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FUTURE LOGIC

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

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CHAPTER 37. NATURAL APODOSIS AND DILEMMA.

1. Apodosis.

2. Dilemma.

1. Apodosis.

Natural apodosis is deductive argument mainly involving (i) a necessary natural conditional as major premise, and (ii) an actual categorical corresponding to the antecedent or to the negation of the consequent as minor premise, with (iii) an actual categorical corresponding to the consequent or to the negation of the antecedent, respectively, as conclusion. Other modalities are less typical, though derivable.

a. Actual Moods.

The premier valid mood, from which all others may be derived, consists in ‘affirming the antecedent’ (modus ponens), as follows. Note well that the conclusion is not ‘This S must be Q’, in spite of our placing the necessary modality of the conditional proposition in the consequent; however, the conclusion ‘this S is P’, although not naturally necessary, is of course logically necessary given the premises; the mode is naa.

When this S is P, it must be Q

and This S is P,

hence, This S is Q.

The major premise informs us that, whatever the surrounding circumstances for this S, its being P is accompanied by its being Q. The apodosis merely takes it at its word, and applies it to the actual circumstance given by the minor premise, to obtain the conclusion.

The following mood follows from this by obversion:

When this S is P, it cannot be Q

and This S is P,

hence, This S is not Q.

The following mood, which consists in ‘denying the consequent’ (modus tollens), may be reduced to the primary one above, ad absurdum: deny the conclusion, while retaining the major premise, and the minor premise is contradicted.

When this S is P, it must be Q

and This S is not Q,

hence, This S is not P.

A complex contraposition underlies this argument, of course. The major premise does not by itself imply the contraposite ‘When this S is not Q, it cannot be P’; but when the major is combined with ‘This S can not-be Q’, as implied by the minor premise, the contraposite is inferable, as we saw in the chapter on eduction. With the contraposite, this mood becomes identical to the one before.

The next follows by obversion from this:

When this S is P, it cannot be Q

and This S is Q,

hence, This S is not P.

The actual moods draw an actual conclusion from a necessary major premise and an actual minor premise, in mode naa. The mode nna is accordingly valid, granting the actuality of the subject. Modes aaa or ana are valid for modus ponens, but their minor premise is redundant; in modus tollens, they are invalid, because the premises are incompatible.

b. Modal Moods.

Modal moods are those with a modal conclusion from modal premises.

Moods with a necessary major and minor premise, affirming the antecedent, yield a necessary conclusion. These moods can be viewed as repetitive applications of the corresponding actual moods, since natural necessity means actuality in all circumstances, and they teach us that if the antecedent is naturally necessary, so must the consequent be. The following are nnn moods valid:

When this S is P, it must be Q

and This S must be P,

hence, This S must be Q.

When this S is P, it cannot be Q

and This S must be P,

hence, This S cannot be Q.

On the other hand, moods with a necessary major and minor premise, denying the consequent (and thus depending on our switching the positions of the events), are not valid, because their conclusion would contrary the major premise. Thus, the following nnn moods are invalid, note well:

When this S is P, it must be Q

and This S cannot be Q,

hence, This S cannot be P.

When this S is P, it cannot be Q

and This S must be Q,

hence, This S cannot be P.

Such apodosis is invalid, even with an unnecessary conclusion (as in nnp), since the major premise requires ‘this S can be P and Q (or nonQ)’ as its basis, and thus formally excludes the logical possibility of the attempted minor premise, let alone any conclusion.

As for mode npp, necessary major premise combined with a potential affirming minor and conclusion, form a valid mood, since as soon as the minor actualizes, so will the conclusion. For instances:

When this S is P, it must be Q

and This S can be P,

hence, This S can be Q.

When this S is P, it cannot be Q

and This S can be P,

hence, This S can not-be Q.

This mood teaches us that the connection together with the base of the antecedent suffice to define a natural conditional, since the base of the consequent (and also the compound basis) follow anyway. But we could also view this mood as redundant, granting that we already know that both the minor premise and conclusion are formally implicit in the major premise.

On the other hand, a necessary major premise combined with a potential denying minor and conclusion, form a more significant, as well as valid, mood of apodosis (mode npp). These arguments have already been encountered in the context of complex contraposition:

When this S is P, it must be Q

and This S can not-be Q,

hence, This S can not-be P.

