions which follow it contain a minor
premise (the antecedent) and a conclusion (the consequent). There is a
positive and a negative mood, the positive one being minor to major and the
negative one major to minor. However, these arguments as they stand are
obviously not in standard form. They need to be reformulated to conform.
If such argument is to be viewed as a variant of a
fortiori, the middle term has to be “the probability of occurrence,” while
the subsidiary term has to be “the actuality of occurrence.” The major
premise, which tells us that “X is in/with Y” occurs more frequently than “X
is in/with Z,” means that the former is more probable than the latter. The
minor premise, which tells us that “X is in/with Z” has occurred, or that “X
is in/with Y” has not occurred, refers to the actuality of occurrence or
lack of it. And the conclusion predicts that “X is in/with Y” has
probably also occurred, or respectively that “X is in/with Z” has probably
also not occurred, again with reference to the actuality or inactuality of
occurrence. We can thus reformulate the arguments as follows to bring out
their ‘a fortiori’ aspect more clearly:
Positive mood (copulative [in] or implicational
[with]):
‘X is in/with Y’ is more probable than ‘X is |
‘X is in/with Z’ was probable enough to |
therefore: ‘X is in/with Y’ is probable enough |
Negative mood (copulative [in] or implicational
[with]):
‘X is in/with Y’ is more probable than ‘X is |
‘X is in/with Y’ was not probable enough to |
therefore: ‘X is in/with Z’ is not probable |
Note the introduction, in this improved formulation, of
the crucial notion of sufficiency (“enough”) or its absence, in accord with
standard a fortiori format. So we can say that we here indeed have a
fortiori arguments. The major and minor items P and Q are in this case the
theses “X is in/with Y” and “X is in/with Z,” respectively. The
middle and subsidiary items R and S are the terms “probably” and
“actually occurs.” So the a fortiori argument involved, mixing theses and
terms, is a hybrid-seeming one (although strictly-speaking it is wholly
copulative, the theses in it being taken as terms). It is a logical a
fortiori argument, probability and actuality being modalities.
Note well that the prediction of the conclusion should
not be taken as a certainty. The argument makes no pretense to yield
anything more than a probable conclusion, the degree of probability being
that specified – clearly or vaguely – in the major premise. Though presented
as a sort of deduction, the argument is essentially inductive. It could well
be that the situation in fact, on the ground, is the opposite of what the
argument predicts. Nevertheless, if the only information we have at
our disposal is that given in the argument, it is reasonable to adopt the
conclusion’s prediction as our ‘best bet’. We have more rational basis for
expecting the outcome that the conclusion predicts than we have for
expecting the contradictory outcome.
Certainty from mere
probability. I would like to draw attention in the present context to
the fallacy inherent in certain probabilistic a fortiori arguments, namely
those that seem to infer a certainty from a mere probability. The following
two arguments, one positive and one negative, illustrate this pitfall:
Thesis P is more probable (R) than thesis Q, |
and, thesis Q is probable (R) enough to imply |
therefore, thesis P is probable (R) enough to |
Thesis P is more probable (R) than thesis Q, |
and, thesis Q is probable (R) enough to deny |
therefore, thesis P is probable (R) enough to |
In these hybrid-looking arguments, the items P, Q and S
are theses and R is a logical-epistemic term (it is logical if ‘probable’
here means ‘demonstrably likely to be true’, but epistemic if it merely means
‘believed by many people’). As we have seen, this apparent
mix is not necessarily a problem, because theses may in such contexts be
intended as (i.e. effectively function as) terms. However, in these two
particular cases, the mix is a problem, because the subsidiary item (S) is
definitely implied (or denied, i.e. its negation is implied). Since the
implication (or denial) is quite intentional, it cannot be written-off as a
badly-worded predication.
At first sight, the proposed argument may seem
meaningful and credible; but upon closer scrutiny it is found fallacious.
The main reason why it is fallacious is that in logic theory no propositions
literally imply others when they (the implying ones) are more or less
probable. In deductive logic, either a proposition X (Q or P in our
example) implies another Y (S or notS, here) or it does not – there is no
such thing as X implying Y if X is probable to a sufficient degree, and X
not implying Y if X is not probable to that degree. Even in inductive logic,
such a concept is unknown – there is only the concept of transmission of
probability, i.e. if X implies Y, then increasing the probability of X being
true increases that of Y being true.
