FUTURE LOGIC

CHAPTER36.NATURAL CONDITIONAL SYLLOGISM AND PRODUCTION.

1.Syllogism.

2.Summary and Quantities.

3.Production.

1.Syllogism.

Syllogism in this context involves three natural conditional propositions, all having a common subject, and whose three predicates are positioned in figures analogous to those found in categorical syllogism. Although the rules of modality, polarity, and quantity are essentially similar, there are interesting differences of detail in the results obtained.

a.The premier valid mood of syllogism involving natural conditionals is the followingfirst figuresingular necessary argument, where M is the middle term. From this mood all others are derivable.

1/nnn

When this S is M, it must be Q

When this S is P, it must be M

so, When this S is P, it must be Q.

This is validated by exposition: consider any random circumstance in which this S is actually P; then, by apodosis from the minor premise, it is also M; and, by apodosis with that from the major premise, it is also Q.

By substituting nonQ for Q, we derive a similar negative-consequent version:

When this S is M, it cannot be Q

When this S is P, it must be M

so, When this S is P, it cannot be Q.

Next, a potential version may be constructed:

1/npp

When this S is M, it must be Q

When this S is P, it can be M

so, When this S is P, it can be Q.

This mood can be validated by reductio ad absurdum to the previous. If the conclusion were denied, then ‘this S cannot be P and Q’ would be true; but the original major premise implies as its basis that ‘this S can be Q’; it follows that:

When this S is Q, it cannot be P;

but When this S is M, it must be Q,

therefore, When this S is M, it cannot be P.

The connection implied by this result, being ‘this S cannot be M and P’, causes the original minor premise to be denied. Ergo, the original conclusion is undeniable.

The negative-consequent version of this mood is the following:

When this S is M, it cannot be Q

When this S is P, it can be M

so, When this S is P, it can not-be Q.

Needless to say, any modes subaltern to the above are also valid. Thus,nnpis implied valid, bynnnornpp.

Syllogism in this figure with a potential major premise are not valid. Consider, for example, the mood below:

1/pnp

When this S is M, it can be Q

When this S is P, it must be M

so, When this S is P, it can be Q.

Although this S is M in all the circumstances relating to this S being P (minor premise), it remains conceivable that there be circumstances in which this S is M without being P (as conversion attests); these latter circumstances may be precisely among the only ones in which this S is Q, as well as M (major premise); so there is no guarantee that this S can be P and Q together (as in the attempted conclusion), indeed it may well be that this S must cease to be P before it is allowed to be Q (in which case, when this S is P, itbecomesQ).

A-fortiori, this invalidation also applies to the mode1/ppp. The argument is essentially that denying the attempted conclusion, by saying ‘This S cannot be P and Q’, does not result in the inconsistency of a denied major or minor premise. Analogous negative-consequent versions are equally spurious, of course.

We can also construct parallel actual moods. But, the following one might be regarded as more akin to apodosis than syllogism, though valid:

1/naa

When this S is M, it must (or cannot) be Q

This S is P and M

so, This S is P and Q (or nonQ).

As for the mood below, it concerns the mechanics of categorical conjunction, and hardly any longer qualifies as conditional argument in the narrow sense.

1/aaa

This S is M and Q (or nonQ), in actual circumstance,

This S is P and M, in the same circumstance,

so, This S is P and Q (or nonQ).

What we have here, of course, are interface situations, where different domains of logic meet.

Note that the modenaais subaltern toaaa(even though necessity does not imply actuality here), because we can also infer that ‘This S is M and Q (or not Q)’ from the combination of major and minor premise. However, an actual conclusion from a necessary minor premise (as in1/nnaor1/ana), and modes involving a mix of actual and potential premises (aporpa), are invalid. This is easily demonstrated.

So much for the first figure. The parallels to categorical syllogism should be obvious; and indeed, categorical syllogism can be viewed as a special case of conditional syllogism, where the subject is ‘thing’ instead of a specific ‘S’.

Note in passing that sorites are possible with natural conditionals, as with categoricals.

b.The valid singular moods of theother figurescan easily be derived from those given so far, using the methods of reduction developed in other contexts. The primary ones are listed below, for the record, without little further discussion, for the sake of brevity.

For thesecondfigure:

2/nnn

When this S is Q, it must be M

When this S is P, it cannot be M

so, When this S is P, it cannot be Q.

