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FUTURE LOGIC

© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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CHAPTER 40. EXTENSIONAL CONDITIONAL DEDUCTION.

1. Syllogism.

2. Production.

3. Apodoses.

4. Dilemma.

1. Syllogism.

We can expect the valid quantity modes of extensional conditional syllogism to be analogous to the valid modality (not quantity) modes of natural or temporal conditional syllogism. The valid polarity modes are bound to be the same in all types of conditioning.

a. Extensional conditional syllogism in the first figure, has the valid plural modes uuu, upp. These may be validated by exposition, or we may reduce the particular version to the general ad absurdum (using the major premise). Negative moods may be derived from positive ones by obversion. The following moods are typical:

1/uuu.

Any S which is M, is Q

Any S which is P, is M

so, Any S which is P, is Q.

No S which is M, is Q

Any S which is P, is M

so, No S which is P, is Q.

1/upp.

Any S which is M, is Q

Some S which are P, are M

so, Some S which are P, are Q.

No S which is M, is Q

Some S which are P, are M

so, Some S which are P, are not Q.

Additionally, the mode 1/uss is valid; that is, the minor premise and conclusion could equally well have been singular (though that would be closer to apodosis than syllogism). Providing the indicated instance of the subject is one and the same, the mode 1/sss is also valid (though more to do with conjunction than conditioning), and indeed is the argument we appeal to repeatedly in exposition.

Subaltern modes are uup, usp, ssp. But the modes uus, sus are invalid, because here u does not imply s (unless we are additionally given that ‘This S is P’). Also, pup, ppp, psp, spp, are not valid: the major premise cannot be particular.

Also note, though the major premise consequent may be negative, the minor premise consequent has to be positive, unless the middle term is negative in both premises. We can further design an equal number of valid moods with a negative minor term, by substituting nonP for P.

Lastly, we could introduce other natural or temporal categories of modality in the premises. It goes without saying that the conclusion must be altered accordingly, in each case. For example:

Any S which can be M, is Q

Any S which must P, can be M

so, Any S which must be P, is Q.

Sorites can be formed with extensional conditionals, as with categoricals.

Note that formal relations are often left tacit in arguments. For instance, in the last example, if the minor premise consequent had been ‘must be M’, we could still draw the same conclusion, since ‘Any S which must be M, can be M’ is formally true (it being already given that ‘Some S must be M’).

b. The remaining figures follow, using the appropriate methods of reduction. Typical examples of each are given below, without further ado. As in the first figure, many variations on these themes are workable:

In the second figure, the mode 2/uuu is valid:

Any S which is Q, is M

No S which is P, is M

so, No S which is P, is Q.

No S which is Q, is M

Any S which is P, is M

so, No S which is P, is Q.

Similarly, with particular or singular minor premise and conclusion; that is, the modes upp and uss are valid. Subaltern modes are uup, usp; but the modes uus, sus do not work. The mode sss is valid, only if the middle term has the same polarity in both premises (contrary to the habitual configuration for this figure). The modes pup, ppp, psp, and spp are not valid, as before.

In the third figure, we have arguments like:

Any S which is M, is Q

Any S which is M, is P

so, Some S which are P, are Q.

No S which is M, is Q

Any S which is M, is P

so, Some S which are P, are not Q.

These uup moods (note the particular conclusion) are of course subaltern to those in upp or pup, in which one or the other premise is particular. Also valid, are moods with two singular premises, in sss; subaltern to this mode, are modes uss and sus, since the middle term is antecedent in both premises. But note that uuu, uus, and ppp, psp, spp, are all invalid modes.

In the fourth figure, we have (the significant mood):

No S which is Q, is M

Some S which is M, is P

so, Some S which are P, are not Q.

This argument is in mode upp; similarly valid is the mode uss, with a singular minor premise. Subaltern to these are uup, usp. All other modes are invalid, namely: uuu, pup, sss, uus, sus, spp, psp, ppp. Note that sss would require a contradictory middle term.

