CHAPTER 28. LOGICAL COMPOSITIONS.
This chapter very briefly describes various processes having to do with the logical composition of conjunctions and disjunctions. The equations developed here, are selected to enable us to deal efficiently, in later chapters, with factorial formulas, especially. They are only a tip of an enormous iceberg, comprising hypothetical relations as well, consideration of all modalities of connections and bases, and full analysis of the negative side.
Although I personally avoid symbolic logic as much as possible, so as not to obscure for myself and others the meaning of what I am doing — in the case of the theorems below, I find that symbolization of logical relations does indeed bring out the processes more clearly. I will use a nomenclature and symbolic representation, which I personally find more comfortable, but which differs slightly from that adopted by modern logic (Copi, 319). It is, of course, to modern logicians that we owe these valuable clarifying formulas.
Let p, q, r, s be any theses, which will be conjoined or disjoined; their antitheses notp, notq, etc., may be symbolized by a so-called ‘curl’ (a curved minus sign like this: ~), as in ~p, ~q, etc. We may write: ‘p and q’ symbolically as ‘p + q’ (with plus sign) or as ‘pq’ (with no separation) or as ‘p.q’ (the dot suggesting a product). Also, we may write ‘p or q’, taken in the weakest ‘and/or’ sense, that ‘one of the theses must be true’, as ‘p v q’ (v for versus, supposedly) or as ‘p,q’ (note the comma); note that some computer programming languages use a vertical bar (like this: ½), instead. Brackets ‘{}’ are used to signify a clause within a larger sentence.
Note that the results for other forms of disjunction, like ‘or else’ or ‘either-or’, are often different; these will not be discussed here.
There are two sets of logical composition processes for us to consider:
a. ‘Addition‘, which is merging of conjunctions, or of disjunctions, with others of the same sign; the reverse process of separation, where it is feasible, might be called ‘subtraction’ (symbolized by a minus sign: –); and
b. ‘Multiplication‘, which is merging a mix of conjunctions and disjunctions with each other; the reverse direction, where it is feasible, might be called ‘division’.
Two or more propositions which are added together are said to form a single, composite proposition. A proposition which is not itself a composite of others is called elementary. Specifically, a proposition consisting of a conjunction of others is called a compound; this is one form of composition.
Note well that this logic is limited to the one sense of disjunction, and to fully problematic bases. Other manners of disjunction, such as ‘two (or more) theses must be true’ and ‘one (or more) theses must be false’, and more specific bases, each have their own logic. Also, we are here going to deal with disjunction with minimal reference to modality, although a more modal approach would be more precise and interesting. Consideration of conditional disjunction would cause the study to spill over into the interplay of hypotheticals with the processes here considered.
So the present research is limited, because its purpose is utilitarian. A fuller theory of logical composition requires additional work. However, the material dealt with here has indirect applications. Many theorems can be derived from those here described. We can, for instance, change the polarity of theses or logical relations in various ways. We can also expect that some of the laws of ‘at least one thesis must be true’ logic, carry over into ‘more than one thesis must be true’ logic.
The theorems below are presented as usually reversible eductions, meaning that given the form on the left, the form on the right follows at will, and usually vice versa. However, many of them can also be classed as deductive processes, and will reappear in later chapters in that guise. Another way to view them is, as rules of transformation, like changes in what we regard as clauses.
