The purpose of this paper is to investigate the many relational sets in which a logical square is the base template. Most logicians are familiar with Aristotle’s famous Square of Opposition; they are not, however, aware of how widely beyond it the template in which its terms are portrayed applies. I here propose that the template of the square is an instance of a logical Ur-structure; applicable to not only the logic of relation, but also to the logic of identity, to the logic of semiotic relations, to the logic of perception and action, and even to the categorization of philosophical disciplines. Let us begin, however, by listing the statements that comprise the Square of Opposition, and the logical relations obtaining between them; once we have done so, we shall then be in a position to generalize to the other sets of relations already mentioned, and thusly demonstrate how the configuration of the square may serve as a template by means of which to situate many basic sets of statements and relations, a fact that indicates the existence of a deep structure that they share between them.
Aristotle’s Square of Opposition situates the four base statements comprising Aristotelian (also known as traditional and syllogistic) logic. Instances of these statements are typified by a subject, S, related to a predicate, P, by means of a relational term. The four categories of relational statements follow.
1) All S is P. These statements are called A statements, and are categorized as Universally Affirmative. They assert that all members of class S are also members of class P.
2) No S is P. These statements are called E statements, and are categorized as Universally Negative. They assert that no members of class S are members of class P.
3) Some S is P. These statements are called I statements, and are categorized as Particular Affirmative. They assert that some members of class S are also members of class P.
4) Some S is not P. These statements are called O statements, and are categorized as Particular Negative. They assert that some members of class S are not members of Class P.
Several possible logical relations obtain between the various pairs that may be formed from these statements. These different possible relations follow.
1) Contradictory Statement pairs. Two statements are considered to be contradictory if both of them cannot be true, but neither can both of them be false. Thus one of the two statements must be true, and whichever of them is true, the other one must be false. A and O statements form a contradictory pair. If all S are P, it cannot be the case that some S are not P, and if some S are not P, it cannot be the case that all S are P. E and I statements also form a contradictory pair. If no S are P, it cannot be the case that some S are P, and if some S are P, it cannot be the case that no S are P.
2) Contrary Statement pairs. Two statements are considered to be contrary if both of them cannot be true, but both of them can be false. A and E statements form a contradictory pair. If all S are P, it cannot be the case that no S are P, and if no S are P, it cannot be the case that all S are P. If some but not all S are P, however, it entails that both the statement that all S are P and the statement that no S are P are false.
3) Subcontrary Statement pairs. Two statements are considered to be subcontrary if both of them cannot be false, but both of them can be true. I and O statements form a subcontrary pair. If some, but not all, S are P, then both the statement that some S are P and the statement that some S are not P are true (this situation also, as noted above, falsifies both the corresponding A and E statements). However, if some S are P, then it cannot be the case that no S are P, and if some S are not P, it cannot be the case that all S are P (they are contrary pairs). Since the statements that no S are P and that all S are P are contradictory, they cannot both be false; thus, the statements that some S are P and that some S are not P also cannot both be false.
4) Superalternate and Subalternate Statements. The superalternate statement is the universal member of the pair, and the subalternate statement is the particular member. If the universal, superalternate member of the pair is true, then the subalternate member of the pair must also be true. A and I statements form such a pair. If all S are P, then it must also be the case that some S are P. E and O statements also form such a pair. If no S are P, then it must also be the case that some S are not P.
Predicate logic, otherwise known as quantifier logic, is also amenable to having its four base statements plotted on the square, and they have the same relations to each other which statements in Aristotelian (or traditional or syllogistic) logic do; this is because they are basically the same system, using different notation. The four base statements of quantifier logic follow.
1) The Universal Inclusive. For all X, if X is a member of set S, then X is a member of set P. This statement is equivalent to the Universally Affirmative A statement in syllogistic logic that all S is P.
2) The Universal Exclusive. For all X, if X is a member of set S, then X is not a member of set P. This statement is equivalent to the Universal Negative E statement in syllogistic logic that no S is P.
3) The Particular Inclusive. There is at least one X that is a member of both set S and set P. This statement is equivalent to the Particular Affirmative statement I in syllogistic logic that some S is P.
4) The Particular Exclusive. There is at least one X that a member of set S but not a member of set P. This statement is equivalent to the Particular Negative statement O in syllogistic logic that some S is not P.
These two logics are logics of relation; let us now look at the logic of identity. It is typified by what are called Aristotle’s Three Laws of Thought. They are also equivalent, and are logically convertible into one another. They are as follows.
1) The Principle of Identity. If A, then A. This statement asserts that if some thing A exists, then it exists; if a statement is true, then it is true.
