FUTURE LOGIC
CHAPTER 29. HYPOTHETICAL SYLLOGISM AND PRODUCTION.
There are several kinds of deductive argument involving hypothetical propositions or their derivatives. They are distinguished according to whether they involve only hypotheticals, or hypotheticals mixed with categorical forms. The main kinds are syllogism, production, apodosis and dilemma. Note that the valid moods are not here listed in symbolic terms, as we did with categoricals, to avoid obscuring their impact.
Hypothetical syllogism is argument whose premises and conclusion are all hypotheticals. It is mediate inference, with minor (symbol P), middle (M), and major (Q) theses, deployed in figures, as was the case in categorical syllogism.
Its most primary valid mood, from which all others may be derived by direct or indirect reduction, is as follows. It tells us, as for the analogue in categorical syllogism, that, as H.W.B. Joseph would say, ‘whatever falls under the condition of a rule, follows the rule’.
This primary mood is valid irrespective of whether the hypotheticals involved are of unspecified base, normal (contingency-based), or abnormal. That is generally true for its primary derivatives, too; but subaltern derivatives are only applicable in cases where both theses are known to be logically contingent (and not just problematic), because the subalterns require eductive processes which depend on this condition for their validity.
If M, then Q
if P, then M
so if P, then Q
This is a first figure syllogism. Its validity obviously follows from the meaning of the operator ‘if-then’ involved. Although the connection in hypotheticality is expressed by modal conjunctive statements, ‘if-then’ underscores an additional, not-tautologous, sense, occurring on a finer level. This teaches us a purely conjunctive argument, from which many laws for the logic of conjunction may be inferred, that:
The premises: {M and nonQ} is impossible,
and {P and nonM} is impossible, together
yield the conclusion: {P and nonQ} is impossible.
This could be written symbolically as 1/H2nH2nH2n, note.
a. Figure One.
(i) From the primary valid mood, we can draw up the following full list of valid, uppercase, perfect moods, in first figure, by substituting antitheses for theses in every possible combination.
If M, then Q |
If nonM, then Q |
if P, then M |
if P, then nonM |
so, if P, then Q |
so, if P, then Q |
If M, then nonQ |
If nonM, then nonQ |
if P, then M |
if P, then nonM |
so, if P, then nonQ |
so, if P, then nonQ |
If M, then Q |
If nonM, then Q |
if nonP, then M |
if nonP, then nonM |
so, if nonP, then Q |
so, if nonP, then Q |
If M, then nonQ |
If nonM, then nonQ |
if nonP, then M |
if nonP, then nonM |
so, if nonP, then nonQ |
so, if nonP, then nonQ |
(ii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, lowercase, perfect mood, by reductio ad absurdum, as follows. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.
If M, then Q |
contrapose major: |
If nonQ, then nonM |
if P, not-then nonM |
deny conclusion: |
if P, then nonQ |
so, if P, not-then nonQ |
get anti-minor |
if P, then nonM |
From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the first figure, by substituting antitheses for theses in every possible combination.
If M, then Q |
If nonM, then Q |
if P, not-then nonM |
if P, not-then M |
so, if P, not-then nonQ |
so, if P, not-then nonQ |
If M, then nonQ |
If nonM, then nonQ |
if P, not-then nonM |
if P, not-then M |
so, if P, not-then Q |
so, if P, not-then Q |
If M, then Q |
If nonM, then Q |
if nonP, not-then nonM |
if nonP, not-then M |
so, if nonP, not-then nonQ |
so, if nonP, not-then nonQ |
If M, then nonQ |
If nonM, then nonQ |
if nonP, not-then nonM |
if nonP, not-then M |
so, if nonP, not-then Q |
so, if nonP, not-then Q |
(iii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, imperfect mood, by reductio ad absurdum, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the moods as imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that the minor premise is uppercase, and the major premise and conclusion are lowercase.
If M, not-then Q |
deny conclusion: |
If nonP, then Q |
if P, then nonM |
contrapose minor: |
if M, then nonP |
so, if nonP, not-then Q |
get anti-major: |
if M, then Q |
From this primary mood, we can draw up the following full list of valid, imperfect moods, in the first figure, by substituting antitheses for theses in every possible combination.