The analogies between mathematics and logic should not be overrated; they only go so far. Logic may be regarded as the manipulation of concepts of any kind, whereas mathematics concerns specifically numerical concepts. Although there is some intervention of mathematics in logic, for the resolution of quantitative issues, and we may be said to think logically when engaged in mathematics, these sciences are very different fields of interest. They have rationalism in common, but their scopes are different and neither is really a subsidiary of the other.
a. Addition of conjuncts:
p + {q + r} = p + q + r
{p + q} + {r + s} = p + q + r + s
…and so on for any number of conjuncts. Proof is that p, q, r, s are all independently true, anyway, on both sides of the equations. The elimination of repetitives is a special case of addition: since p + p = p, it follows that p + {p + q} = p + q.
b. Addition of alternatives:
p v {q v r} = p v q v r
{p v q} v {r v s} = p v q v r v s
…and so on for any number of disjuncts. Proof is in such cases best sought by matrixual analysis; that is, testing each and every eventual combination of theses and antitheses, to see whether or not it obeys the demands of the given composite, then comparing the results on the two sides of the equation. Thus, with three theses:
p v {q v r} | p v q v r |
p + q + r | p + q + r |
p + q + notr | p + q + notr |
p + notq + r | p + notq + r |
p + notq + notr | p + notq + notr |
notp + q + r | notp + q + r |
notp + q + notr | notp + q + notr |
notp + notq + r | notp + notq + r |
(notp + notq + notr) | (notp + notq + notr) |
The conjunctions shown in brackets are those which, having been tried out on the given disjunctions, failed the test. Both sides evidently mean that, so far, any conjunction of p, q, r and their antitheses is a conceivable outcome, to the exception of ‘notp and notq and notr’. Equations involving more theses are similarly dealt with.
As with conjuncts, the elimination of repetitives is a special case of addition. Since p v p = p (meaning, p will be affirmed in either case), it follows that p v {p v q} = p v q. Note that in the case of ‘p or else p’, notp would follow; the form ‘either p or p’ is of course inconceivable.
c. With regard to subtraction, the equations above are to be reread from right to left.
Addition followed by subtraction is useful to remove a common factor from two brackets:
{p + q} + {p + r} = p + p + q + r = p + {q + r}
{p v q} v {p v r} = p v p v q v r = p v {q v r}
…or to reshuffle brackets:
{p + q} + {r + s} = p + q + r + s = {p + r} + {r + s}
{p v q} v {r v s} = p v q v r v s = {p v r} v {r v s}
d. But the idea of ‘subtraction’ more precisely fits the equation ‘p + {~q} = p – q’, of course. This suggests implications like the following, which shall be seen again in the context of ‘logical apodosis’:
{p v q} – q implies p
{p v q v r} – r implies p v q
…and so on for any number of theses. Note that these implications are valid only in one direction. For instance, in the first case, p alone cannot tell us whether q or notq is true, and therefore cannot yield the conclusion that ‘{p v q} and notq’ are both true.
Nor may one push the analogy to mathematics so far as to move q to the other side of the implication and claim that ‘p v q’ and ‘p + q’ are equal. It only follows that ~p + ~q together imply ~{p v q}, and ~p + {p v q} together imply ‘q’.
The significance of multiplication in practise, is to clarify the logically possible combinations of theses, into compounds or other composites, which are implied by various interplays of conjunction and disjunction. This is mixed-form logic. Any impossible combinations are put aside.
a. Conjunctive multiplication:
p + {q v r} = {p + q} v {p + r} = pq v pr
{p v q} + {r v s} = pr v ps v qr v qs
Proof is best sought by matrixual analysis. Thus, with three theses, we find that only three of the conjunctions in each matrix are allowed so far, and those three are the same on both sides. Each side excludes the five bracketed conjunctions. So the two statements are equivalent.
p + {q v r} | {p + q} v {p + r} |
p + q + r | p + q + r |
p + q + notr | p + q + notr |
p + notq + r | p + notq + r |
(p + notq + notr) | (p + notq + notr) |
(notp + q + r) | (notp + q + r) |
(notp + q + notr) | (notp + q + notr) |
(notp + notq + r) | (notp + notq + r) |
(notp + notq + notr) | (notp + notq + notr) |
Equations involving more theses are similarly dealt with. Note well that if the result of such a multiplication contains an inconsistent clause, it is simply canceled out; for instance, if q = ~p, then the ‘p + q + r’ and ‘p + q + notr’ combinations are automatically eliminated, leaving only ‘p + notq + r’ as possible.
