**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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| if |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| if |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| so, |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| if |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| if |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| if |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 |
| If |

if |
| if |

so, |
| so, |

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 | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

If | If |

if | if |

so, | so, |

(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 | Q |

P | P |

P | P |

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.