If M, not-then Q |
If nonM, not-then Q |
if P, then nonM |
if P, then M |
so, if nonP, not-then Q |
so, if nonP, not-then Q |
If M, not-then nonQ |
If nonM, not-then nonQ |
if P, then nonM |
if P, then M |
so, if nonP, not-then nonQ |
so, if nonP, not-then nonQ |
If M, not-then Q |
If nonM, not-then Q |
if nonP, then nonM |
if nonP, then M |
so, if P, not-then Q |
so, if P, not-then Q |
If M, not-then nonQ |
If nonM, not-then nonQ |
if nonP, then nonM |
if nonP, then M |
so, if P, not-then nonQ |
so, if P, not-then nonQ |
(iv) Subaltern moods. These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.
The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.
If M, then Q
if P, then M
so, if P, not-then nonQ.
The following sample can be derived from moods of type (i) by obvert-inverting the conclusion, or equally well from moods of type (iii) by replacing the major premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.
If M, then Q
if P, then M
so, if nonP, not-then Q.
In summary, we thus have a total of 3X8 = 24 primary valid moods in the first figure, plus 2X8 = 16 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 possibilities.
b. Figure Two.
(i) From one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, uppercase, perfect mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, uppercase, perfect mood, of the first figure.
If Q, then M |
with same major: |
If Q, then M |
if P, then nonM |
deny conclusion: |
if P, not-then nonQ |
so, if P, then nonQ |
get anti-minor: |
so, if P, not-then nonM |
From this primary, valid mood, we can draw up the following full list of valid, uppercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.
If Q, then M |
If Q, then nonM |
if P, then nonM |
if P, then M |
so, if P, then nonQ |
so, if P, then nonQ |
If nonQ, then M |
If nonQ, then nonM |
if P, then nonM |
if P, then M |
so, if P, then Q |
so, if P, then Q |
If Q, then M |
If Q, then nonM |
if nonP, then nonM |
if nonP, then M |
so, if nonP, then nonQ |
so, if nonP, then nonQ |
If nonQ, then M |
If nonQ, then nonM |
if nonP, then nonM |
if nonP, then M |
so, if nonP, then Q |
so, if nonP, then Q |
(ii) Next, from one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, lowercase, perfect mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, lowercase, perfect mood, of the first figure. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.
If Q, then M |
with same major: |
If Q, then M |
if P, not-then M |
deny conclusion: |
if P, then Q |
so, if P, not-then Q |
get anti-minor: |
if P, then M |
From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.
If Q, then M |
If Q, then nonM |
if P, not-then M |
if P, not-then nonM |
so, if P, not-then Q |
so, if P, not-then Q |
If nonQ, then M |
If nonQ, then nonM |
if P, not-then M |
if P, not-then nonM |
so, if P, not-then nonQ |
so, if P, not-then nonQ |
If Q, then M |
If Q, then nonM |
if nonP, not-then M |
if nonP, not-then nonM |
so, if nonP, not-then Q |
so, if nonP, not-then Q |
If nonQ, then M |
If nonQ, then nonM |
if nonP, not-then M |
if nonP, not-then nonM |
so, if nonP, not-then nonQ |
so, if nonP, not-then nonQ |
(iii) Subaltern moods. These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.
The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.
If Q, then M
if P, then nonM
so, if P, not-then Q.
The following sample can be derived from moods of type (i) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.
If Q, then M
if P, then nonM
so, if nonP, not-then nonQ.
The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct uniform polarity of the middle thesis in the two premises.
If Q, then M
if P, then M
so, if nonP, not-then Q.
In summary, we thus have a total of 2X8 = 16 primary valid moods in the second figure, plus 3X8 = 24 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 possibilities.
c. Figure Three.
(i) From one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, perfect mood, with lowercase major premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, and transposing, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.
If M, not-then nonQ |
deny conclusion: |
If P, then nonQ |
if M, then P |
with same minor: |
if M, then P |
so, if P, not-then nonQ |
get anti-major: |
if M, then nonQ |
From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase major premise, in the third figure, by substituting antitheses for theses in every possible combination.
If M, not-then nonQ |
If nonM, not-then nonQ |
if M, then P |
if nonM, then P |
so, if P, not-then nonQ |
so, if P, not-then nonQ |
If M, not-then Q |
If nonM, not-then Q |
if M, then P |
if nonM, then P |
so, if P, not-then Q |
so, if P, not-then Q |
If M, not-then nonQ |
If nonM, not-then nonQ |
if M, then nonP |
if nonM, then nonP |
so, if nonP, not-then nonQ |
so, if nonP, not-then nonQ |
If M, not-then Q |
If nonM, not-then Q |
if M, then nonP |
if nonM, then nonP |
so, if nonP, not-then Q |
so, if nonP, not-then Q |
(ii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, perfect mood, with lowercase minor premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the minor premise, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.
