Paradoxes and Their Resolutions

A Thematic Compilation by Avi Sion

2. Clarifying contradiction


1.    Dialectic

The three “Laws of Thought” may be briefly explicated as follows:

  1. Thesis: there are certain appearances; appearances appear.
  2. Antithesis: there are incompatibilities between certain of these appearances; in such cases, one or both of them must be false.
  3. Synthesis: some remaining appearances must be true; find out which!

We can in this perspective consider dialectic as a fundamental form of thought, through which knowledge is made to progress on and on. It is not a mere detail, an occasional thought-process, but a driving force, an engine, of thought.

The laws are not mere information, but calls to cognitive action. They enjoin proactive and curative cognitive measures, to ensure (as much as possible at any given time) continued verification, consistency and completeness.

(i) The law of identity tells us to seek out the facts and sort them out as well as we can. The purpose of this law is to instill in people a healthy respect for facts, in the course of observation and judgment. It is essentially a call to honesty, and submission to the verdict of truth. People often think, or act as if they think, that ignoring or denying unpleasant facts or arguments will make them ‘go away’ – the law of identity says ‘no, they will not disappear, you must take them into consideration’.

Some people think that it is impossible for us to ignore that “A is A”. Far from it! All of us often do so – as when we refuse to look at or admit the evidence or a logical demonstration; when we avoid reality or evade it having glimpsed it; when we lie to ourselves or to others; and so forth. If the law of identity were always obeyed by us, there would be no need to formulate it. Logic states the obvious, because it is often shunned.

(ii) When the law of non-contradiction says to us “you cannot at once both affirm and deny a proposition”, it is also telling us that if we ever in the course of discourse encounter a situation where a proposition seems both true (for some reason) and false (for other reasons), we have to go back upstream in our discourse and find out where we went wrong in the course of it[1], and we have to effect an appropriate correction such as to eliminate the difficulty.

We are not just saying: “ah, there is a contradiction”, and leaving it at that, nonplussed. No, we are impelled to seek a solution to the problem, i.e. to resolve the contradiction. We are inferring that there must be something wrong in our earlier thinking that led us to this conundrum, some error of observation or reasoning that requires treatment. So long as this situation is tolerated, and we cannot pinpoint the source of error, the credibility of all related knowledge is proportionately diminished. Consistency must be restored as soon as possible, or we risk putting all subsequent knowledge in doubt.

(iii) Similarly, the law of the excluded middle does not just inform us that “no proposition can be claimed neither true nor false”. This law insists that if we find ourselves in such a situation, and it is indeed the case that both a proposition and its exact negation both seem false, we cannot let the matter rest or hope to find some compromise position – we have to eventually, as soon as possible, find good reason to opt for one side or the other. There is no logically acceptable middle ground, no avenue of escape.

These action implications inherent in the laws of thought may also be characterized as dialectical thinking. In this perspective, the “thesis” is our knowledge (or opinion) as it happens to be at a given time; the “antithesis” is the discovery of a logical flaw in that thesis, which causes us to have doubts about it and seek its review; and finally, the “synthesis” is the corrections we make in our premises, so as to resolve the difficulty encountered and obtain a less problematic new state of knowledge.

2.    Contradiction

Many people misunderstand what we logicians mean by ‘contradiction’. The contradictory of a term ‘A’ is its negation, ‘not A’, which refers to anything and everything in the universe other than A, i.e. wherever precisely A is absent in the world. The relation of contradiction between A and not-A is mutual, reversible, perfectly symmetrical.

The presence of something (A) excludes its absence (i.e. not A) in that very same thing, and vice versa, if all coordinates of space and time are identical. However, this does not exclude the logical possibility that the same thing may be partly A and partly not A. Thus, the law of thought ‘either A or not A’ can also be stated more quantitatively as “either ‘all A’ or ‘all not A’ or ‘part A and part not A”.

Some people appeal to this possibility of three alternatives as an argument against the laws of thought! But that is a misunderstanding – or worse, deliberate sophistry.

