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The Laws of Thought

A Thematic Compilation by Avi Sion

12. On Contradiction


1.     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[1]. This irreconcilability is not a contradiction, however, because it is possible for a surface to be neither black nor white.


2.     Varieties of Contradiction


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.


3.     Double Standards


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”[2].

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.


Drawn from Ruminations (2005), Chapter 1 (sections 4-6).



[1]           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.

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

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