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FUTURE LOGIC

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

 

CHAPTER 51.  ELEMENTS AND COMPOUNDS.

 

            Our inquiry must now turn to a new doctrine, which may be called factorial analysis. This doctrine is to some extent an offshoot of that of opposition, and interesting for its own sake. Its essential value, however, is to prepare us for the investigation of modal induction, although some information of relevance to deduction is to be found in it. This doctrine is new, because modal logic involves a lot more forms than the traditional logic, and so an issue which was obvious and minor now looms large.

 

1.    Elements and Compounds.

2.    Gross Formulas.

3.    Oppositions.

4.    Double Syllogisms.

5.    Complements.

 

1.      Elements and Compounds.

 

            The various categorical propositions, A, E, I, O, and their modal counterparts, were presented as the building blocks or elements of knowledge. Elementary propositions are relatively abstract items of knowledge, which intersect in various combinations.

            The conjunctions of two or more such elementary propositions, concerning the same subject and predicate, may be referred to as compounds. A compound is in a sense a unit of information too, although it is expressed by us as a sum of elements. Knowledge could conceivably have been constructed by giving each compound a distinct form, but then the elements of data they contain in common would have remained hidden. We wisely, even if instinctively, chose to limit the number of forms in our thoughts, and deal with compounds in terms of their constituent elements.

            Not all elementary propositions may be conjoined, of course; some are incompatible, for example 'A and O'. Some conjunctions are redundant, as when a proposition is conjoined with another which is in any case implicit in it; for instance, A and I together mean no more than A alone. However, some compounds are significant, and our task will now be to identify these.

 

2.      Gross Formulas.

 

            At any given stage in the development of knowledge we may have no or partial or complete information, concerning the relation between a specific subject and predicate pair. The sum of information available may be called a formula. A formula may consist of one or more elementary propositions. The elementary propositions taken individually may all be formulas, if they happen to summarize the state of knowledge at that point. Their combinations in distinct consistent compounds, summing up the known without redundancies, are also possible formulas.

            Any information not included in a formula is to be considered unavailable in the context of knowledge; thus a formula must contain all known data concerning the two terms in question.

            We will express compound formulas in the briefest way, e.g. 'AIn' signifying 'A and In', without use of extraneous words or symbols for conjunction; it being understood that propositions so fused concern the same subject and predicate, of course.

            This study will concern itself only with plural propositions, although some comments about singulars will be made when useful. This is done for the sake of simplicity and clarity, but also in recognition that science is primarily interested in broad statements, and only incidentally in minutiae.

            Within the closed system of actual propositions, that covered effectively by classical logic, only five formulas were conceivable: A, I, E, O, and IO. This in a sense resolves the issue of formulas with regard to extensional modality taken in isolation.

            When the other types of modality are introduced, the issue becomes less obvious and more complex. The following table shows methodically what combinations, of the 20 elementary propositions (singulars ignored), can occur consistently and without redundancy. It results that there are a total of 195 distinct formulas, 20 of which are of course elementary, and the rest compounds of up to 4 propositions (75X2 + 79X3 + 21X4).

            Note that compounds are expressed in their most compressed form (e.g. AI is included in A). In practise, we do not always compress compound statements; sometimes we prefer to stress an implication. For instances, 'can never be' stresses that 'cannot be' implies 'is never'; 'can sometimes be' suggests a compound of 'is sometimes' and 'can'.

  Table 51.1       Consistent Conjunctions of Categoricals.