If M, then Q |
deny conclusion: |
If P, then nonQ |
if M, not-then nonP |
with same minor: |
if M, not-then nonP |
so, if P, not-then nonQ |
get anti-major: |
if M, not-then Q |
From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase minor premise, in the third figure, by substituting antitheses for theses in every possible combination.
If M, then Q |
If nonM, then Q |
if M, not-then nonP |
if nonM, not-then nonP |
so, if P, not-then nonQ |
so, if P, not-then nonQ |
If M, then nonQ |
If nonM, then nonQ |
if M, not-then nonP |
if nonM, not-then nonP |
so, if P, not-then Q |
so, if P, not-then Q |
If M, then Q |
If nonM, then Q |
if M, not-then P |
if nonM, not-then P |
so, if nonP, not-then nonQ |
so, if nonP, not-then nonQ |
If M, then nonQ |
If nonM, then nonQ |
if M, not-then P |
if nonM, not-then P |
so, if nonP, not-then Q |
so, if nonP, not-then Q |
(iii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, imperfect mood, of the third figure, by direct reduction, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the mood as imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that both premises and the conclusion are uppercase.
If M, then Q |
with same major: |
If M, then Q |
if nonM, then P |
contrapose minor: |
if nonP, then M |
so, if nonP, then Q |
get conclusion: |
so, if nonP, then Q |
From this primary mood, we can draw up the following full list of valid, imperfect moods, in the third figure, by substituting antitheses for theses in every possible combination.
If M, then Q |
If nonM, then Q |
if nonM, then P |
if M, then P |
so, if nonP, then Q |
so, if nonP, then Q |
If M, then nonQ |
If nonM, then nonQ |
if nonM, then P |
if M, then P |
so, if nonP, then nonQ |
so, if nonP, then nonQ |
If M, then Q |
If nonM, then Q |
if nonM, then nonP |
if M, then nonP |
so, if P, then Q |
so, if P, then Q |
If M, then nonQ |
If nonM, then nonQ |
if nonM, then nonP |
if M, then nonP |
so, if P, then nonQ |
so, if P, then nonQ |
(iv) Subaltern moods. These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.
The following sample can be derived from moods of type (i) by replacing the major premise with its obvertend, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.
If M, then Q
if M, then P
so, if P, not-then nonQ.
The following sample can be derived from moods of type (i) by replacing the major premise with its obvert-invertend, or equally well from moods of type (iii) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature, but note the distinct mixed polarity of the middle thesis in the two premises.
If M, then Q
if nonM, then P
so, if P, not-then Q.
The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend, or equally well from moods of type (iii) by obverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct mixed polarity of the middle thesis in the two premises.
If M, then Q
if nonM, then P
so, if nonP, not-then nonQ.
In summary, we thus have a total of 3X8 = 24 primary valid moods in the third figure, plus 3X8 = 24 subaltern valid moods. Or a total of 48 valid moods, out of 8X8X8 = 512 possibilities.
d. With regard to the fourth figure, it can be ignored in hypothetical syllogism. Since the first figure here (unlike with categorical syllogism) includes imperfect moods, the fourth figure here would introduce no new valid moods for us. Its valid moods can of course all be reduced directly to the first figure, by transposing or contraposing the premises, but they do not represent a movement of thought of practical value.
We therefore have, in the three significant figures taken together, a total of 24+16+24 = 64 primary valid moods, plus 16+24+24 = 64 subaltern valid moods. Or a total of 128 valid moods, out of 3X512 = 1536 possibilities; meaning a validity rate of 8.33%.
The chaining of syllogisms into a series forming a sorites is possible with hypothetical syllogism, similarly to categorical syllogism. This is used in practise, of course, and applies irrespective of basis. The typical sorites looks as follows:
If A, then B
if B, then C
…
if G, then H
therefore, if A, then H.
Note that we are in the figure one, and we state the most minor premise first, and successively work up to the most major premise, and lastly the conclusion. A sorites should be reducible to valid syllogisms to be valid.
Of course, sorites is only the most regular form of continuous argument, the easiest to think without aid of paper and pencil. More broadly, any succession of premises, in any combination of figures, yielding a valid final conclusion, may be viewed as continuous, even though we have to think out the intermediate conclusions, zigzagging from figure to figure, to reach the result.