When this S is Q, it cannot be M

When this S is P, it must be M

so, When this S is P, it cannot be Q.

2/npp

When this S is Q, it must be M

When this S is P, it can not-be M

so, When this S is P, it can not-be Q.

When this S is Q, it cannot be M

When this S is P, it can be M

so, When this S is P, it can not-be Q.

Note the change of polarity of the major event, in this figure. Modennpis subaltern tonnnornpp; butpnpis not valid. Also valid, in the fig. 2, is mode2/naa; though notnna,ana. Two actual premises (aa), with the polarities of the events as shown above, are naturally impossible, since the middle term would have mixed polarity; however, if the middle event has exceptionally the same polarity in the two premises,aaabecomes feasible, though the minor premise is useless to the inference. Also invalid, as before, areap,paorpp.

For thethirdfigure:

3/npp

When this S is M, it must be Q

When this S is M, it can be P

so, When this S is P, it can be Q.

When this S is M, it cannot be Q

When this S is M, it can be P

so, When this S is P, it can not-be Q.

3/pnp

When this S is M, it can be Q

When this S is M, it must be P

so, When this S is P, it can be Q.

When this S is M, it can not-be Q

When this S is M, it must be P

so, When this S is P, it can not-be Q.

Subaltern tonpporpnp, is mode3/nnp; but modennnis invalid. Also valid, in the fig. 3, is modeaaa; and its subalternsnaaandana, though notnna. Also invalid, areap,paorpp, as always.

For thefourthfigure (significant mood):

4/npp

When this S is Q, it cannot be M

When this S is M, it can be P

so, When this S is P, it can not-be Q.

Note the change of polarity of the major event, in this figure; also, the mixed polarity of the middle event. Modennpis subaltern tonpp; butnnnorpnpare not valid. Also valid, in the fig. 4, is mode4/naa; though notnna,ana. Two actual premises (aa) are naturally impossible, unless the middle event has exceptionally the same polarity in the two premises. Also invalid, areap,paorpp.

c.In addition to all the above, we could construct an equal number of valid moods, whose premises and/or conclusions involve anegative antecedent, obviously. Such moods are easily validated by substituting the negation of a term for a term, in various ways. Some interesting results emerge, as the samples below show.

In figure one, all the primary moods can be reiterated, with a negative middle term (as in the sample below) and/or a negative minor term.

1/nnn

When this S is not M, it must be Q

When this S is P, it cannot be M

so, When this S is P, it must be Q.

In figure two, all the primary moods can be reiterated, with a negative major term (as in the sample below) and/or a negative minor term.

2/nnn

When this S is not Q, it must be M

When this S is P, it cannot be M

so, When this S is P, it must be Q.

In figure three, all the primary moods can be reiterated, with a negative minor term (as in the sample below) and/or a negative middle term.

3/npp

When this S is M, it must be Q

When this S is M, it can not-be P

so, When this S is not P, it can be Q.

In the fourth figure, we may switch the (mixed) polarities of the middle term, and/or of the major term, and/or insert a negative minor term. We thus have a total of 8 valid modes of polarity in each of the 4 figures.

These random examples demonstrate that the rules of polarity may seemingly be by-passed. Thus, for examples, we seem to process a negative minor premise in the first figure, or to obtain a positive conclusion in the second figure, or to draw a positive conclusion from a negative premise in the third figure. But of course, the rules of polarity are still essentially operative, the changes are illusory.

Still, such moods have practical significance. Without their clarification, we might miss out on possible inferences from data, or make errors. The reader is therefore advised to develop a full list of such syllogisms, as an exercise.

2.Summary and Quantities.

The following table neatly summarizes the results obtained in the previous section. Note the similarities and differences between the modes of modality here, and those for categorical syllogism.

Table 36.1Natural Conditional Syllogisms.

Figure One.

plus 4 with negative minor term.

Figure Two.

plus 4 with negative minor term.

Figure Three.

plus 4 with negative middle term.

Figure Four.

plus 4 with negative minor term.

In the first figure, 2 modal modes, and 1 actual mode, are valid (and these have 2 subalterns). For 8 polarity modes, this means a total of 24 (+16) valid moods. Similarly, in fig. 2, there are at least 24 (+8) valid moods, not counting the special cases ofaaa. In fig. 3, the total is 24 (+24). In fig. 4, it is at least 16 (+8), not counting the special cases ofaaa.