Also note that ‘p + {p v q}’ implies p (with the status of q left open), since p is already affirmed independently; this incidentally limits the ‘p v q’ clause to the two roots ‘p + q’ and ‘p + ~q’. As for ‘{p v q} + {p v r}’, it does not imply p, since notp may concur with q and r, without disobeying the premise.
b. Disjunctive multiplication:
p v {q + r} = {p v q} + {p v r} = p,q + p,r
{p + q} v {r + s} = p,r + p,s + q,r + q,s
Proof is again best sought by matrix logic. The two sides of the equation yield five identical allowances, and three identical exclusions (in brackets).
p v {q + r} | {p v q} + {p v r} |
p + q + r | p + q + r |
p + q + notr | p + q + notr |
p + notq + r | p + notq + r |
p + notq + notr | p + notq + notr |
notp + q + r | notp + q + r |
(notp + q + notr) | (notp + q + notr) |
(notp + notq + r) | (notp + notq + r) |
(notp + notq + notr) | (notp + notq + notr) |
Note that the special composite ‘p v {p + q}’ implies p (with the status of q left open), since p v p = p. In contrast, the special composite ‘{p v q} + {p v r}’ does not imply p, since notp may concur with q and r,
More complex cases are proved similarly, by testing the various roots, by exposing implied possibilities of conjunctions between all the theses and antitheses, and seeing if they correspond on both sides of the equations. In practise, multiplication of more than two clauses is best dealt with by successive multiplication of pairs of clauses.
Observe, incidentally, that the matrix of ‘p v {q + r}’ includes the three roots of ‘p + {q v r}’, and an additional two alternatives.
c. With regard to division, the equations above are to be reread from right to left. The idea of division lies in our seeming to take the common factor p out of the brackets, as in mathematics.
a. The various equations developed thus far can be used to analyze more complex mixtures of conjunction and disjunction. Processes like the following may be called ‘expansions’:
p = p + {q v ~q} = {p.q} v {p.~q}
This equation teaches us that any proposition p may be logically composed, in the way of conjunctive multiplication, with any other meaningful proposition q and its negation ~q, since the proposition ‘q or notq’ is always true by the law of the excluded middle. We can repeat the process as often as we wish, as in:
p = p + {q v ~q} + {r v ~r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r}
b. The purpose of logical composition, is to reduce any given formula to a disjunction of conjunctions. It appears that our faculty of understanding requires such reduction, to fully grasp the significance of any complex formula. This means that the results we obtained earlier for multiplication are not final, because they do not satisfy the mind’s requirement.
Expressions like ‘pq v pr’ or ‘p,q + p,r’ are not satisfactory, because they do not specify the truth or falsehood of every proposition involved. They must be expanded further, as follows:
(i) Conjunctive multiplication.
p + {q v r} = {p + q} v {p + r} = pq v pr
but, pq = {p.q.r} v {p.q.~r}
pr = {p.q.r} v {p.~q.r}
therefore, p + {q v r} = {p.q.r} v {p.q.~r} v {p.~q.r}
(ii) Disjunctive multiplication.
p v {q + r} = {p v q} + {p v r} = pp v pr v qp v qr = p v pq v pr v qr
but, p = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r}
pq = {p.q.r} v {p.q.~r}
pr = {p.q.r} v {p.~q.r}
qr = {p.q.r} v {~p.q.r}
therefore, p v {q + r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r} v {~p.q.r}
Notice the elimination of repetitive conjuncts or disjuncts, and the reordering of clauses, in accordance with the principles of addition earlier established. (That is, since xx = x and x,x = x and xy = yx and x,y = y,x.)
The above expansions are the ultimate solutions of the problems of multiplication: the most informative interpretations. The object of such process is to express the original formula in less ambiguous form. The preceding results do not clearly define the status of each of the propositions p, q, r. The components have to always be fully expanded to become comprehensible.