FORMULA

An Ac A At Ap In Ic I It Ip En Ec E Et Ep On Oc O Ot Op
An An Ac A At Ap In Ic I It Ip                    
AcIn   Ac A At Ap In Ic I It Ip                    
Ac   Ac A At Ap   Ic I It Ip                    
AIn     A At Ap In Ic I It Ip                    
AIc     A At Ap   Ic I It Ip                    
A     A At Ap     I It Ip                    
AtIn       At Ap In Ic I It Ip                    
AtIc       At Ap   Ic I It Ip                    
AtI       At Ap     I It Ip                    
At       At Ap       It Ip                    
ApIn         Ap In Ic I It Ip                    
ApIc         Ap   Ic I It Ip                    
ApI         Ap     I It Ip                    
ApIt         Ap       It Ip                    
Ap         Ap         Ip                    
In           In Ic I It Ip                    
Ic             Ic I It Ip                    
I               I It Ip                    
It                 It Ip                    
Ip                   Ip                    
AcInOp   Ac A At Ap In Ic I It Ip                   Op
AcOp   Ac A At Ap   Ic I It Ip                   Op
AInOp     A At Ap In Ic I It Ip                   Op
AIcOp     A At Ap   Ic I It Ip                   Op
AOp     A At Ap     I It Ip                   Op
AtInOp       At Ap In Ic I It Ip                   Op
AtIcOp       At Ap   Ic I It Ip                   Op
AtIOp       At Ap     I It Ip                   Op
AtOp       At Ap       It Ip                   Op
ApInOp         Ap In Ic I It Ip                   Op
ApIcOp         Ap   Ic I It Ip                   Op
ApIOp         Ap     I It Ip                   Op
ApItOp         Ap       It Ip                   Op
ApOp         Ap         Ip                   Op
InOp           In Ic I It Ip                   Op
IcOp             Ic I It Ip                   Op
IOp               I It Ip                   Op
ItOp                 It Ip                   Op
IpOp                   Ip                   Op
Op                                       Op
AInOt     A At Ap In Ic I It Ip                 Ot Op
AIcOt     A At Ap   Ic I It Ip                 Ot Op
AOt     A At Ap     I It Ip                 Ot Op
AtInOt       At Ap In Ic I It Ip                 Ot Op
AtIcOt       At Ap   Ic I It Ip                 Ot Op
AtIOt       At Ap     I It Ip                 Ot Op
AtOt       At Ap       It Ip                 Ot Op
ApInOt         Ap In Ic I It Ip                 Ot Op
ApIcOt         Ap   Ic I It Ip                 Ot Op
ApIOt         Ap     I It Ip                 Ot Op
ApItOt         Ap       It Ip                 Ot Op
ApOt         Ap         Ip                 Ot Op
InOt           In Ic I It Ip                 Ot Op
IcOt             Ic I It Ip                 Ot Op
IOt               I It Ip                 Ot Op
ItOt                 It Ip                 Ot Op
IpOt                   Ip                 Ot Op
Ot                                     Ot Op
AtInO       At Ap In Ic I It Ip               O Ot Op
AtIcO       At Ap   Ic I It Ip               O Ot Op
AtIO       At Ap     I It Ip               O Ot Op
AtO       At Ap       It Ip               O Ot Op
ApInO         Ap In Ic I It Ip               O Ot Op
ApIcO         Ap   Ic I It Ip               O Ot Op
ApIO         Ap     I It Ip               O Ot Op
ApItO         Ap       It Ip               O Ot Op
ApO         Ap         Ip               O Ot Op
InO           In Ic I It Ip               O Ot Op
IcO             Ic I It Ip               O Ot Op
IO               I It Ip               O Ot Op
ItO                 It Ip               O Ot Op
IpO                   Ip               O Ot Op
O                                   O Ot Op
ApInOc         Ap In Ic I It Ip             Oc O Ot Op
ApIcOc         Ap   Ic I It Ip             Oc O Ot Op
ApIOc         Ap     I It Ip             Oc O Ot Op
ApItOc         Ap       It Ip             Oc O Ot Op
ApOc         Ap         Ip             Oc O Ot Op
InOc           In Ic I It Ip             Oc O Ot Op
IcOc             Ic I It Ip             Oc O Ot Op
IOc               I It Ip             Oc O Ot Op
ItOc                 It Ip             Oc O Ot Op
IpOc                   Ip             Oc O Ot Op
Oc                                 Oc O Ot Op
InOn           In Ic I It Ip           On Oc O Ot Op
IcOn             Ic I It Ip           On Oc O Ot Op
IOn               I It Ip           On Oc O Ot Op
ItOn                 It Ip           On Oc O Ot Op
IpOn                   Ip           On Oc O Ot Op
On                               On Oc O Ot Op
AcEp   Ac A At Ap   Ic I It Ip         Ep         Op
AIcEp     A At Ap   Ic I It Ip         Ep         Op
AEp     A At Ap     I It Ip         Ep         Op
AtIcEp       At Ap   Ic I It Ip         Ep         Op
AtIEp       At Ap     I It Ip         Ep         Op
AtEp       At Ap       It Ip         Ep         Op
ApIcEp         Ap   Ic I It Ip         Ep         Op
ApIEp         Ap     I It Ip         Ep         Op
ApItEp         Ap       It Ip         Ep         Op