When this S is P, it cannot be Q

and This S can be Q,

hence, This S can not-be P.

What of moods with a potential, instead of necessary, major premise? Modus ponens cases are redundant, and modus tollens cases are invalid, as shown below:

If the associated minor premise affirms the antecedent, whether necessarily or potentially, a necessary conclusion (mode pnn or ppn) is of course out of the question. Drawing a potential conclusion (in pnp or ppp) would teach us nothing new, since that is already implied in the basis of the major, anyway. For instance:

When this S is P, it can be Q

and This S can or must be P,

hence, This S can be Q.

If the associated minor premise denies the consequent necessarily, we cannot draw a conclusion, because the two premises are anyway contrary to each other. Thus, for instance, the following is invalid (mode pnp):

When this S is P, it can be Q

and This S cannot be Q,

hence, This S can not-be P.

If the associated minor premise denies the consequent potentially, we cannot draw a conclusion, because we have no guarantee that the some circumstances referred to by the major overlap with those referred to by the minor. Thus, for instance, the following is invalid (mode ppp):

When this S is P, it can be Q

and This S can not-be Q,

hence, This S can not-be P.

c. We can construct further valid moods, analogous and equal in number to the above described, by substituting a negative antecedent, ‘When this S is not P,…’ in all the majors. The polarity of the corresponding minor or conclusion must of course be changed to match, in every case.

d. Also, we can quantify all the valid moods. One of the premises must be general, to guarantee overlap; the quantity of the conclusion then follows that of the other premise. Thus, we have two sets of quantified moods, with some overlapping cases (both premises general).

Those with a general major premise, and any quantity in the minor, like:

When any S is P, it must be Q

and All/This/Some S is/are P,

hence, All/This/Some S is/are Q.

When any S is P, it must be Q

and All/This/Some S is/are not Q,

hence, All/This/Some S is/are not P.

And those with any quantity in the major premise, and a general minor, like:

When any/this/some S is/are P, it/they must be Q

and All S are P,

hence, All/This/Some S is/are Q.

When any/this/some S is/are P, it/they must be Q

and No S is Q,

hence, All/This/Some S is/are not P.

Similarly with the allowable changes in modality, and with negative consequents and/or antecedents, of course. Clearly, the rules of quantity here are less restrictive than those of modality; this is because the quantity of antecedent and consequent is one and the same, whereas the modality concerns their relationship.

Moods such as those below are, of course, not valid, because they go beyond the brief of the forms concerned. However, if we regard the minor premise as an adduction of evidence or counterevidence, we may view the suggested conclusion as tending to be confirmed.

When this S is P, it must be Q

and This S is Q (is given as evidence),

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

When this S is P, it must be Q

and This S is not P (is given as counterevidence),

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

Compare such natural adduction to logical adduction. Here, we are assuming that the actual set of circumstances surrounding the minor premise, is among the sets of natural circumstances in which the major premise holds, namely all the circumstances when this S is P or all the circumstances when this S is not Q.

2. Dilemma.

Natural disjunctive arguments are reducible to natural conditional processes, at least in the case of two alternatives. For example, the following apodosis could be validated by inferring ‘When this S is not P, it must be Q’ from the major premise.

This S must be P or Q

This S is not P

so, This S is Q.

Or again, the following sample of ‘syllogistic’ argument, admittedly somewhat forced and not likely to be used as such in practise, could be validated in a similar way.

This S must be M or Q

This S must be P or not M

so, This S must be P or Q

Likewise, we can develop arguments for production of natural disjunctives.

These are of course only the simplest samples. Other polarities, other modalities, other manners of disjunction, and multiple disjunction, would need be considered for full treatment of the field. But these topics will not be analyzed further, here.

Natural dilemma, however, deserves some attention, because of the improved insight into the meaning of natural necessity which it provides, and to stress its distinction from logical dilemma.

a. Simple constructive natural dilemma consists, as shown below, of premises and conclusion all of which are necessary; the major premise consists of a conditional whose antecedent is a natural disjunction (or, alternatively, of the equivalent conditionals in conjunction), the minor premise is disjunctive, and the conclusion is categorical.