If something, e.g. ‘B’, implies but is not implied by not-A, it (i.e. B) is as ‘incompatible’ with A as not-A is, but it is not contradictory to A: it is merely contrary to A. The contradictory not-A of A differs from A’s contraries in that the absence of not-A implies A, whereas in the case of mere contraries like B (or B1 or B2… etc.) this added logical relation of ‘exhaustiveness’ does not apply.

When contradictories are placed in a disjunction, ‘either A or not-A’, the disjunction involved signifies both mutual exclusion (‘or’, meaning ‘not together’) and exhaustiveness (‘either’, meaning ‘and there is no other alternative’). It intends: if ‘A’, then not ‘not-A’; and if not ‘A’, then ‘not-A’.

On the other hand, any number of contraries can be placed in a disjunction: ‘A or B or B1 or B2… etc.’, so that the presence of any disjunct implies the absence of all the others; but such disjunction is not exhaustive, unless we specify that the list of contraries in it is complete. If that list is indeed complete, then the negation of all but one of the disjuncts implies the affirmation of the remaining one. Thus, ‘not-A’ can be equated to the exhaustive disjunction of all things in the world ‘contrary to A’.

Something different from A, e.g. ‘C’, is not necessarily contradictory or even contrary to A. The mere fact of difference does not imply incompatibility. Different things (like A and C) may be compatible, i.e. capable of coexistence in the same thing, at the same time and place. ‘Difference’ simply signifies that we are able to distinguish between the things concerned: i.e. they are not one and the same when they appear before our consciousness. ‘Similar’ things may be the same in appearance, but not one (e.g. two instances of the same kind); or they may be one (i.e. parts of a single whole), yet not the same.

Thus, for example, the logical relation between the colors black and white depends on how precisely we focus on them. They are different, since distinguishable. Since they may coexist on different parts of the same surface, they are broadly compatible. However, as such or per se, they are contrary; that is to say: if I perceive a surface or part of surface as totally white, and you perceive the very same place and time as totally black, our claims are incompatible[2]. This irreconcilability is not a contradiction, however, because it is possible for a surface to be neither black nor white.

The expression ‘contradiction in terms’ refers to a compound term composed of incompatible elements, such as ‘A and not A’ or ‘A and B (where B is contrary to A)’. Such a mixed-up term may be said to be paradoxical, as well as internally inconsistent, since it implies that contradiction is possible, so that the laws of thought are denied by it, and then (by generalization, if you like) ‘anything goes’ including denial of the ‘A and not A’ conjunction.

For example, the term “illusory reality” is a contradiction in terms. On the other hand, note, terms like ‘an inhuman human’ or ‘an anti-Semitic Jew’ are not strictly speaking contradictions in terms; they refer to natural possibilities of conjunction, only the terminology used makes them superficially seem contradictory (i.e. there are people who behave inhumanly, or Jews that hate their own people).

The proposition ‘A is not A’ (or ‘some thing that is A is also not A’), being self-contradictory, implies ‘A is A’, its contradictory form. This statement should be explicitly acknowledged, though obvious, because it correlates two important concepts, viz. ‘internal inconsistency’ and ‘the logic of paradoxes’.

The statement ‘A is not A’ is logically impossible, because it both affirms and denies the same thing. Therefore, the opposite statement is true. That statement, i.e. ‘A is A’, is logically necessary, because even its contradictory ‘A is not A’ implies it.

Whoever claims ‘A is not A’ is admitting ‘A is A’ – ipse dixit, he himself said it! Whereas, whoever claims ‘A is A’ is consistent with himself.

Self-contradiction consists of three items:

  1. The proposition in question, call it P.
  2. The admission that it is an assertoric statement, i.e. one that affirms or denies something.
  3. The admission that all assertoric statements involve claims to consciousness, to knowledge, to truth, etc.