2) The Principle of Contradiction. Either A or Not A. This statement asserts that either A exists or it does not; if a statement is true, then its negation is untrue.
3) The Principle of Excluded Middle. Not both A and Not A. This statement asserts that A cannot both exist and not exist; a statement cannot be both true and false.
Of course, you can count as well as I can; therefore you are wondering how three laws of thought can occupy all four corners on a square. I contend that there is actually a fourth law of thought. It is easier to illustrate this if I first rename one of them. Let us rename the first statement the Principle of Positive Identity. Then we shall add a fourth statement, as follows.
4) The Principle of Negative Identity. If Not A, then Not A. This statement asserts that if something A does not exist, then it does not exist; if a statement is false, then it is false.
Now, we may position the four statements on the square. The Principle of Positive Identity corresponds to the A statement, and the Principle of Negative Identity corresponds to the E statement. The Principle of Contradiction and the Principle of Excluded Middle would correspond to the I and O statements, respectively. However, since we are dealing with one term rather than two, and because these four statements are logically convertible to each other, and therefore equivalent, the relations of contradictory, contrary, subcontrary, and superalternate/subalternate do not apply to any pair of them; rather, each statement logically entails the other three.
Next, we shall peruse the Semiotic Square, a structure created by Algirdas Julien Greimas to delineate the possible semantic meanings that may be expressed with a pair of verbal linguistic terms. It is therefore a structure not of identity or of relation, but of signification.
The significative expressions in the Semiotic Square are formed from a modal utterance M, followed by either an utterance of doing, D, or an utterance of state, S. Some examples of modal utterances are: having to, wanting to, being able to, knowing to, knowing how to, knowing what to, knowing where to, knowing when to, causing to, telling to, telling how to, telling what to, telling where to, telling when to, etc. Utterances of doing include do, say, make, believe, etc., utterances of state include be, know, have, etc. The four different statements are formed from any pair of these terms by varying the negations applied to them. If we unite the modal term having to with the utterance of state be, we can derive four possible significative expressions; these statements follow (~ signifies negation in this notation).
1) Having-to-Be. This is a statement of Necessity, and corresponds with the A statement of syllogistic logic. It is represented by ms or md.
2) Having-Not-to-Be. This is a statement of Impossibility, and corresponds with the E statement of syllogistic logic. It is represented by m~s or m~d.
3) Not-Having-Not-to-Be. This is a statement of Possibility, and corresponds with the I statement of syllogistic logic. It is represented by ~m~s or ~m~d.
4) Not-Having-to-Be. This is a statement of Contingency, and corresponds with the O statement of syllogistic logic. It is represented by ~ms or ~md.
Now we can examine the relations that obtain between pairs of these statements. It becomes obvious after such a perusal that the same relations that apply between pairs of statements in the Square of Opposition also apply in the Semiotic Square.
1) Contradictory pairs. The expressions Having-to-Be and Not-Having-to-Be comprise a contradictory expression pair, just like their A and O statement isomorphs do in the Square of Opposition. As was stated above, contradictory statement pairs cannot both be true, but neither can they both be false – and so it is with these two expressions. It cannot be the case for some X that it both has to be and does not have to be. Neither can it be the case that it neither has to be nor does not have to be. This relation also holds for the expressions Having-Not-to-Be and Not-Having-Not-to-Be, just as it does for their homologous E and I statements in the Square of Opposition. It cannot be the case for some X that it both has not to be and does not have not to be; neither can it be the case that it neither has not to be nor does not have not to be.
2) Contrary pairs. The expressions Having-to-Be and Having-Not-to-Be comprise a contrary expression pair, just like their A and E statement isomorphs in the Square of Opposition. As was stated above, contrary statement pairs cannot both be true, but the can both be false – and that is also the case with these two expressions. It cannot be the case for some X that it both has to be and has not to be; it can, however be the case that it neither has to be nor has not to be.
3) Subcontrary pairs. The expressions Not-Having-Not-to-Be and Not-Having-to-Be comprise a subcontrary expression pair, just like their homologue I and O statement homologues do in the Square of Opposition. As was stated above, subcontrary statements cannot both be false, but may both be true – and the same relation applies between this expression pair. It cannot be the case for some X that it neither does not have not to be nor does it have not to be. It can be the case, however, that it both does not have not to be and does not have to be.