For all these figures, as in the first, other combinations of polarities may be introduced; see our treatment of this issue in the context of natural modality for full details. Likewise, as in the first figure, the occurrences may have any combinations of natural and/or temporal modalities.

c. Note well, in all the figures, the analogies between the valid modes of extensional syllogism in quantitative issues (with u, s, p symbols), and the valid modes of modality in natural (n, a, p symbols) or temporal (c, m, t symbols) conditional syllogism. These uniformities facilitate remembering.

However, note also, the differences between their respective treatments of quantity and modality. The valid quantity modes for extensionals differ from the valid quantity modes for naturals or temporals. And likewise, modality inferences differ. It is therefore important to be aware of the modal type of any conditional proposition.

2. Production.

The situations and results for extensional production are again clearly different from those concerning natural or temporal production.

a. In the first figure, production of extensional conditionals from categorical premises proceeds as in the following samples, mode 1/upu:

All P are Q

Some S are P

therefore, Any S which is P, is Q.

No P is Q

Some S are P

therefore, No S which is P, is Q.

We thus are able to infer, given a universal major premise, a conditional universal from a particular minor; also of course inferable is the categorical ‘Some S are Q (or not Q)’. In thinking of the natural or temporal type, our conclusion would have been ‘Some S are P and Q (or nonQ)’, instead.

The above minor premise could equally be universal, with the same conditional conclusion, in the subaltern mode uuu; but here a better conclusion could be drawn, the categorical ‘All S are Q (or nonQ)’, which subalternates the extensional conditional. This again shows us the essential continuity between categorical and conditional argument.

With a singular minor premise, a singular conclusion can be drawn, the conjunction ‘This S is P and Q (or nonQ)’, so the mode 1/uss is valid.

When the premises can have modalities besides actuality, the conclusion is also modal, but it must reflect a valid categorical syllogism, as in the following samples with natural modalities. Note how, in some cases, the conclusion retains similar occurrences, whereas in other cases the conclusion may alter modality and even copula, in accordance with earlier findings.

All P must be Q

Some S must be P

(whence, Some S must be Q)

so, Any S which must be P, must be Q.

All P must be Q

Some S can be P

(whence, Some S can be Q)

so, Any S which can be P, can be Q.

All P can be Q

Some S must be P

(whence, Some S can be or become Q)

so, Any S which must be P, can be or become Q.

All P can be Q

Some S can be P

(whence, Some S can be or become Q)

so, Any S which can be P, can be or become Q.

Contrast the extensional conditional conclusion in the second of these samples, to the natural conditional conclusion which could also be drawn from the same premises, namely ‘When certain S are P, they must be Q’). Their concerns are clearly distinct.

Similarly, for moods with a negative major term. And again similarly with temporal modalities, or with mixtures of modal and actual premises, or premises of mixed modal type. In every case, the rules of modal categorical must be respected, to produce a valid extensional conditional.

b. The valid moods of the other figures follow from those of the first figure, as usual.

In figure two, we have mainly (mode 2/upu):

No Q is P

Some S are P,

therefore, No S which is P, is Q.

All Q are P

Some S are not P,

therefore, No S which is not P, is Q.

Note the polarity of the antecedent of the conclusion, in the latter case. With a singular minor premise, a singular conclusion could also be drawn (mode 2/uss), of the form ‘This S is P (or nonP), and Q’.

In the third figure, we can draw a general extensional, if the major premise is general and the minor particular, singular or general (3/upu, usu, uuu); but we can draw only a particular extensional, if the major is particular or singular, and the minor premise is general (3/pup, sup). The main moods are thus:

All P are Q (or nonQ)

Some P are S,

therefore, Any S which is P, is Q (or nonQ).

Some P are Q (or nonQ)

All P are S,

therefore, Some S which are P, are Q (or nonQ).