Thus, given P (e.g. “reality is unknowable”), admit that P implies “this is an assertion” – but all assertions imply some knowledge of reality – therefore, P implies non-P. There is a process from P to its negation, which Logic demands we acknowledge. That demand cannot be refused without committing the very same self-contradiction. This is not a circular or ad infinitum proof, but an appeal to honesty, without which no dialogue is possible.

That all assertoric propositions assert is an aspect of the Law of Identity. The Law of Non-contradiction may be discerned in the argument: All assertions assert something; P is an assertion; therefore, P asserts; whence, if P denies asserting, P implies non-P. The Law of the Excluded Middle is also implicit here, in the awareness that we have no choice but to firmly disown P.

Disguised contradictions. Contradictions appear in discourse in many guises. They are not always overt, but may be hidden in the fact of making a statement or in the standards of judgment used.

A claim may be paradoxical because it inherently entails its own contradiction, although it does not on the surface seem to be self-inconsistent. Such implication is not always formal but requires awareness of the meaning of the terms used. This form of indirect self-contradiction has been called “the Stolen Concept fallacy”[3].

For instance, the skeptical claim “I know nothing” may be rejected as self-contradictory, because as soon as someone makes it – someone who understands and intends the meaning of the terms “I”, “know” and “nothing” – that is by itself proof absolute that the person concerned “knows” something, whence the original claim (of total ignorance) is shown up to be unavoidably contradictory and thus necessarily false.

Thus, in cases of this sort, the tacit implication involved is that one of the terms used (knowing nothing) implicitly includes the act in question (knowing that I know nothing), as a case in point contradictory to the explicit claim. (Rephrasing the said statement as “I do not know anything” does not change its underlying assumptions, needless to say.)

There are countless examples of such inherent self-contradiction. Saying “I have nothing to say” is saying something. Claiming “We have no memory” is self-contradictory, because each term in it presupposes a word, concept and background experiences remembered by the speaker – and the hearer too. An amusing common example is “I do not speak a word of English”!

Another important form of covert self-inconsistency is the use of a double standard. This consists in applying less stringent standards of judgment to one’s own discourse than to the discourse of one’s intellectual opponents. A lot of philosophical, and particularly political and religious, discourse resorts to such inequitable methodology.

The contradiction involved in a double standard is apparent the moment we step back and view its user’s knowledge and methodology as a whole. In this wider perspective, the user of a double standard is clearly inconsistent with himself, even if his discourse viewed piecemeal may superficially seem self-consistent.

Whole philosophies may be based on such fallacious reasoning. For instance, Phenomenalism sets as a general standard a limitation of knowledge to sensory data without allowing extrapolations from them to assumed external material objects – yet it does not criticize its own adductions using the same rigid standard.

There are two ways this fallacy may be committed: one may use relaxed standards on one’s own discourse, while seemingly applying universal norms to one’s opponents’ discourse; or one may appear to apply universal norms to oneself, while concocting overly strict norms for them. One may exempt oneself from the usual logical rules, or one may make unusual logical demands on others.

In either case, the holder of a double standard is in conflict with logic’s requirement of uniformity. An assumption of reason is that all humans are epistemologically on the same plane. Equity is an aspect of ‘common sense’. Experience and logic have to be used to convince oneself and others, not sophistical manipulation or authority.

Standards of judgment have to be fair and universal; all discourse must be equally treated. If differences are advocated, they have to be convincingly justified. The principle of equality admittedly involves generalization; but the onus of proof is on any proposed particularization of it.

An example of a double standard is the appeal to cultural relativism. One may seek to rationalize ideas or thought processes that are contrary to ordinary reason, by claiming them to belong to a different cultural framework. Such tolerance seems on the surface friendly and open-minded, but it is proposed without full consideration of its negative human and epistemological implications.

3.    Consistency is natural

It is important to here reiterate the principle that consistency is natural; whereas inconsistency is exceptional.