4) Superalternate and Subalternate pairs. The expressions Having-to-Be and Not-Having-Not-to-Be comprise a superalternate/subalternate expression pair, just like their A and I statement isomorphs in the Square of Opposition. If it is the case for some X that it has to be, it must also be the case that X does not have not to be. This relation also holds for the expressions Having-Not-to-Be and Not-Having-to-Be, just as it does for their homologous E and O statements ion the Square of Opposition. If it is the case for some X that it has not to be, it must also be the case that X does not have to be.
Next, we shall investigate the logic of perception and action. This logic is applied to terms what signify modes by means of which a subject, or self, interrelates with objects in its environing world (this can include mental objects, such as ideas). Instances of this structure include perceiver-perceiving-perceived for perception, conceiver-conceiving-conceived for conception, knower-knowing-known for knowledge, signifier-signifying-signified for signification, and so forth. Other interrelating modalities include willing, doing, having, making, remembering, imagining, showing, telling, believing, and so on. Once again, this logical structure has been thought to have three terms – the self, the interrelating modality, and the object - but I see a fourth. In each case, this line that travels from the subject to the object via an actively interrelating mode cannot be the whole story; there must be a complementary actively verifying mode that flows back from the object to the subject, completing a feedback loop by means of which the subject may ascertain whether or not its interrelation has indeed achieved its object.
Let us peruse some of these interrelating modalities and examine the relations between the terms that comprise such a structure.
An object, the perceived, comes into the subject’s field of awareness. This causes the subject to perceive the object. But now what happens? In the cases that are significant for the subject, the subject then actively responds to this perception by attending to the perceived. The focusing of this attention informs the subject more fully concerning the object; not just the fact that it is, but also what it is. Attended-to objects are perceptually investigated by significantly perceiving subjects; their defining and differentiating characteristics are sought and noted. Attention thus actively lifts the object from an anonymous, marginal and incidental existence in the subject’s perceptual field into the realm of a particularly attended significant object among presently insignificant others to which attention is not being so focused. Thus, for this interrelating modality, our fourth term is attending.
The perceiver and the perceived are correlative terms. For the subject to exist as a perceiver, an object must exist as perceived. Likewise, for an object to exist as perceived, a subject must exist as a perceiver. For this situation to obtain, in other words, perceiving must occur. However, in the realm of bare existence, at least, perceiving and attending are not mutually grounding, but rather, they share a superalternate/subalternate relation. For attending to occur, perceiving must occur; however, the obverse is not the case. Passive perceiving may occur in the absence of attending. Such marginal and incidental perceptions are not, however, significant perceptions for the subject. Subjects bestow significance upon objects by actively attending to them. It may therefore be said that when the interrelation transcends the realm of bare existence and enters the realm of significance, the character of the interrelation between these two terms changes from superalternate/subalternate to mutually grounding. The significantly perceived is that to which attention is paid.
The same point may be made concerning conceiving in the presence or absence of a contemplation of the conceived, knowing in the presence or absence of the known, etc. But what can then be said concerning the signifier-signifying-signified interrelation? In this case, the deceptive single middle term is actually a conflation of two: for signifying to occur, the signifying subject must not only have perceived or conceived of the signified object, but also have attended to or contemplated it.
On to our next examination: the categorization of philosophical disciplines. First, the following questions must be posed: which of the various philosophical disciplines may be placed upon the Square of Opposition, and what are the interrelations between them?
The philosophical disciplines of phenomenology, genetic epistemology, semiotics and memetics may be placed upon such a square, precisely because certain interrelations obtain between them.
Phenomenology and genetic epistemology are philosophical stances in relation to the realm of the being of consciousness, while semiotics and memetics are philosophical stances in relation to the realm of conscious meaning, or that which the being of consciousness contains. As phenomenology and genetic epistemology are complementary disciplines with relation to the being of consciousness, so semiotics and memetics complement each other in relation to consciously held meanings. Phenomenology and semiotics are synchronic, or statically structural (the focus-field-fringe structure of perception is one such phenomenological structure; the signifier-sign-signified structure of signification is a semiotic structure); genetic epistemology and memetics are diachronic, or dynamically functional, developmental and evolutionary. Since the objects of study for phenomenology and semiotics are static, universal and apodictically self-evident structures, their forms may be astatisticsally deduced with precision, whereas, since the objects of study for genetic epistemology and memetics are particular dynamic functions that differ depending upon circumstances such as age or content, they may only be statistically and inductively inductively approximated. Phenomenology does not address the question of how self-conscious awareness could have evolved, but accepts its developmentally matured structures as ground conditions to be derived and described; genetic epistemology, by studying the emergence of the structures of self-conscious awareness in the developing child, can offer insights into how such an awareness might have evolved during the evolution of the species. Likewise, semiotics does not address the question of how symbolic capacity could have evolved, accepting its fully matured structures as ground conditions to be extracted and diagrammatically delineated, while memetics, by studying the transmission of symbolicity from the caregiver to the child and its progressive internalization by that child, implicitly offers ontogeny/phylogeny type suggestions as to how a species possessing symbolic capacity could have evolved - suggestions which I pursued in my paper Tools, Language and Text.