In this figure, a singular premise does not yield a singular conclusion, because of the inappropriate positions of the terms.

In the fourth figure, we have:

No Q is P

Some P are S,

so, Any S which is P, is not Q.

Likewise with a singular or general minor premise. Again, a singular premise does not yield a singular conclusion, due to the position of terms.

Production of modal extensional conditionals in these figures, is also feasible — keeping in mind the rules of categorical syllogism, as well as the above models for each figure. The reader should explore some examples.

c. Lastly, note that we can combine syllogism and production to form arguments involving a categorical major premise and a conditional minor premise and conclusion, as in the following example:

All M are Q

Any S which is P, is M,

so, Any S which is P, is Q.

The minor implies that ‘Some S are M’; this, together with the major premise, produces ‘Any S which is M, is Q’; which, in a syllogism with the original minor premise, in turn yields the required conclusion.

Similarly with modals, as for instance in:

All M must be Q

Any S which must be P, can be M,

so, Any S which must be P, can be Q.

Note also the following derivative argument, involving a categorical minor premise and a conditional major premise and conclusion. The fact that necessity implies possibility, because necessity is one of the species of possibility, gives us the hidden premise in parentheses, provided the categorical minor is true.

Any S which can be P, can be Q,

and Some S must be P

(whence, any S which must be P, can be P),

therefore, Any S which must be P, can be Q.

Here, we ‘produce’ a new, narrower, conditional from a given conditional, instead of the categorical ‘All P must be Q’; or this process could be viewed as ‘eduction’ complicated by a proviso.

Note that extensional conditionals can also be arrived at by inductive means (observation and generalization); they do not have to be deduced by syllogism or production.

3. Apodosis.

a. Extensional apodosis follows the pattern set by the primary moods presented below. These are modus ponens (affirming the antecedent) arguments, in mode uss; they simply apply the principle expressed in the major premise to a singular case:

Any S which is P, is Q, No S which is P, is Q,

and This S is P, and This S is P,

hence, This S is Q. hence, This S is not Q.

The major premise cannot be particular. But the minor can be universal or particular, and the conclusion will have the same quantity. The plural moods are:

Any S which is P, is Q, No S which is P, is Q,

and All S are P, and All S are P,

hence, All S are Q. hence, No S is Q.

Note, with regard to the mode uuu, modus ponens, that the major premise is compatible with the minor and conclusion; general extensional conditionals only imply particular bases, and particularity means contingency or generality. Also:

Any S which is P, is Q, No S which is P, is Q,

and Some S are P, and Some S are P,

hence, Some S are Q. hence, Some S are not Q.

The mode upp, modus ponens, may be regarded in two ways: (i) it teaches us that all you need for definition of a general conditional is the base of the antecedent plus the connection, because the base of the consequent follows by such apodosis; or (ii) since we know that the two bases are formally implicit, such argument is in practise redundant.

However, the latter viewpoint is incorrect, because not all conditionals are formulated from knowledge of the basis and connection, but some are arrived at obliquely, as by syllogism, so that modus ponens in upp is informative, it aids understanding of the data in hand.

b. The following moods are modus tollens (denying the consequent) arguments, in mode uss. These may be validated directly, by contraposition of the major premise on the basis of the minor; the conclusion is new information, emerging from the contraposite and the base of its antecedent in a modus ponens apodosis. Or we may validate them by reductio ad absurdum, contradicting the conclusion results in denial of the minor premise, by modus ponens.

Any S which is P, is Q, No S which is P, is Q,

and This S is not Q, and This S is Q,

hence, This S is not P. hence, This S is not P.

The major premise again cannot be particular. The minor and conclusion can be particular; but note well that they cannot be general, since they would contradict the bases of the major. The valid plural moods of modus tollens are, therefore, only the following:

Any S which is P, is Q, No S which is P, is Q,

and Some S are not Q, and Some S are Q,

hence, Some S are not P. hence, Some S are not P.