ApEp         Ap         Ip         Ep         Op
IcEp             Ic I It Ip         Ep         Op
IEp               I It Ip         Ep         Op
ItEp                 It Ip         Ep         Op
IpEp                   Ip         Ep         Op
Ep                             Ep         Op
AIcEpOt     A At Ap   Ic I It Ip         Ep       Ot Op
AEpOt     A At Ap     I It Ip         Ep       Ot Op
AtIcEpOt       At Ap   Ic I It Ip         Ep       Ot Op
AtIEpOt       At Ap     I It Ip         Ep       Ot Op
AtEpOt       At Ap       It Ip         Ep       Ot Op
ApIcEpOt         Ap   Ic I It Ip         Ep       Ot Op
ApIEpOt         Ap     I It Ip         Ep       Ot Op
ApItEpOt         Ap       It Ip         Ep       Ot Op
ApEpOt         Ap         Ip         Ep       Ot Op
IcEpOt             Ic I It Ip         Ep       Ot Op
IEpOt               I It Ip         Ep       Ot Op
ItEpOt                 It Ip         Ep       Ot Op
IpEpOt                   Ip         Ep       Ot Op
EpOt                             Ep       Ot Op
AtIcEpO       At Ap   Ic I It Ip         Ep     O Ot Op
AtIEpO       At Ap     I It Ip         Ep     O Ot Op
AtEpO       At Ap       It Ip         Ep     O Ot Op
ApIcEpO         Ap   Ic I It Ip         Ep     O Ot Op
ApIEpO         Ap     I It Ip         Ep     O Ot Op
ApItEpO         Ap       It Ip         Ep     O Ot Op
ApEpO         Ap         Ip         Ep     O Ot Op
IcEpO             Ic I It Ip         Ep     O Ot Op
IEpO               I It Ip         Ep     O Ot Op
ItEpO                 It Ip         Ep     O Ot Op
IpEpO                   Ip         Ep     O Ot Op
EpO                             Ep     O Ot Op
ApIcEpOc         Ap   Ic I It Ip         Ep   Oc O Ot Op
ApIEpOc         Ap     I It Ip         Ep   Oc O Ot Op
ApItEpOc         Ap       It Ip         Ep   Oc O Ot Op
ApEpOc         Ap         Ip         Ep   Oc O Ot Op
IcEpOc             Ic I It Ip         Ep   Oc O Ot Op
IEpOc               I It Ip         Ep   Oc O Ot Op
ItEpOc                 It Ip         Ep   Oc O Ot Op
IpEpOc                   Ip         Ep   Oc O Ot Op
EpOc                             Ep   Oc O Ot Op
IcEpOn             Ic I It Ip         Ep On Oc O Ot Op
IEpOn               I It Ip         Ep On Oc O Ot Op
ItEpOn                 It Ip         Ep On Oc O Ot Op
IpEpOn                   Ip         Ep On Oc O Ot Op
EpOn                             Ep On Oc O Ot Op
AEt     A At Ap     I It Ip       Et Ep       Ot Op
AtIEt       At Ap     I It Ip       Et Ep       Ot Op
AtEt       At Ap       It Ip       Et Ep       Ot Op
ApIEt         Ap     I It Ip       Et Ep       Ot Op
ApItEt         Ap       It Ip       Et Ep       Ot Op
ApEt         Ap         Ip       Et Ep       Ot Op
IEt               I It Ip       Et Ep       Ot Op
ItEt                 It Ip       Et Ep       Ot Op
IpEt                   Ip       Et Ep       Ot Op
Et                           Et Ep       Ot Op
AtIEtO       At Ap     I It Ip       Et Ep     O Ot Op
AtEtO       At Ap       It Ip       Et Ep     O Ot Op
ApIEtO         Ap     I It Ip       Et Ep     O Ot Op
ApItEtO         Ap       It Ip       Et Ep     O Ot Op
ApEtO         Ap         Ip       Et Ep     O Ot Op
IEtO               I It Ip       Et Ep     O Ot Op
ItEtO                 It Ip       Et Ep     O Ot Op
IpEtO                   Ip       Et Ep     O Ot Op
EtO                           Et Ep     O Ot Op
ApIEtOc         Ap     I It Ip       Et Ep   Oc O Ot Op
ApItEtOc         Ap       It Ip       Et Ep   Oc O Ot Op
ApEtOc         Ap         Ip       Et Ep   Oc O Ot Op
IEtOc               I It Ip       Et Ep   Oc O Ot Op
ItEtOc                 It Ip       Et Ep   Oc O Ot Op
IpEtOc                   Ip       Et Ep   Oc O Ot Op
EtOc                           Et Ep   Oc O Ot Op
IEtOn               I It Ip       Et Ep On Oc O Ot Op
ItEtOn                 It Ip       Et Ep On Oc O Ot Op
IpEtOn                   Ip       Et Ep On Oc O Ot Op
EtOn                           Et Ep On Oc O Ot Op
AtE       At Ap       It Ip     E Et Ep     O Ot Op
ApItE         Ap       It Ip     E Et Ep     O Ot Op
ApE         Ap         Ip     E Et Ep     O Ot Op
ItE                 It Ip     E Et Ep     O Ot Op
IpE                   Ip     E Et Ep     O Ot Op
E                         E Et Ep     O Ot Op
ApItEOc         Ap       It Ip     E Et Ep   Oc O Ot Op
ApEOc         Ap         Ip     E Et Ep   Oc O Ot Op
ItEOc                 It Ip     E Et Ep   Oc O Ot Op
IpEOc                   Ip     E Et Ep   Oc O Ot Op
EOc                         E Et Ep   Oc O Ot Op
ItEOn                 It Ip     E Et Ep On Oc O Ot Op
IpEOn                   Ip     E Et Ep On Oc O Ot Op
EOn                         E Et Ep On Oc O Ot Op
ApEc         Ap         Ip   Ec E Et Ep   Oc O Ot Op
IpEc                   Ip   Ec E Et Ep   Oc O Ot Op
Ec                       Ec E Et Ep   Oc O Ot Op
IpEcOn                   Ip   Ec E Et Ep On Oc O Ot Op
EcOn                       Ec E Et Ep On Oc O Ot Op
En                     En Ec E Et Ep On Oc O Ot Op
 