Both phenomenology and semiotics aspired to the status of rigorous theoretical science, but both failed; phenomenology due to its inability to plumb the depths of sedimentation - the preconscious structures underlying the structures of self-conscious awareness, semiotics due to its inability to extricate itself from the semiotic web wherein meanings mutually define in relation to each other and get beneath the sign to anchor those meanings in actual concrete lived world referents. They both, therefore, remain philosophical doctrines. Genetic epistemology is considered a soft science, and is, in fact, a main branch of developmental psychology, due to its grounding in the logical induction of successions of developing cognitive structures from the experimentally controlled observation of behavior in children; it therefore does not have to concern itself with the so far unobservable particular sets of neural traces which encode specific memetic structures. If and when direct observation of particular neural traces becomes possible, memetics may then enter the realm of science.
We could thus, by characterizing A and E disciplines are belonging to the realm of Being, and I and O disciplines as belonging to the realm of meaning, and by further categorizing A and I disciplines as being synchronic and E and O disciplines as being diachronic. Given such definitional stipulations, phenomenology would be an A discipline, genetic epistemology would be an E discipline, semiotics would be an I discipline, and memetics would be an O discipline.
The philosophical discipline of hermeneutics would fit everywhere in general on this square but nowhere in exclusive particular, as it mainly concerns itself with the optimum observational distance that an observer must adopt with regard to the observed in order to derive maximum information of a particular type from it; the resolution of the hermeneutic dialectic between distanciation and appropriation for a particular type of information derivation is the attainment, by the observer, of this optimum observational distance from the observed for that particular informational type. Once again, we have a matched pair, the observer and the observed, which mutually define by their correlative opposition, resolving a dialectical struggle between two tendencies which mutually define by their correlative opposition, distanciation and appropriation, in order to attain the optimal observational distance applicable in a particular case; for the observer to follow either tendency exclusively leads to the complete impossibility of observation – either by the observer being absolutely close to the observed, or by being infinitely far away from it.
Do the foregoing considerations qualify the logical square as an Ur-structure? Perhaps some further considerations might strengthen the case.
Currently, the three most dynamic fields in science and technology are genetics, cognitive science and computer technology; in fact, advancements in the third facilitate advances in the first two. The principles governing these fields are typically expressed is mathematical language There are four basic building blocks of genes present in our DNA; they are the amino acids adenine, guanine, cytosine, and thymine. Adenine only links with guanine, and cytosine only links with thymine; this yields two base pairs. The patterns of alternation between these pairs in the genome encode the commands by means of which it acts. The four basic logic gates in computer chips are the AND gate, the OR gate, the NAND (not-and) gate, and the NOR (not-or) gate. Human brains are composed of neurons that are connected to each other via axons and dendrites that interface in synapses; the signals sent via these pathways may be either excitatory or inhibitory. We thus have two basic structures, nodes (neurons) and pathways (dendrites and axons, linked via synapses), and these nodes communicate between each other along these pathways utilizing two basic signal types; either excitatory or inhibitory. Even in mathematics itself, four basic operations obtain; addition, subtraction, multiplication, and division - and these four operations form two pairs that are reversals of each other; subtraction is the reverse of addition, and division is the reverse of multiplication.
Why would such similarities obtain? To gain insight into this question, let us engage in a cursory examination of physics.
On the microphysical scale, it is said that there are three particles; protons, neutrons, and electrons. However, a neutron is simply a proton and electron combined. On the quantum level, it is likewise said that there are three constituents; quarks, leptons, and gluons; however, these constituents are further categorized as either bosons or mesons.
On the macrophysical scale, we have, not space and time as separable entities, since neither infinitesimal space nor instantaneous time can exist, but as different aspects of a spatiotemporal manifold, and we have matter and energy, which are comprised of the same basic ‘stuff’, except that one mode of this stuff is bound into particulate structures, and the other is free of it and flows in waves. This leaves us with two inextricably intertwined pairs. And the ways in which they affect each other are four; the strong nuclear force, the weak nuclear force, electromagnetism, and gravity, two of which operate on the microphysical level, and two of which operate on the macrophysical level.
If Aristotle’s Square of Opposition is indeed a particular surface instance of a deeper, more general structure, that structure may apply very widely and go very deep.
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