APPENDIX2:

Redefining Majority and Minority inFuture Logic

This essay was written end March 2011 to correct certain errors spotted inFuture Logic.

Introduction

In chapter 5.2, about Propositions, I introduce the quantities ‘most’ and ‘few’ as follows:

Other quantifiers define ‘some’ more precisely. Thus, ‘a few’ or ‘many’ mean, a small or large number; ‘few’ or ‘most’ mean, a minority or majority, a small or large proportion. These for most purposes have the same logical properties as particulars, though the latter two sometimes require special treatment.

This statement is correct, though rather vague. In chapter 6.2, concerning Oppositions, I define them more precisely as follows:

Also note in passing the position of forms quantified by ‘most’ or ‘few’, which we mentioned earlier. For a given polarity, the former includes the latter. Further, these quantities are intermediates between ‘all’ (which includes them) and ‘some’ (which they include). If their polarity is different, most (over 50%) and few (defined as 50% or less) are contradictory to each other. So that majority and minority are both contrary to the universal of opposite polarity, and both subcontrary to the particular of opposite polarity. They are unconnected to the singular forms.

The paragraph just quoted is unfortunatelyfilled with errors, which we shall return to further on. The problem was brought to my attention a couple of days ago by a loyal reader and perspicacious critic from Trinidad and Tobago with the alias “zahc” – whom I warmly thank. I can only plead in my defense that this book, like all others, was written in a race against time.

As regards Eductions, I apparently said nothing. But there is more on these quantities in chapter 9.4 on Syllogisms: Applications, specifically:

Regarding syllogisms involving propositions which concern a majority or minority of a class, we get results similar to those obtained with singular moods. Thus, in the first figure, there are four main valid moods, their form being: ‘If All M are (or are-not) P, and Most (or Few) S are M, then Most (or Few) S are (or are-not) P’. In the second figure, there are four main valid moods, too, with the form: ‘If All P are (or are-not) M, and Most (or Few) S are-not (or are) M, then Most (or Few) S are not P’. In the third figure, we have only two main valid moods. They are especially noteworthy in that they manage without a universal premise. Their form is: ‘Most M are (or are-not) P, Most M are S, therefore Some S are (or are-not) P’. Note that the two premises are majoritive, and the conclusion is only particular. The validity of these is due to the assumption that ‘most’ includes more than half of the middle term class, so that there is overlap in some instances.

The validity of these moods is not affected by the above mentioned problem; i.e. what is said here is okay.

Definitions

Let us try now to construct an accurate theory of the quantities ‘most’ and ‘few’ (not to confuse with ‘a few’, which is contrasted with ‘many’). The table below clarifies them enormously. We see from this table that, to deal with all eventualities, we need no less thanfivespecifications of quantity, viz.all, most, half, few,andnone. ‘Most’ and ‘few’ can also be stated as ‘in the majority of cases’ and ‘in the minority of cases’.

If we have ‘all’ on the positive side, we have ‘none’ on the negative side; and vice versa. If we have ‘most’ on the positive side, we have ‘few’ on the negative side; and vice versa. If we have ‘half’ on the positive side, we have ‘half’ on the negative side, too; and vice versa. The positive and negative sides always add up to 100%, covering all possible cases.

Thus, ‘all’ and ‘none’ mean, of course, 100 and 0 percent respectively; ‘half’ must be defined asexactly50 percent, no more and no less; ‘most’ must bedefinedas less than ‘all’ and more than ‘half’ (meaning, roughly, from 99 to 51 percent); ‘few’ must bedefinedas less than ‘half’ and more than ‘none’ (meaning, roughly, from 49 to 1 percent).

 Quantity of S that are P % of instances of S that are P % of instances of S that are not P Quantity of S that are not P Sum total of % all 100 0 none 100 most 99 1 few 100 most 98 2 few 100 most 97 3 few 100 And so on… most 53 47 few 100 most 52 48 few 100 most 51 49 few 100 half 50 50 half 100 few 49 51 most 100 few 48 52 most 100 few 47 53 most 100 And so on… few 3 97 most 100 few 2 98 most 100 few 1 99 most 100 none 0 100 all 100

Note that, in the past, I did not separately account for the ‘half-half’ situation, but stuffed it under ‘few’. I tried to define ‘few’ as ‘50% or less’ and ‘most’ as ‘over 50%’. These definitions in terms of open ranges (‘or less’, ‘over’) caused me some confusion and led me into error. The definitions now proposed are much clearer, because they are made with reference to exact quantities only, viz. 100%, 50% and 0%, even though rough ranges can be inferred from them. It should be said that the exact quantity ‘half’ does not always exist; for example, if the number of voters in an election is odd, there will be no possibility of a tie.

We can from these various considerations now propose precise eductions and oppositions.

Eductions

• ‘Most S are P’ implies ‘Few S are not P’, and vice versa.
• ‘Half S are P’ implies ‘Half S are not P’, and vice versa.
• ‘Few S are P’ implies ‘Most S are not P’, and vice versa.

Similarly, any more precise quantity on one side implies another on the other side. The quantities always come in pairs adding up to 100%, i.e. 99+1, 98+2, 97+3, etc. to 3+97, 2+98, 1+99, so that if one is known so is the other.

Each specific value between 100 and 0 implies either ‘most’ or ‘half’ or ‘few’. For instance, if 70% of S are P and (as is implied anyway) 30% of S are not P, then it is true to say that ‘Most S are P’ and (as is implied anyway) ‘Few S are not P’.

Note well that ‘Most S are P’ doesnotimply ‘Few S are P’ – they are mutually exclusive. Likewise, ‘All S are P’ does not include the lesser quantities ‘Most S are P’, ‘Half S are P’, and ‘Few S are P’.