 

            Note in passing that if we considered either natural or temporal modality as a closed system, we would find ourselves in each case with a total of 49 formulas, 12 of which were elementaries, and the remaining 37 were compounds of up to 4 propositions. Formulas involving actual propositions only are 5 in number, and formulas which mix modality types number 102.

            Now although this list of formulas is complete in itself, it will become apparent that it does not in fact exhaust the possible states of knowledge. We shall see that formulas of this kind are gross assertions, which do not clarify all the issues involved.

 

3.      Oppositions.

 

            Once we view a compound as a unit, one complex proposition, we may ask what oppositional relations exist between compounds. Consider, for example, the affirmative compounds AIn, ApIn, ApI. They may be placed in a hierarchy relative to each other and to the cognate elements, as follows:

 

Diagram 51.1     Hierarchy of Compounds.

 

            Looking at the arrows of subalternation, we see a gradual softening of position, ranging from An to Ip. There is a continuum of affirmative statements, in which temporal modality could also be inserted. A similar hierarchy may be developed for the analogous negatives. More complex, bipolar compounds also have their inter-oppositions, including many such subalternations.

            The contradictory of any compound is a disjunctive proposition, note well; it disjoins the contradictories of the various elements involved, in an 'and/or' manner. Thus, for examples:

 

AIn is contradicted by 'O and/or Ep',

ApIn is contradicted by 'On and/or Ep',

ApI is contradicted by 'On and/or E'.

 

            If AIn is false, then one of O or EpO or Ep must be true; each of the latter is by itself only contrary to AIn: it is the disjunction as a whole which is contradictory.

            Similarly for all other compounds. Note that some 'ands' yield impossible combinations; these are as such automatically eliminated. For example: the contradictory of ApIOc is 'On and/or E and/or At', in which any combination of On with At is rejectable at once, meaning that only the alternatives 'On or EOn or E or AtE or At' are viable.