Thus, modes uss and upp are valid in both ponens and tollens extensional apodosis. But the mode uuu is only valid in ponens; in tollens, it is invalid, note well:

Any S which is P, is Q, No S which is P, is Q,

and No S is Q, and All S are Q,

hence, No S is P. hence, No S is P.

c. We may of course introduce a negative antecedent into any of the arguments above or below; just replace P with nonP throughout. For examples:

Any S which is not P, is Q, No S which is not P, is Q,

and This S is not P, and This S is not P,

hence, This S is Q. hence, This S is not Q.

Also, any natural or temporal modality, or mixture of them, may be involved, provided we adhere to the set interpretations of extensional conditionals. The rules of quantity of the extensional apodosis process are the same with modals, as with actuals.

The following are some examples of modal modus ponens. Note the faithful transmission of natural modality from consequent to conclusion. If the antecedent is necessary, nothing less than a necessary minor will activate it.

Any S which can be P, can be Q,

and This S can be (or is or must be) P,

hence, This S can be Q.

Any S which must be P, can be Q,

and This S must be P

hence, This S can be Q.

Any S which can be P, must be Q,

and This S can be (or is or must be) P,

hence, This S must be Q.

Any S which must be P, must be Q,

and This S must be P,

hence, This S must be Q.

The following are some examples of modal modus tollens. It is interesting how, granting the premises, we are able to draw a conclusion of opposite natural modality, as well as polarity, to the antecedent. If the consequent is potential, nothing less than a necessary minor will activate it.

Any S which can be P, can be Q,

and This S cannot be Q,

hence, This S cannot be P.

Any S which must be P, can be Q,

and This S cannot be Q,

hence, This S can not-be P.

Any S which can be P, must be Q,

and This S can not-be (or is not or cannot be) Q,

hence, This S cannot be P.

Any S which must be P, must be Q,

and This S can not-be (or is not or cannot be) Q,

hence, This S can not-be P.

d. Note well in all the above arguments, the differences between extensional apodosis, and natural or temporal such arguments.

Thus, in natural (or temporal) apodosis the major premise may be particular if the minor is general; but not here: in extensional apodosis the major must be general. On the other hand, in natural (or temporal) apodosis the consequent cannot be potential (or temporary), whereas here it can.

Such differences in process are due to the switched roles of the features of quantity and modality, from one type of conditioning to the next. In naturals or temporals, the conditioning is defined by the modality, and the quantity is incidental. In extensionals, the conditioning is defined by the quantity, and the modalities involved are incidental.

Lastly, note the existence here too of adductive arguments, which merely suggest a result, with some degree of probability, though not certainty:

Any S which is P, is Q

and This S is Q (is given as evidence)

hence, This S is P (is somewhat confirmed).

Any S which is P, is Q

and This S is not P (is given as counter-evidence)

hence, This S is not Q (is somewhat confirmed).

Compare extensional adduction, to logical, natural or temporal adduction. Here, we are expressing a likelihood that the indicated instance of the subject, is indeed one of the instances of the subject covered by the major premise. Note that though the conditional is general, it may be based on a very limited number of cases.

4. Extensional Dilemma.

Extensional disjunctive arguments are reducible to extensional conditional processes. It is important to always clarify just what we intend by the disjunction, because often different interpretations are feasible.

An example of an extensional disjunctive apodosis

All S are P or Q (implying Any S which is not P, is Q)

This S is not P

hence, This S is Q.

An example of disjunctive syllogism (reduced to two conditional syllogisms):

All S are M or Q (all S-nonM are Q, all S-nonQ are M)

All S are P or nonM (all S-nonP are nonM, all S-M are P)

hence, All S are P or Q (all S-nonP are Q, all S-nonQ are P)

Production of extensional disjunctives may likewise be achieved by production and reconstruction of extensional conditionals.