We can readily reformulate all the above syllogisms using derivative forms, such as simple disjunctions. For examples, the following arguments, taken at random, are easily validated by transforming the disjunctives into standard hypotheticals:
M and/or Q |
Q or else M |
P or else M |
P not and/or nonM |
P or else nonQ |
P not and/or Q. |
Here again, I would not regard these as distinct valid moods. Even if they are used in practise, we are mentally required to restate them in ‘If/then’ form to understand them. It will however be seen, in the context of dilemma, that there are certain arguments, which mix ‘If/then’ forms with disjunctives, which are comprehensible on their own merit, and used in everyday discourse.
Such arguments may also be regarded as ‘logical compositions’. With multiple alternatives, the possible number of arguments increases and so does the mental confusion. When translating the given disjunctions into ‘If-then’ statements causes us as much confusion, the best course is to express each proposition in terms of the conjunctions is allows and forbids; then we can best see what conclusion, if any, may be drawn.
We can also, it is noted, appeal to the above valid moods of the syllogism to clarify reasoning involving compound forms. That is, when one or both premises signifies implicance or subalternation or contradiction or contrariety or subcontrariety, we may be able to fuse the results of two or more simple syllogisms, and get a compound conclusion.
Lastly, arguments may be fashioned in conditional frameworks, so that we have nested hypotheticals for premise(s) and conclusion. This may be viewed as a wider logic, concerning composite antecedents or consequents, conjunctive or even disjunctive ones. Researching the mechanics of partial or alternative theses is an area that deserves eventual attention, but presumably it can be reduced to the findings of unconditional logic.
Subaltern moods are implicitly conditional; they have as hidden premises, the categorical propositions that the theses are logically contingent, rather than merely problematic or partly or wholly incontingent. The tacitly understood premises are: ‘P (and nonP) is contingent, and Q (and nonQ) is contingent’. I have made no effort to develop subaltern moods with abnormal bases, because once a thesis is known to be incontingent it is rarely thereafter used in hypothetical propositions.
How are hypothetical propositions produced? By their very nature they do not presuppose the reality of their theses, so how do we know that the antecedent does (or does not) engage the consequence? This question will be answered in this section.
Hypothetical propositions signify a logical connection between the theses, so that any argument which is logically valid may be recast in hypothetical form.
The theses involved may of course have any form, including themselves hypothetical. The term ‘connection’ here is to be understood in its widest sense, including any logical relationship, positive or negative, normal or abnormal. Thus, all oppositions, eductions, deductions, are included here; overall, a valid inference of any kind produces a positive hypothetical, an invalid inference produces a negative hypothetical.
Also, the expression ‘logically valid’ should be taken as comprehensive of the known and the unknown; there is no presumption here that the science of logic as we know it to date is complete. It is important to stress this; while all established logical truths are capable of producing hypotheticals, it does not follow that hypotheticals cannot be produced by means not yet clarified by this science. No claim to omniscience is required.
An example of production would be recasting a categorical syllogism in hypothetical form: e.g. ‘If all S are M and all M are P, then all S are P’. This is a conclusion, whose premises are the process of validation of that mood of the syllogism via the laws of logic.
If we instead produced the briefer conclusion ‘If all S are M, all S are P’, the process to be valid must have included, after the above, a nesting (to ‘If all M are P, then if all S are M, all S are P’) and an apodosis (with minor premise ‘All M are P’). Thus enthymeme need not be viewed as merely syllogism with a suppressed (tacit) premise, but as the end product of a series of definite arguments.
However, production is not limited to relationships in terms of variables, but is especially useful for application to specific values. Using a formal relationship as major premise, we may, through the act of substitution as minor premise, produce a hypothetical with particular contents as conclusion. Continuing the above example, we might for instance produce, ‘If all men were wise, they would not make war’.
In short, any logical series which is incomplete, may be made to at least yield a hypothetical conclusion, and thus constitute a productive process.
The missing information may simply be the exact quantity involved. Thus, if in the above example we do not know whether all or only some S are M, we can still conclude from ‘All M are P’ that ‘If any S is M, it is P’. This produces a hypothetical proposition which seems general, but in fact only suggests that some S may be M. Incidentally, the expression ‘whether’ may itself be viewed as a derivative form of hypothetical, concealing a dilemma.
Similarly, a negative hypothetical would express a nonsequitur. For example, ‘If no S are M and all M are P, it does not follow that no S are P’. Likewise, with particular contents or indefinite quantities, as above.
Clearly, the possibilities are virtually infinite. Any formal or informal sequence permitted or forbidden by the laws of logic constitutes a productive process. Ordinarily, a hypothetical would not be formed, unless information was missing or already known wrong, and only problematic elements would be included in it as theses; but there is nothing illicit in forming one even with definite theses of known truth.