Also, ‘Most S are P’, ‘Half S are P’, and ‘Few S are P’ tell us nothing about individual cases (i.e. do not imply ‘This S is P’).

Oppositions

‘Most S are P’, ‘Half S are P’, ‘Few S are P’, as well as ‘all S are P’, each implies ‘Some S are P’ and denies ‘No S is P’.

‘Most S are P’, ‘Half S are P’, and ‘Few S are P’, each denies ‘all S are P’.

Indeed, ‘Most S are P’, ‘Half S are P’, and ‘Few S are P’, deny each other.

Similarly, of course, on the negative side (i.e. regarding ‘S are not P’).

Granting that ‘Some S are P’, it follows that these Some must be All or Most or Half or Few; i.e. these four quantities are exhaustive. But of course, ‘No S is P’ is an alternative to ‘Some S are P’ and its four subalterns.

Thus, we see that ‘Most S are P’ and ‘Few S are P’ are contrary to each other, and not contradictory as previously suggested. That is, they are incompatible but not exhaustive.

Syllogism

What has been said earlier regarding syllogisms with quantities ‘Most’ and ‘Few’ remains unchanged. But what of the quantity ‘Half’? In the first and second figures, if the minor premise concerns ‘Half S’, so will the conclusion. In the third figure, we still have overlap for the middle term if one premise concerns ‘Most M’ (>50%) and the other concerns ‘Half M’ (=50%), so a particular conclusion (for some S) is valid. However, if both premises concern ‘Half M’, there is no overlap and no conclusion can be drawn.

Evaluation of my past treatment

When I wrote chapter 6.2 ofFuture Logic, I defined ‘most’ and ‘few’ respectively as ‘over 50%’ and ‘50% or less’. Let us now temporarily label these two concepts ‘mmost’ and ‘ffew’, for the sake of the present discussion. If we compare these concepts to those here defined, here is what we get:

• ‘mmost’ means ‘all or most’.
• ‘ffew’ means ‘half or few or none’.

I was clearly wrong to say in the past that “For a given polarity, the former includes the latter.” They complement each other, but exclude each other. I was also clearly wrong to depict ‘mmost’ and ‘ffew’ as “intermediates between ‘all’ (which includes them) and ‘some’ (which they include).” From the definitions just proposed, it is clear that ‘all’ is included in ‘mmost’ and excluded from ‘ffew’; and moreover, ‘some’ includes ‘mmost’ and somewhat conflicts with ‘ffew’ (since the later includes the possibility of ‘none’). Conversely, ‘all’ implies ‘mmost’ and denies ‘ffew’; and ‘some’ is implied by ‘mmost’ but not implied by ‘ffew’.

Moreover, since the five quantities ‘all or most or half or few or none’ are together exhaustive and contrary to each other, it follows that the two quantities ‘mmost’ and ‘ffew’ are also both exhaustive and incompatible, i.e. they are contradictory. However, this is true for a given polarity, and (contrary to what I said in the past) not foroppositepolarities. What is the opposition between, say, ‘mmost S are P’ and ‘ffew S arenotP’? Well, ‘mmost S are P’ means ‘all or most S are P’, and this is equivalent to ‘few or no S are not P’; and, ‘ffew S arenotP’ means ‘half or few or no S are not P’. Since the former disjunction (viz. ‘few or no S are not P’) is included in the latter one (viz. ‘half or few or no S are not P’), it follows that ‘mmost S are P’ (being more specific) logically implies ‘ffew S arenotP’ (which is more generic). Thus, their opposition is one of subalternation (and not contradiction, obviously).

No need to say more; it is evident that there were indeed errors in my past treatment.

Probability

‘Most’, ‘Half’, and ‘Few’ are first of all quantities, but of course, more broadly,modalitiesand degrees of probability. As quantities of the subject, they are extensional modalities. But they can also qualify the context of knowledge (logical modalities), the circumstances (natural modalities), the times and places (temporal and spatial modalities, respectively), and so forth. What we have said above must therefore be carried over to these other fields.

This concerns, principally, chapter 14.2 on Modal Oppositions and Eductions, where I wrote:

As for the oppositions of probability forms. Probability (most cases) subalternates improbability-not (few cases), and probability-not subalternates improbability; which is why we speak of degrees or levels of probability. By definition, probability (covering over half the times or circumstances) and improbability (half or less of them, let us say) are contradictory; likewise probability of negation and improbability of negation. Therefore, probability and probability-not are contrary, and their negations are subcontrary. These relations could be illustrated by a square.

This paragraph contains errors, as we shall now show. To avoid confusion here, let us use the above redefined quantities again. Let us use the term “cases”  to refer to individual contexts, instances, circumstances, times, places, whatever, in accord with the mode of modality concerned. Then:

• Probably X means “In most cases, X exists.”
• Probably not X means “In most cases, X does not exist.”
• Improbably X means “In few cases, X exists.”
• Improbably not X means “In few cases, X does not exist.”

We can also introduce the idea of ‘fifty-fifty’, say, where the number of cases is the same either way. Now, the main eductions and oppositions will be as follows:

• Probably X implies and is implied by Improbably not X.
• Fifty-fifty X implies and is implied by Fifty-fifty not X.
• Improbably X implies and is implied by Probably not X.
• Probably X, Fifty-fifty X, and Improbably X, all imply (but are not implied by) Possibly X, and are all contrary to each other and to Necessarily X and Impossibly X.
• Probably not X, Fifty-fifty not X, and Improbably not X, all imply (but are not implied by) Possibly not X, and are all contrary to each other and to Necessarily not X and Impossibly not X.

This should set the record straight.

Avi Sion