## 1. Dialectical Reasoning

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

*Thesis*: there are certain appearances; appearances appear.*Antithesis*: there are incompatibilities between certain of these appearances; in such cases, one or both of them must be false.*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. Genesis of Axioms

Axioms are not arbitrary, a-priori starting points of true human knowledge*. *They may be deductive or inductive, but in either case are to some extent *empirical *(in the large sense of ‘phenomenological’, i.e. without depending on any materialist or mentalist assumption concerning what is experienced).

**Deductive axioms** are established using certain positive or negative logical *arguments*, which we naturally find convincing. But even a deductive axiom relies on certain experiences, those *that gave rise* *to *the concepts and logical techniques involved in the proposition and its acknowledgment as an axiom.

The positive argument for an axiom is essentially dilemmatic: “whether this or that, so and so is true”. An example is the axiom that diversity exists. The mere *seeming* of diversity is itself a case of diversity, sufficient to establish the fact of diversity. It is no use arguing (like Parmenides or the Buddha) that this apparent diversity is an “illusion”, and that “all is really one” – because the coexistence of illusion and reality is itself an event of diversity. Thus, diversity truly exists, and cannot just be ignored. We might still try to uphold the thesis that reality is ultimately unitary, but only if we convincingly account for the fact of diversity.

Deductive axioms are also justified negatively through paradoxical logic, i.e. by showing that their contradictories are *self-contradictory*. For example, “There is no diversity” is a claim to diversity (since it involves many words, many letters, many sounds, etc.), and therefore self-contradictory; whence, it is *self-evident* that “There is some diversity”. This argument may also be construed (as above) as dilemmatic in form: “whether you deny or affirm diversity, you affirm it”.

**Inductive axioms** rely on some generalization, or (more broadly) adduction, from experience; but such inductive process in their case is not ever likely to be in need of revision. Many truths of utility to epistemology are inductive, and yet once realized remain immutable; they thus behave largely like deductive axioms, and may by analogy be classed as inductive axioms.[2]

For example, the fact that most of our beliefs are contextual is a non-contextual truth, though based on common observation. The awareness that most of our knowledge is empirical, and subject to revision as new experiences are encountered, that it is in constant flux, altering and growing – this is a broad observation that once realized will not be affected by any further empirical data. This observation is not useless, note well: it logically affects pursuit of knowledge, teaching us to remain aware of the non-finality of most of our beliefs.

But note also, the said principle of contextuality is pretty vague; it cannot by itself put specific knowledge in doubt (i.e. without some other more specific reason for doubt). Another example of such general but unspecific truth is the principle (derived from the law of the excluded middle) that “there is always some explanation”. This optimistic principle serves to encourage research, but does not tell us what the solution of the problem is specifically.

## 3. Paradoxical Propositions

A (single) paradoxical proposition has the form “if P, then notP” or “if notP, then P”, where P is any form of proposition. It is important to understand that ** such propositions are logically quite legitimate within discourse: a (single) paradox is not a contradiction**. On the other hand, a double paradox, i.e. a claim that both “if P, then notP”

*and*“if notP, then P” are true in a given case of P, is indeed a contradiction.

The law of non-contradiction states that the conjunction “P and notP” is logically impossible; i.e. contradictory propositions cannot both be true. Likewise, the law of the excluded middle states that “notP and not-notP” is logically unacceptable. The reason for these laws is that such situations of antinomy put us in a cognitive quandary – we are left with no way out of the logical difficulty, no solution to the inherent problem.

On the other hand, single paradox poses no such threat to rational thought. It leaves us with a logical way out – namely, denial of the antecedent (as self-contradictory) and affirmation of the consequent (as self-evident). The proposition “if P, then notP” logically implies “notP”, and the proposition “if notP, then P” logically implies “P”. Thus, barring double paradox, *a proposition that implies its own negation is necessarily false, and a proposition that is implied by its own negation is necessarily true*.

It follows, by the way, that the conjunction of these two hypothetical propositions, i.e. double paradox, is a breach of the law of non-contradiction, since it results in the compound conclusion that “P and notP are both true”. Double paradox also breaches the law of the excluded middle, since it equally implies “P and notP are both false”.

These various inferences may be proved and elucidated in a variety of ways:

- Since a hypothetical proposition like “if x, then y” means “x and not y is impossible” – it follows that “if P, then notP” means “P and not notP are impossible” (i.e. P is impossible), and “if notP, then P” means “notP and not P are impossible” (i.e. notP is impossible). Note this explanation well.

We know that the negation of P is the same as notP, and the negation of notP equals P, thanks to the laws of non-contradiction and of the excluded middle. Also, by the law of identity, repeating the name of an object does not double up the object: it remains one and the same; therefore, the conjunction “P and P” is equivalent to “P” and the conjunction “notP and notP” is equivalent to “notP”.