Some modern logicians have come up with the notion of “proving consistency” – but this notion is misconceived. Consistency is the natural state of affairs in knowledge; it requires no (deductive) proof and we are incapable of providing such proof, since it would be ‘placing the cart before the horse’. The only possible ‘proof’ of consistency is that no inconsistency has been encountered. Consistency is an inductive given, which is very rarely overturned. All our knowledge may be and must be assumed consistent, unless and until there is reason to believe otherwise.

In short: harmony generally reigns unnoticed, while conflicts erupt occasionally to our surprise. One might well wonder now if this principle is itself consistent with the principle herein defended that negatives are never per se objects of cognition, but only exist by denial of the corresponding positives. Our principle that consistency is taken for granted seems to imply that we on occasion have logical insights of inconsistency, something negative!

To resolve this issue, we must again emphasize the distinction between pure experience and the interpretations of experience that we, wordlessly (by mere intention) or explicitly, habitually infuse into our experiences. Generally, almost as soon as we experience something, we immediately start interpreting it, dynamically relating it to the rest of our knowledge thus far. Every experience almost unavoidably generates in us strings of associations, explanations, etc.

The contradictions we sometimes come across in our knowledge do not concern our pure experiences (which are necessarily harmonious, since they in fact exist side by side – we might add, quite ‘happily’). Our contradictions are necessarily contradictions between an interpretation and a pure experience, or between two interpretations. Contradictions do not, strictly speaking, reveal difficulties in the raw data of knowledge, but merely in the hypotheses that we conceived concerning such data.

Contradictions are thus to be blamed on reason, not on experience. This does not mean that reason is necessarily faulty, but only that it is fallible. Contradictions ought not be viewed as tragic proofs of our ignorance and stupidity – but as helpful indicators that we have misinterpreted something somewhere, and that this needs reinterpretation. These indicators are precisely one of the main tools used by the faculty of reason to control the quality of beliefs. The resolution of a contradiction is just new interpretation.

How we know that two theories, or a theory and some raw data, are ‘in contradiction’ with each other is a moot question. We dismiss this query rather facilely by referring to “logical insight”. Such insight is partly ‘experiential’, since it is based on scrutiny of the evidence and doctrines at hand. But it is clearly not entirely empirical and involves abstract factors. ‘Contradiction’ is, after all, an abstraction. I believe the answer to this question is largely given in the psychological analysis of negation.

There is an introspective sense that conflicting intentions are involved. Thus, the ‘logical insight’ that there is inconsistency is not essentially insight into a negative (a non-consistency), but into a positive (the intuitive experience of conflict of intentions). Although the word inconsistency involves a negative prefix, it brings to mind something empirically positive – a felt tension between two theses or a thesis and some data.

For this reason, to say that ‘consistency is assumable, until if ever inconsistency be found’ is consistent with our claim that ‘negations are not purely empirical’. (Notice incidentally that we did not here “prove” consistency, but merely recovered it by clarifying the theses involved.)

The above analysis also further clarifies how the law of non-contradiction is expressed in practice. It does not sort out experiences as such, but concerns more abstract items of knowledge. To understand it fully, we must be aware of the underlying intentions. A similar analysis may be proposed to explain the law of the excluded middle.

In the latter case, we would insist that (by the law of identity) ‘things are something, what they are, whatever that happen to be’. Things cannot be said to be neither this nor the negation of this, because such characterizations are negative (and, respectively, doubly negative) – and therefore cannot constitute or be claimed as positive experience. Such situations refer to uncertainties in the knower, which he is called upon to eventually fill-in. They cannot be proclaimed final knowledge (as some modern sophists have tried to do), but must be considered temporary postures in the pursuit of knowledge.

From Ruminations 1, 5, & 9.


[1]              “Check your premises”, Ayn Rand would say.

[2]              Our disagreement is not terminological, note. We have in the past agreed as to what experiences ‘black’ and ‘white’ correspond to; here, we suddenly diverge.

[3]              By Ayn Rand and (I think) Nathaniel Branden.


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