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1. Variations in form and content

2. Logical-epistemic a fortiori

3. Ethical-legal a fortiori

4. There are no really hybrid forms

5. Probable inferences

6. Correlating ontical and probabilistic forms

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 being D does,

and, B being D implies enough E in it to imply it to be F;

therefore, A being C implies enough E in it to imply it to be F.

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 ‘comprehensible’ (S); therefore,

all the more, term P is well defined (R) enough to be comprehensible (S).

‘Better definition’ (R) is required of a term to ‘pinpoint its instances’ (P) than to ‘be comprehensible’ (Q),

and, term S is well defined (R) enough to pinpoint its instances (P); therefore,

all the more, term S is well defined (R) enough to be comprehensible (Q).

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) than thesis Q is,

and, Q implies correct predictions (R) enough to imply that ‘thesis A is probably true’ (S); therefore,

all the more, P implies correct predictions (R) enough to imply that thesis A is probably true (S).

More ‘correct predictions’ (R) are required to imply ‘thesis A probably true’ (P) than to imply ‘thesis B probably true’ (Q),

and, thesis S implies correct predictions (R) enough to imply that thesis A is probably true (P); therefore,

all the more, S implies correct predictions (R) enough imply that thesis B is probably true (Q).

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 imperative (S);

therefore, all the more, P is valuable (R) enough to make A imperative (S).

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 RP’, and ‘Q produces R to degree RQ’, and

‘RP is greater than RQ’ – whence, ‘if RP then RQ’.

· Minor premise means: ‘Q produces R to degree RQ’, and

‘if RQ then S (= the term ‘makes A is imperative’)’.

· Conclusion means: ‘P produces R to degree RP’ (given), and

‘if RP then S’ (since RP implies RQ, and RQ 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 nonRnonP’, and ‘nonQ inhibits R to degree nonRnonQ’ , and

‘nonRnonP is greater than nonRnonQ’ – whence, ‘if nonRnonP then nonRnonQ’.

· Minor premise means: ‘nonQ inhibits R to degree nonRnonQ’, and

‘if nonRnonQ then S (= the term ‘makes A is imperative’)’.

· Conclusion means: ‘nonP inhibits R to degree nonRnonP’ (given), and

‘if nonRnonP then S’ (since nonRP implies nonRQ, and nonRQ 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 do) P than to be (or have or do) Q,

and, S is virtuous enough to be (or have or do) P;

therefore, all the more, S is virtuous enough to be (or have or do) Q.

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 (positive mood), and

If X is not found in Y, it is probably also not in Z (negative mood).

Implicational form (X, Y, Z are theses):

X more often occurs in conjunction with Y than with Z; therefore:

If X is found in conjunction with Z, it is probably also with Y (positive mood), and

If X is not found in conjunction with Y, it is probably also not with Z (negative mood).

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 in/with Z’, and

‘X is in/with Z’ was probable enough to actually occur (at a certain time);

therefore: ‘X is in/with Y’ is probable enough to actually have occurred or to later occur.

Negative mood (copulative [in] or implicational [with]):

‘X is in/with Y’ is more probable than ‘X is in/with Z’, and

‘X is in/with Y’ was not probable enough to actually occur (by a certain time);

therefore: ‘X is in/with Z’ is not probable enough to actually have occurred or to later occur.

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 thesis S;

therefore, thesis P is probable (R) enough to imply thesis S.

Thesis P is more probable (R) than thesis Q,

and, thesis Q is probable (R) enough to deny thesis S;

therefore, thesis P is probable (R) enough to deny thesis S.

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 than to strike his father:

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 one’s father than to strike one’s neighbor:

if a man is antisocial enough to strike his father,

then he is antisocial enough to strike his neighbor.

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.