Extensional dilemma is more complicated, and worth exploring more deeply. The reader should compare it to logical, natural and temporal dilemma, to see the analogies and differences.

a. The simple dilemmas look as follows (taking ‘or’ to mean that one of the alternatives has to be applicable):

The simple constructive form:

Any S which is M or N, is P

but, All S are M or N

therefore, All S are P.

In this argument, the major premise tells us that those S which are not M, are N and P; and those S which are not N, are M and P; and none of all these S are both nonM and nonN. The minor premise tells us that all S fit the preconditions expressed in the major, yielding the conclusion that all S are also subsumed by the consequent, categorically.

The simple destructive form:

Any S which is P, is M or N

but, Some S are not M and not N

therefore, Some S are not P.

The major premise informs us that of all the S which are P, none is also both nonM and nonN. The minor premise presents us with some cases of S which are indeed both nonM and nonN. The conclusion is, therefore the latter S cannot be counted among the former, and there must be some S which are not P.

Note that the constructive version has a general minor premise and conclusion (mode, uuu), whereas the destructive version only works with a particular minor premise and conclusion (mode, upp). A constructive dilemma with a merely particular minor (upp) would yield a conclusion already known, since it is a base of the major; a destructive dilemma with a universal minor (uuu) would yield a conclusion contradictory to a base of the major.

In the singular, uss mode, the minor disjunction cannot be meant extensionally; where it happens in ordinary discourse, we intend a logical basis; the conclusion would still be valid on that basis, however. Logical basis disjunction is of course also sometimes intended within universals or particulars. More on this topic in the chapter on condensed propositions.

b. If we look at the special case of antithetical antecedent predicates:

Any S which is M or not M, is P,

but All S are M or not M,

whence All S are P.

…we see this means that ‘Whether any S is M or not M, it is P’ implies ‘All S are P’. This reflects the compatibility of the propositions ‘Any S which is M, is P’ and ‘Any S which is not M, is P’, provided ‘All S are P’ to prevent their contraposability.

We can reword it as ‘Though some S are M and some others not M, all S are P’. The ‘though’ stresses the independence of the general consequent from the contingent antecedent: their ‘link’ is so strong, that it is effectively absent. This model allows us to understand universality as a type of necessity; what is found in all the cases of a subject is viewed as more ingrained in their nature, than attributes which differ from similar case to case.

c. The complex constructive and destructive extensional dilemmas, respectively, look as follows.

Any S which is M is P, and any S which is N is P,

but All S are M or N,

therefore, All S are P.

In this constructive version, the extensional disjunction in the minor premise ensures that all S fit the preconditions of one or the other of the horns of the major premise, and so make the antecedents exhaustive, and their common consequent general.

Any S which is P is M, and any S which is P is N,

but Some S are not M or not N,

therefore, Some S are not P.

In this destructive version, notice that the minor premise is disjunctive. It could have been, more narrowly, ‘Some S are not M and not N’; but since the conclusion may be drawn by apodosis from either of these negatives without the other, we can broaden the applicability of the argument by saying ‘or’. However, this disjunction may be intended as merely logical.

Note that the valid quantity modes here are uuu for the constructive, and upp for the destructive; as for simple dilemma, a constructive upp is uninformative, and a destructive uuu is logically impossible.

The singular uss mode is conceivable with a logical, rather than extensional, disjunctive minor, constructively or destructively; it is also conceivable in a destructive mood with ‘This S is not M and not N’ as the minor premise.

e. All the above forms of dilemma may of course involve antecedents and consequents of other polarities, and of natural or temporal modalities other than actuality. The rules of modality here are similar to those of modal extensional apodosis. The reader should construct some examples of modal dilemma, to get acquainted with it.

Lastly note, there is no argument in extensional dilemma, equivalent to rebuttal of a logical dilemma by an ‘equally cogent’ dilemma. The minor premises required for that would be contradictory. The reader should experiment, and find out if this statement is correct.

2016-08-20T07:19:44+00:00