Notice that the meaning of “if P, then notP” is “(P and not notP) is *impossible*”. Thus, although this implies “notP is true”, it does *not* follow that “if notP is true, P implies notP”. Similarly, *mutatis mutandis*, for “if notP, then P”. We are here concerned with strict implication (logical necessity), not with so-called material implication.

The reason why this strict position is necessary is that in practice, truth and falsehood are contextual – most of what we believe true today might tomorrow turn out to be false, and vice-versa. On the other hand, logical necessity or impossibility refer to a much stronger relation, which in principle once established should not vary with changes in knowledge context: it applies to *all* conceivable contexts.

- Since a hypothetical proposition like “if x, then y” can be recast as “if x, then (x and y)” - it follows that “if P, then notP” equals “if P, then (P and notP)”, and “if notP, then P” equals “if notP, then (notP and P)”. In this perspective, a self-contradictory proposition implies a contradiction; since contradiction is logically impermissible, it follows that such a proposition must be false and its contradictory must be true. This can be expressed by way of apodosis, in which the laws of thought provide the categorical minor premise, making it possible for us to exceptionally draw a categorical conclusion from a hypothetical premise.

If P, then (P and notP) |

but: not(P and notP) |

therefore, not P |

If notP, then (notP and P) |

but: not(notP and P) |

therefore, not notP |

- We can also treat these inferences by way of dilemma, combining the given “if P, then notP” with “if notP, then notP” (the latter from the law of identity); or likewise, “if notP, then P” with “if P, then P”. This gives us, constructively:

If P then notP – and if notP then notP |

but: either P or notP |

therefore, notP |

If notP then P – and if P then P |

but: either notP or P |

therefore, P |

Paradox sometimes has remote outcomes. For instance, suppose Q implies P, and P implies notP (which as we saw can be rewritten as P implies both P and notP). Combining these propositions in a syllogism we obtain the conclusion “if Q, then P and notP”. The latter is also a paradoxical proposition, whose conclusion is “notQ”, even though the contradiction in the consequent does not directly concern the antecedent. Similarly, non-exclusion of the middle may appear in the form “if Q, then neither P nor notP”. Such propositions are also encountered in practice.

It is interesting that these forms, “Q implies (P and notP), therefore Q is false” and “Q implies (not P and not notP), therefore Q is false”, are the arguments implicit in our application of the corresponding laws of thought. When we come across an antinomy in knowledge, we dialectically seek to rid ourselves of it **by finding and repairing some earlier error(s) of observation or reasoning**. Thus, paradoxical argument is not only a derivative of the laws of thought, but more broadly the very way in which we regularly apply them in practice.

That is, the dialectical process we use following discovery of a contradiction or an excluded middle (or for that matter a breach of the law of identity) means that we believe that:

**Every apparent occurrence of antinomy is in reality an illusion.**

It is an illusion *due to paradox*, i.e. it means that *some of the premise(s)* that led to this apparently contradictory or middle-excluding conclusion are in error and in need of correction. The antinomy is never categorical, but hypothetical; it is a sign of and dependent on some wrong previous supposition or assumption. The apparent antinomy serves knowledge by revealing some flaw in its totality, and encouraging us to review our past thinking.

Contradiction and paradox are closely related, but not the same thing. Paradox (i.e. single not double paradox) is not equivalent to antinomy. We may look upon them as cognitive difficulties of different degrees. In this perspective, whereas categorical antinomy would be a dead-end, blocking any further thought––paradox is a milder (more hypothetical) degree of contradiction, one open to resolution.

We see from all the preceding (and from other observations below) the crucial role that paradox plays in logic. The logic of paradoxical propositions does not merely concern some far out special cases like the liar paradox. It is an essential tool in the enterprise of knowledge, helping us to establish the fundaments of thought and generally keeping our thinking free of logical impurities.

Understanding of the paradoxical forms is not a discovery of modern logic[3], although relatively recent (dating perhaps from 14^{th} Cent. CE Scholastic logic).

Drawn from *Ruminations* (2005), Chapter 1 (sections 1-3).

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

[2] Indeed, it could be argued that, since ‘deductive’ axioms all have some empirical basis (as already explicated), they are ultimately just a special case of ‘inductive’ axiom.

[3] For instance, Charles Pierce (USA, 1839-1914) noticed that some propositions imply all others. I do not know if he realized this is a property of self-contradictory or logically impossible propositions; and that self-evident or necessary propositions have the opposite property of being implied by all others. I suspect he was thinking in terms of material rather than strict implication.