As for degrees of implication, they are formally
conceivable; but given that ‘X probably implies Y’, it does not follow that
‘if X is probable to some high degree it implies Y to be certain’.
Rather, probable implication is to be treated as a weakened form of
implication, meaning that whereas the form ‘X fully implies Y’ transmits the
high probability of X to Y (and in the limiting case, if X is certain, then
Y is also certain), the form ‘X only probably implies Y’ transmits only a
fraction of X’s probability on to Y (i.e. here, if X is probable, then Y is
‘probably probable’). This can be expressed quantitatively: if X implies Y
with probability m%, say; and X is itself only probable to degree
n%, say; then the resulting probability of Y is only m% of n%.[11]
It should be added that it makes no difference whether
the hybrid-seeming a fortiori argument involves an implication or a
denial. It is fallacious either way. The principle of adduction that “no
amount of right prediction ever definitely proves a hypothesis, but all it
takes is a single wrong prediction to disprove it” has no relevance in the
present context. Here, whether S is implied or denied the argument is
invalid, because a probability cannot imply a certainty, whether positive or
negative.
The lesson these examples teach us is that if we use a
logical-epistemic middle term like ‘probable’, then we must also have a
logical-epistemic term like ‘reliable’ contained in one or more of the other
items of the a fortiori argument. Such terms occur together, not by chance
but because their meanings have some rational relation (as probability
rating is related to reliability). We cannot combine the middle term
‘probable’ with an assertion of the subsidiary item’s implication or denial.
There is in fact no logical discourse corresponding to that schema. It is
artificial and conceptually faulty, for the reason already stated that a
certainty cannot be implied by a mere probability.
6.
Correlating ontical and probabilistic forms
Having examined the general forms of ontical a fortiori
argument and various cases of more specifically logical-epistemic a fortiori
argument, the question arises: can logical-epistemic arguments be
constructed from ontical ones and/or vice versa? This question immediately
comes to mind when we read Aristotle’s descriptions of a fortiori argument,
of which the following are some extracts:
Rhetoric, book II, chapter 23:
“…if a quality does not in
fact exist where it is more likely to exist, it clearly does not
exist where it is less likely. Again, … if the less likely thing is
true, the more likely thing is true also.”
Topics, book II, chapter 10:
“If one predicate be
attributed to two subjects; then supposing it does not belong to the subject
to which it is the more likely to belong, neither does it belong where it is
less likely to belong; while if it does belong where it is less likely to
belong, then it belongs as well where it is more likely.”
Here, Aristotle’s emphasis is clearly
‘epistemological’, since he repeatedly uses the word ‘likely’ as his middle
term, yet judging by the examples he there gives the underlying
subject-matter is arguably rather ‘ontological’. This suggests that there
are natural bridges between the ontical and logical-epistemic expressions of
a fortiori argument. Let us look into the matter with reference to one of
Aristotle’s own examples, namely:
A man is more likely to strike his neighbors |
if a man strikes his father, |
then he is likely to strike his neighbors too. |
This example is clearly intended as logical-epistemic,
since it uses the relative likelihood of events to achieve its
inference. But one senses that underlying it is another, more ontical
argument, such as the following (others could of course be suggested):
More antisocial attitude is required to strike |
if a man is antisocial enough to strike his |
then he is antisocial enough to strike his |
Aristotle’s logical-epistemic wording does not reveal
to us precisely why the concluding event (man striking neighbors) is more
likely than the given event (man striking father), whereas my proposed
ontical wording attempts to explain these events and their connection
through some psychological attribute (being antisocial) of the subject (a
man). Aristotle’s effective middle term is a vague, unexplained
‘likelihood’ – whereas my ontical middle term (antisocial mentality) is more
specifically informative as to the causes (different degrees of
antisocial mentality) of the events (striking father or neighbor). One finds
Aristotle’s argument convincing especially because one (consciously or
unconsciously) assumes that there are ontical reasons (such as those I
propose) behind the probabilities he declares.