The grand total of primary moods is thus 88 (not counting specials alluded to in parentheses), of which 56 are modal and 32 are actual; plus 56 subalterns.

All the valid moods listed above are in the singular mode of quantity ‘sss‘, but they may of course bequantified. However, the rules of quantity are less stringent for conditional syllogism than with categorical syllogism.

This is due tosssbeing here valid throughout, because an individual instance of the subject, indicated by ‘this S’, effectively stands outside the syllogistic procedure as such, and remains recognizable independently of the three predicates, P, Q, and M which are being manipulated.

It follows that,so long as one premise is universal, a conclusion can be drawn, having the same quantity as the other premise; but no conclusion is possible from two particular premises, and the conclusion cannot be higher than the lower of the two premises.

In other words:uuu,upp,pup,uss,sus, are all valid, in all the figures, for all the moods established insss. The only invalid inferences with regard to quantity, are thereforeupu,ups,puu,pus,ppp,ppu,pps,usu,suu, obviously.

Below are the modes of quantity for each figure, with a minimum of examples, to illustrate some of the deviations from previous rules.

Thus, in the first and second figures, whileuuu,upp, anduss, remain valid, we have additionallypupandsus. For examples,

1/sus

When this S is M, it must be Q

When any S is P, it must be M

so, When this S are P, it must be Q.

2/pup

When certain S are Q, they must be M

When any S is P, it can not-be M

so, When certain S are P, they can not-be Q.

In the third figure, in addition touppandpup, the modesuuu,ussandsusare valid. For example,

3/uuu

When any S is M, it must be Q

When any S is M, it must be P

so, When any S is P, it can be Q.

In the fourth figure, for the significant mood listed above, instead of justupp, we also haveuuu,pup,uss,sus. For example,

4/pup

When certain S are Q, they cannot be M

When any S is M, it can be P

so, When certain S are P, they can not-be Q.

The reader is invited to develop a full list of plural syllogisms, as an exercise.

3.Production.

Production of natural conditionals is their inference from categorical propositions. This shows us how to construct natural conditionals deductively, rather than empirically. The structure of the premises follows the model of categorical syllogism, while the conclusion encompasses all the original terms.

a.The chief mood of such argument is in thefirst figure; it involves a necessary major, a potential minor, and a necessary conclusion, as follows:

All P must be Q

This S can be P

therefore, When this S is P, it must be Q.

We manage, exceptionally, to reason in thenpnmode, note, because the conclusion, though stronger than the minor premise, concerns a narrower set of circumstances (SP instead of just S).

This argument can be validated by exposition; for any circumstance in which this S is actually P, we know that it will also be Q according to the categorical syllogism1/AnRR. Note well that we are exceptionally drawing a necessary, though conditional, conclusion from a merely potential minor premise.

Alternatively, we can use reduction ad absurdum. Denying the conclusion means either that ‘this S cannot be P’, which contradicts the minor premise, or that ‘this S can be P and not Q’, which implies that, for this S at least, some P can not-be Q, in contradiction to the major premise. Thus, the conclusion is indubitable.

Note well that ‘When this S is P, it must be Q’ does not imply ‘All P must be Q’. Although natural conditionals may be inferred from categorical premises, it does not follow that that is the only way we can get to reach such conclusions. Natural conditionals can also be known by induction; so, they do not logically imply categoricals other than their bases and connections.

The negative version of the above mood is:

No P can be Q

This S can be P

therefore, When this S is P, it cannot be Q.

Note that if the major premise is necessary, and the minor premise is the actual or necessary ‘This S is or must be P’, then the conditional conclusion as such is unaffected; so these are subaltern moods of production.

If both premises are actual, concerning the same circumstances, the conclusion is a categorical conjunction of all three terms, which represents the actual form of natural conditional. The positive and negative versions of thisaaamode, still in the first figure, are:

All P are Q

This S is P

therefore, This S is P and Q.

No P is Q

This S is P

therefore, This S is P and not Q.

We may also have, with the same actual major, a necessary minor ‘This S must be P’, without change of conclusion (mode,ana).