We see that the final equations for multiplications, are simply restatements of the matrixes of {p + [q v r]}, and {p v [q + r]}, respectively. They provide us with a set of roots, like ‘p and q and r’ or ‘p and q and notr’ or ‘p and notq and r’.
The various conjunctions in disjunction represent all the possible outcomes of the original formula. They tell us the various ways it can be read, making a list of its alternative meanings. These are the eventual inferences which can be drawn from it. Any combination which is not mentioned in the conclusion, is not inferable from the premise.
Thus, just as ordinary disjunction is best understood with reference to a matrix, so in more complex situations we must reassemble the components of our proposition into more mentally accessible results. Two formulas with the same matrix, are logically equal.
c. These findings allow us to deal with still more intricate combinations of addition and multiplication. Consider, for instance, the puzzle: What does ‘p and q or r’ mean? Using the symbolic techniques introduced thus far, we can ‘expand’ that proposition as follows. q is in an ambiguous position, between an ‘and’ and an ‘or’, so:
p + q v r may mean p + {q v r}, or may mean {p + q} v r
that is, conjunctive or disjunctive multiplication.
so, {p + q v r} = {p + [q v r]} v {[p + q] v r}
but, {p + [q v r]} = {p.q.r} v {p.q.~r} v {p.~q.r}
and, {[p + q] v r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {~p.q.r} v {~p.~q.r}
then, by addition of alternatives and elimination of repetitives, it follows that:
{p + q v r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {~p.q.r} v {~p.~q.r}
This teaches us incidentally that, since the roots of {p + q v r} are all among the roots of {[p + q] v r}, and vice versa, these two composites are no more nor less informative than each other. That is, the following equation is valid:
p + q v r = {p + q} v r
On the other hand, the composite {p + [q v r]} is more specific and restrictive than either of the composites {p + q v r} or {[p + q] v r}, because it makes allowance for less possibilities. It writes off the alternative outcomes ‘notp and q and r’ and ‘notp and notq and r’, at the outset.
d. Still more complex puzzles can be resolved. These are interesting training exercises, like ladders. The easiest course is to apply already known and simpler processes, successively.
For instance, to expand the formula: {p v q} + {r v s}, let {p v q} = x, say. Then, by substitution and conjunctive multiplication, x + {r v s} = xr v xs. This means {[p v q] + r} v {[p v q] + s}. These clauses can now be expanded, and the resulting alternatives added together. Similarly, we can clarify the formula: {p + q} v {r + s} in stages. Try doing it.
In conclusion, we see that symbolic logic can be a valuable tool for untangling perplexing statements. Modern logicians have also developed similar techniques for compositions involving hypothetical relations, as already mentioned.
However, it should also be apparent that the more intricate the formula, the less likely are we to come across it in practise. This is why modern, symbolic logic tends to degenerate into irrelevancy, and give logic as a whole a bad name.
The value of the main equations, is to show us that our sentences should be clearly formulated, so that the phrases we intend as our clauses are apparent to all. Otherwise, we might be misunderstood. This is especially important when drawing up legal documents, or making scientific statements. Perhaps the best practical applications are in computer and robot programming.
However, beyond a certain point, there is no utility in studying complex formulas, because they are sure to be misinterpreted by the uninitiated, anyway. Likewise, when interpreting texts written by other people, we cannot always be sure that they formulated them with expert knowledge and total awareness of their logical significance.[1]
Even if overly intricate logic is of limited practical utility, it is an important enough doctrine. It is a part of the grand enterprise of pursuit of consistency in Knowledge as a whole. It describes for us, how to make peace within or between large bodies of information.
[1] Perhaps we may use such logic to understand religious precepts, their extents and limits, since what is of Divine origin is theoretically bound to be consistent. Though our motive may be purely to implement these precepts, there is always a danger here of not-knowing all the rules of exegesis G-d intended.