Thus, our first question arises: can we always, in
formal terms, similarly infer an underlying ontical a fortiori argument from
a given logical-epistemic (probabilistic) one? The answer, I would say, is
that we cannot formally infer one, but we can hope to construct
one that would seemingly fit the bill, i.e. explain the predicated
likelihoods by means of some material property or properties. That is to
say, with reference to the following forms, given the one on the left we
may, using our knowledge and intelligence to propose an appropriate middle
term R, construct the one on the right:
Given probabilistic argument | Constructed ontical a fortiori argument |
‘S is P’ is more likely than ‘S is Q’: | More R is required to be P than to be Q; |
if ‘S is P’ occurs, | and, S is R enough to be P; |
then ‘S is Q’ is likely to occur too. | therefore, S is R enough to be Q. |
This reconstruction seems reasonable, at least where
the original middle term is the degree of ‘likelihood’. But let us look into
it more deeply. The given argument compares the likelihood of two events
(theses) ‘S is P’ and ‘S is Q’ and tells us that if the more likely one
indeed occurs then the less likely one is likely to occur too. Note well: it
gives no guarantees as to this outcome, its conclusion being only probable
though the minor premise is actual. Our proposed construct introduces a
new term R that was not given in the original argument. R serves as
middle term of our a fortiori, relative to which the predicates P and Q are
compared in the major premise. R is a predicate of S. If the magnitude of R
in S is large enough, then S is Q; and if its magnitude is even larger, then
S is P. Whence, if S is P, it has to be Q.
Note that the conclusion ‘S is Q’ here is definite – it
is not a mere probability as before. However, the proposed construct as a
whole certainly cannot be inferred from the given argument. We can only
posit our construct in the way of an inductive hypothesis that is hopefully
fitting (if we have thought about it sufficiently), but which may turn out
upon further experience and reflection to be inadequate (in which case it
must be adapted or abandoned). So our new conclusion is not as sure as it
appears. Still, once we have a seemingly credible construct, we can claim it
(on inductive, not deductive grounds – to repeat) to be the underlying
ontical explanation of the given logical-epistemic argument.
Can such ontical explanation be provided for all
logical-epistemic a fortiori arguments, or only for middle terms like
‘likelihood’? I would offhand answer yes, arguing that we never use
logical-epistemic characterizations entirely without reference to more
ontical characteristics. That is, if we ask ourselves why we think a term or
thesis deserves logical or epistemic evaluation X, we will argue the point
ultimately with reference to some sort of more ontical information. Of
course, we may have some such explanation in mind, but be unable to clearly
put it in so many words, so this is difficult to prove in every case. Also
of course, I am generalizing, since I cannot foresee all cases – so I may be
found wrong eventually.
Now, let us turn the initial question around, and ask
the reciprocal question: given an ontical a fortiori argument, can we
formally derive from it a corresponding logical-epistemic argument (meaning,
at least, a probabilistic argument similar to Aristotle’s)? And if so, we
might additionally ask, is there great utility in doing so – or is valuable
information lost in the process?
If we reflect a moment, it is clear that behind my
contention that underlying Aristotle’s probabilistic argument there must be
a more ontical argument that explains it – was the thought that Aristotle
was really thinking in terms of the ontical argument even if he only
verbalized a probabilistic one. So in fact the mental process we were
looking for was in the reverse direction: not from logical-epistemic to
ontical, but rather from ontical to logical-epistemic. We were not so much
asking what ontical information can be drawn from Aristotle’s probabilistic
argument (not a lot, as we have just seen), but what ontical argument
Aristotle had in mind even as he spoke in probabilistic terminology. We want
to retrace his thought process from the pre-verbal ontical thought to its
verbal probabilistic expression.