Note that thennnmode is also valid, by subalternation fromnpn. It is interesting to note, however, that given the premises ‘This S must be P and all P must be Q’ we would rather draw the categorical conclusion ‘This S must be Q’, than the inferior conditional ‘When this S is P, it must be Q’. It shows the essential continuity between categorical and conditional syllogism. Given that ‘Some S can be P’ (which is the base of the minor premise) the conditional conclusion is a subaltern of the categorical one.

Also, two necessary categorical premises, with adequate modality of subsumption, may also be used to draw an actual conjunctive conclusion (nna). All the above conclusions of course further imply that ‘When this S is P, it can be Q’ or ‘… nonQ’, respectively (as in the subalternaapmode).

However, althoughnpnandnnnare valid, the modesnpaornnaare invalid, since a necessary conditional does not imply an actual conjunction. Also, the major premise could not be merely potential, since the middle term P would then not be distributive in respect of modality, even if the minor premise were necessary (pnp, orppp). For the same reason, an actual major cannot be combined with a potential minor (ap), or vice versa (pa).

With regard to quantity, the rules of categorical syllogism remain applicable here, so that the major premise must be universal, while the minor may be universal or particular, as well as singular; the conclusion has the same quantity as the minor.

b.So much for the first figure; the valid moods of theother figuresfollow from these, using the usual methods. Below is a quick overview, ignoring actuals and subalterns, which are obvious.

In figuretwo, the model moods are in thenpnmode:

No Q can be P

This S can be P

therefore, When this S is P, it cannot be Q.

All Q must be P

This S can not-be P

therefore, When this S is not P, it cannot be Q.

Observe, in the latter case, the production of a natural conditional with negative antecedent, exceptionally.

We can in both cases introduce different modalities and quantities, as we did in figure one. Note that only the minor premise may be potential or particular.

In figurethree, the process seems rather contrived, though formally supportable, because of the change of position of the minor term. The model moods are:

This P must be Q

This P can be S

therefore, When certain S are P, they must be Q.

This P cannot be Q

This P can be S

therefore, When certain S are P, they cannot be Q.

This P can be Q

This P must be S

therefore, When certain S are P, they can be Q.

This P can not-be Q

This P must be S

therefore, When certain S are P, they can not-be Q.

Note the necessary conditional conclusion from a necessary major coupled with a merely potential minor, in contrast to the conclusion being no better than potential if the major is only potential, even though the minor is necessary. Thus, though modesnpnandpnpare valid, thepnnmode is invalid.

With regard to other modalities and quantities, the rules of categorical syllogism apply here. Only the major premise may be negative; one of the premises must be necessary (or both actual); one of the premises must be particular (or both singular); and the conclusion is in any case particular.

For thefourthfigure, again the impression of artificiality, but here is the significant model mood (mode,npn) for the record, anyway, without further comment:

No Q can be P

This P can be S

therefore, When certain S are P, they cannot be Q.

c.Lastly, note thatthe combination of syllogism and productionallows us to form arguments involving four terms, in a categorical major premise and a natural conditional minor premise and conclusion. For example,

All M must be Q

When this S is P, it must be M

therefore, When this S is P, it must be Q.

Such argument need not be considered as a distinct process. We draw the proposition ‘This S can be M’ from the minor premise, and use this with the major premise to produce ‘When this S is M, it must be Q’, which is then coupled with the minor in a syllogism with the said conclusion.

d.Some additional comments on production. Consider the first figure valid mood,

All P must be Q

All S can be P

therefore, When any S is P, it must be Q.

Note well the difference between this production of a natural conditional, and the production of a logical hypothetical from the same premises: in the latter case, the conclusion would be ‘If all P must be Q and all S can be P, then all S can be Q’, or even ‘If all P must be Q and all S can be P, then when any S is P, it must be Q’. The focus in natural production is on concrete actualities, whereas logical production is concerned with formal truths.

It is also well, in this context, to keep in mind the difference between a dispensive natural conditional, ‘When any S is P, it must be Q’, which implies a number of independent singulars; and a collectional one, ‘WhenallS are P, they are Q’, which refers to the conjunction of singulars as the required condition.

In the former case, we mean: ‘When this S is P, it is Q, and when that S is P, it is Q, and …’; whereas in the latter, ‘When this S is Pandthat S is Pand…, they are Q’. The same can be said about particulars.