When this S is M or N, it must be P,

but, This S must be M or N,

hence, This S must be P.

Whereas in apodosis to draw such a necessary conclusion, the minor premise had to be a categorical necessity, here we are taught that a necessary conclusion may still be drawn from a slightly less demanding minor premise, namely a disjunctive necessity — provided, of course, that the conditional major premise(s) is/are necessary.

We learn from this that if some event P is ‘bound to’ follow each of circumstances M, N, etc. (however many there be), and the set of circumstances M, N, etc. is exhaustive, then the event P is immovable and effectively independent of any circumstance. Thus, the dilemma as a whole tells us ‘Whether this S is M or N, it must be P’.

Note well that the minor premise and conclusion could not have been actual, as in apodosis. There is no actual form of natural disjunction; the proposition ‘This S is M or N’, taken literally, is a logical disjunctive, based on a doubt as to whether ‘This S is M’ or ‘This S is N’ is true, without implying that both these actualities are potential in the real world in the present circumstances.

b. Note well the special case of simple constructive natural dilemma:

‘When this S is M, it must be P, and when it is not M, it must be P’.

but, This S must be M or not M,

hence, This S must be P.

Or, more briefly, ‘Whether this S is or is not M, it must be P’. There is nothing in the structure of natural conditionals preventing contradictory antecedents from having the same necessary consequent. In such case, the consequent is absolutely, and not just relatively, necessary, so that the antecedents are redundant.

(This is the nearest thing to logical paradox, which we find in natural conditioning; there is of course no exact equivalent, since ‘When this S is not P, it must be P’ would imply that ‘This S both can not-be P, and must be P’, a natural impossibility.)

It is with this phenomenon in mind that we developed our original definition of natural necessity as ‘actuality in every circumstance, whatever the actual circumstance’. Strictly-speaking the concept of ‘in’ is more primitive than of ‘when’ or ‘or’; but the above dilemma serves as a clarification, anyway.

(Any seeming circularity is due to the fundamentality of the concepts involved; there is no inconsistency in that; nor is it redundant, because it aids our understanding, and the development of a formal logic of modality.)

c. In contrast, the simple destructive natural dilemma has to consist of conditional major premise with a disjunctive consequent, which combined with an actual minor premise, yields a categorical actual conclusion, as shown below.

When this S is P, it must be M or N

but, This S is not M and not N;

hence, This S is not P.

Why so? Because the other alternatives are meaningless. Had we formulated destructive dilemma as follows:

When this S is P, it must be M, and when this S is P, it must be N;

but, This S is not M and not N;

hence, This S is not P.

…we would be faced with two ordinary apodoses, each one of which would suffice to obtain the required conclusion.

If, on the other hand, we had formulated it as follows:

When this S is P, it must be M, and when this S is P, it must be N;

but, This S cannot be M or N

or even, This S must be not M or not N;

…we would be faced with incompatible premises, since the majors imply that ‘This S can be M and N’, and yet the minors deny that. Also, concluding that ‘This S cannot be P’ would deny the implication of the major that ‘This S can be P’.

(If the minor premise said ‘This S can no longer be M or N’, instead of ‘cannot’, then we might assume a similar loss of power for the antecedent, and conclude ‘This S can no longer be P’; however, that interpretation is far from certain: for it is conceivable that the major premise relationships are entirely different in such eventuality.)

d. With regard to complex natural dilemma, it takes the following constructive and destructive forms, for similar reasons.

When this S is M, it must be P, and when this S is N, it must be Q;

but, This S must be M or N,

hence, This S must be P or Q.

When this S is P, it must be M, and when this S is Q, it must be N;

but, This S must be not M and not N;

hence, This S must be not P or not Q.

Note that, in complex natural dilemma, there is a destructive form which is an exact analogue of the constructive, i.e. having a necessary disjunctive minor premise and conclusion. We can reduce the destructive to the constructive, by contraposing its major premise’s horns, on the basis of its minor premise.

e. Lastly note, rebuttal of a natural dilemma, by a seemingly ‘equally cogent’ dilemma involving antithetical terms, is in no case logically possible, in view of the formal incompatibility between the needed minor premises. Try it and see.

2016-08-20T07:31:46+00:00