            There is no need for us to work out all the interrelationships in advance. The work can be done ad hoc, as specific need arises.

 

4.      Double Syllogisms.

 

            Once we regard a compound proposition as a single whole in its own right, we are enticed to ask whether there are corresponding compound syllogisms. Consider, for example, the closed system of actuals. Here, we have one conjunctive formula, 'I and O'; its contradictory is 'A or E', since not-{I and O} means notI and/or notO, which means E and O (= E) or I and A (= A) or E and A (impossible).

            With regard to the conjunctive compound 'I and O'. Compound syllogism is impossible in the first figure, since we would need both an A and an E major premise with the same terms. It is also impossible in the second figure, since this figure only yields negative conclusions. However, in the third figure, we have the following valid double syllogism, merging 3/IAI and 3/OAO:

 

            Some M are P and some M are not P,

            and All M are S,

            therefore, Some S are P and some S are not P.

 

            It must follow that the disjunctive compound 'A or E' (which contradicts IO) also has a valid mood of the syllogism. It must be in the first figure, disjoining 1/AAA and 1/EAE, so that denial of its conclusion causes denial of its major premise, by reductio ad absurdum to the above one:

 

            All M are P or No M is P,

            and All S are M,

            therefore, All S are P or No S is P.

 

            This shows that the compound IO, and its contradictory, have a deductive life of their own. These are the only Aristotelean syllogisms capable of processing compounds.

            The same can be done with modal compounds. I will not go into detail but simply give a pair of examples:

 

            Some M can be P and some M can not-be P,

            and All M must be S,

            therefore, Some S can be P and some S can not-be P.

 

            All M must be P or No M can be P,

            and All S must be M,

            therefore, All S must be P or No S can be P.

 

            Other quantities and modalities than these can similarly be processed. The reader is encouraged to try and evolve a full list of compound syllogisms, as an exercise, with reference to the full list of compound propositions given earlier. Are there tandems involving triple or quadruple compounds?

 

5.      Complements.

 

            To fully understand how any two terms, S and P, are related, we must know their relations in both directions: from S to P and from P to S. These may be called the front and reverse side of the overall relation. The S-P side alone can only provide us with a 'flat' picture of the intersection of the terms; the reality is 'stereoscopic', and to express it entirely we need to specify the P-S side as well.

            The S to P and P to S relations may be called complementary. The possible complements of any S to P relation are the propositions compatible with its converse. Thus, the doctrine of complements is an offshoot of the doctrines of eduction and opposition.

            Consider actual categoricals. Since 'All/This/Some S are P' are convertible to 'Some P are S' only, the possible complements of A, R, or I (in S-P), are A or O (in P-S). Since 'No S is P' is convertible fully to 'No P is S', the latter is the only possible complement of the former. Lastly, since G and O are not at all convertible, they are compatible with any of A, E, or IO, on the reverse side.

            Similarly for modals. Since An, Rn, In, Ap, Rp, Ip, are all convertible to Ip only, their possible complements are all the propositions compatible with this converse, namely any form but En. For En, which converts fully to En, the only possible complement is En. Lastly, since Gn, On, Ep, Gp, Op, are none of them at all convertible, any form may complement them. As with naturals, so with temporals.

            Just as we developed a list of possible gross formulas for the S to P relationship, we could additionally work out the compatible P to S gross formulas for each S to P gross formula. This would provide us with more complex, 'two-way' gross formulas, yielding a fuller picture of reality than heretofore available.

            (If the complement is identical in form to the original proposition, then the relation may be said to be reciprocal; otherwise, it is nonreciprocal. Thus, for instance, if 'all S are P' and 'all P are S' are both true, the relation of S and P is reciprocal; in contrast, A complemented by IO is nonreciprocal.)

            Note in passing that we could go a step further, and consider not only P-S relations as complements to S-P, but also relations involving the antitheses of one or both of the terms. In that case, obversion, obverted conversion, conversion by negation, contraposition, inversion, and obverted inversion, all become significant, telling us more about the possible combinations of S and P in all their facets.

            To get deeper still, we would perhaps have to take transitives into consideration, looking into their possible conjunctions, as 'supplements', on the S-P and P-S sides, and indeed, on every other side.

            However, all these complications will be ignored in this treatise.

 

 

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