Consider therefore the following pair of arguments,
this time the one on the left being a given ontical a fortiori argument and
the one on the right a proposed probabilistic construct:
Given ontical a fortiori argument | Constructed probabilistic argument |
More R is required to be P than to be Q; | ‘S is P’ is more likely than ‘S is Q’: |
and, S is R enough to be P; | if ‘S is P’ occurs, |
therefore, S is R enough to be Q. | then ‘S is Q’ is likely to occur too. |
If we examine these arguments carefully, we see that
the latter cannot be inferred from the former. Of course, all information
concerning R is lost in transition. But moreover, we have no basis for
believing the major premise that ‘S is P’ is more likely than ‘S is Q’; for,
given that ‘More R is required to be P than to be Q’, it could still be true
that ‘S is P’ is less likely (i.e. occurs less frequently) than ‘S is
Q’. The two minor premises are in agreement that ‘S is P’; but, whereas the
conclusion of the given a fortiori is definite that ‘S is Q’, the conclusion
of the construct is that ‘S is Q’ is merely probable. Thus, not only does
the proposed construct’s major premise not follow from the given major
premise, but the conclusion of the construct is less informative and sure
than that of the original argument.
So, there is in fact no justification for supposing
that an ontical a fortiori argument gives rise to a probabilistic argument
as above proposed. The ontical argument does not formally tell us anything
about the likelihood of the events it discusses. If such likelihood is
asserted in an analogous probabilistic argument, it is new
information (just as the middle term R was new information, in the opposite
direction), which must be separately justified or admitted as a hypothesis
to be assessed inductively (e.g. we would have to ask in Aristotle’s above
example whether it is empirically true that men strike their neighbors more
often than they strike their fathers). Moreover, to repeat, the proposed new
argument involves loss of information (about R) and has a less certain
conclusion (about S being Q).
So, if we suppose that Aristotle really had an ontical
argument in mind when he formulated his probabilistic one, we may say that
such discourse on his part was inaccurate and wasteful. Conversely, granting
that he meant only what he said, we could read more into it provided we
realize that such interpretation on our part is not deductive inference but
inductive hypothesis. In short, the relation between ontical and
logical-epistemic a fortiori arguments can be described as hermeneutical
rather than strictly logical. We often in practice do blithely hop from
ontical to probabilistic form or vice versa – but we ought to be careful
doing so, because formal analysis shows that it is not always licit. In
logic, even the word ‘likely’ means something specific and cannot be used at
will.
[1] This is a question I have not (as I
recall) previously asked myself.
[2] See my Future Logic, chapter
18, on this topic.
[3] That is, although permutation of
modalities is not formally permissible, as the modality of a proposition
concerns its copula rather than its predicate (i.e. ‘can be [X]’ cannot
always be read as ‘is [able to be X]’ – to do so leads to errors of
inference), we may nevertheless conceivably do so, provided the
predication as a whole (i.e. copula plus predicate) is carried over, as
occurs in extensional conditioning.
[4] Again see Future Logic part
IV, concerning de re modes of conditioning.
[5] I personally prefer using the word
‘ontal’ (which I originally found used in the Enc. Brit.); but
‘ontic’ (used, e.g., by W. Windelband) and ‘ontical’ (commonly found in
the Internet) seem more widely used; so, I have here opted for the
latter as a compromise.
[6] Note well this point – modalities
like necessity or possibility do not strictly-speaking have degrees.
When we assign them degrees we really refer to high or low
probabilities. Necessity refers to maximum (100%) probability, while
possibility refers to some unspecified (from >0 to100%) probability.
[7] Though we state the middle item more
briefly as a term, ‘correct predictions’, because this term is the
essential variable in it, it is better to think of it as a proposition,
‘many of its predictions are correct’, so as to avoid making the a
fortiori argument ‘hybrid’ in form. Notice also that the propositional
form brings out more clearly that the predictions are made by a specific
thesis (P or Q in the antecedental case and S in the consequental case).
[8] Observe that logical-epistemic terms
in this context come in ‘suitably related’ pairs, as e.g. ‘better
defined’ comes with ‘more comprehensible’. This does not mean, of
course, that each term can only be paired off with one other term – it
may well have many possible companions. But it does mean that not just
any two such terms may be paired off.
[9] In chapter 4.5.
[10] This refers to my list of eight
primary moods – ignoring here corresponding secondary moods, which may
be viewed as mere derivatives of the primary ones.
[11] See my Future Logic, chapters
30.1 and 46.2.