A very important field of logic is that dealing with paradox, for it provides us with a powerful tool for establishing some of the most fundamental certainties of this science. It allows us to claim for epistemology and ontology the status of true sciences, instead of mere speculative digressions. This elegant doctrine may be viewed as part of the study of axioms.

## 1.     Internal Inconsistency

Consider the hypothetical form ‘If P, then Q’, which is an essential part of the language of logic. It was defined as ‘P and nonQ is an impossible conjunction’.

It is axiomatic that the conjunction of any proposition P and its negation nonP is impossible; thus, a proposition P and its negation nonP cannot be both true. An obvious corollary of this, obtained by regarding nonP as the proposition under consideration instead of P, is that the conjunction of any proposition nonP and its negation not-nonP is impossible; thus, a proposition P and its negation nonP cannot be both false.

So, the Law of Identity could be formulated as, “For any proposition, ‘If P, then P’ is true, and ‘If nonP, then nonP’ is true”. The Laws of Contradiction and of the Excluded Middle could be stated: “For any proposition, ‘If P, then not-nonP’ is true (P and nonP are incompatible), and ‘If not-nonP, then P’ is true (nonP and P are exhaustive)”.

Now, consider the paradoxical propositions ‘If P, then nonP’ or ‘If nonP, then P’. Such propositions appear at first sight to be obviously impossible, necessarily false, antinomies.

But let us inspect their meanings more closely. The former states ‘P and (not not)P is impossible’, which simply means ‘P is impossible’. The latter states ‘nonP and not P is impossible’, which simply means ‘nonP is impossible’. Put in this defining format, these statements no longer seem antinomial! They merely inform us that the proposition P, or nonP, as the case may be, contains an intrinsic flaw, an internal contradiction, a property of self-denial.

From this we see that there may be propositions which are logically self-destructive, and which logically support their own negations. Let us then put forward the following definitions. A proposition is self-contradictory if it denies itself, i.e. implies its own negation. A proposition is therefore self-evident if its negation is self-contradictory, i.e. if it is implied by its own negation.

Thus, the proposition ‘If P, then nonP’ informs us that P is self-contradictory (and so logically impossible), and that nonP is self-evident (and so logically necessary). Likewise, the proposition ‘If nonP, then P’ informs us that nonP is self-contradictory, and that P is self-evident.

The existence of paradoxes is not in any way indicative of a formal flaw. The paradox, the hypothetical proposition itself, is not antinomial. It may be true or false, like any other proposition. Granting its truth, it is its antecedent thesis which is antinomial, and false, as it denies itself; the consequent thesis is then true.

If the paradoxical proposition ‘If P, then nonP’ is true, then its contradictory ‘If P, not-then nonP’, meaning ‘P is not impossible’, is false; and if the latter is true, the former is false. Likewise, ‘If nonP, then P’ may be contradicted by ‘If nonP, not-then P’, meaning ‘nonP is not impossible’.

The two paradoxes ‘If P, then nonP’ and ‘If nonP, then P’ are contrary to each other, since they imply the necessity of incompatibles, respectively nonP and P. Thus, although such propositions taken singly are not antinomial, double paradox, a situation where both of these paradoxical propositions are true at once, is unacceptable to logic.

In contrast to positive hypotheticals, negative hypotheticals do not have the capability of expressing paradoxes. The propositions ‘If P, not-then P’ and ‘If nonP, not-then nonP’ are not meaningful or logically conceivable or ever true. Note this well, such propositions are formally false. Since a form like ‘If P, not-then Q’ is defined with reference to a positive conjunction as ‘{P and nonQ} is possible’, we cannot without antinomy substitute P for Q here (to say ‘{P and nonP} is possible’), or nonP for P and Q (to say ‘{nonP and not-nonP} is possible’).

It follows that the proposition ‘if P, then nonP’ does not imply the lowercase form ‘if P, not-then P’, and the proposition ‘if nonP, then P’ does not imply the lowercase form ‘if nonP, not-then nonP’. That is, in the context of paradox, hypothetical propositions behave abnormally, and not like contingency-based forms.

This should not surprise us, since the self-contradictory is logically impossible and the self-evident is logically necessary. Since paradoxical propositions involve incontingent theses and antitheses, they are subject to the laws specific to such basis.

The implications and consistency of all this will be looked into presently.

## 2.     The Stolen Concept Fallacy

Paradoxical propositions actually occur in practice; moreover, they provide us with some highly significant results. Here are some examples:

• denial, or even doubt, of the laws of logic conceals an appeal to those very axioms, implying that the denial rather than the assertion is to be believed;
• denial of man’s ability to know any reality objectively, itself constitutes a claim to knowledge of a fact of reality;
• denial of validity to man’s perception, or his conceptual power, or reasoning, all such skeptical claims presuppose the utilization of and trust in the very faculties put in doubt;
• denial on principle of all generalization, necessity, or absolutes, is itself a claim to a general, necessary, and absolute, truth.
• denial of the existence of ‘universals’, does not itself bypass the problem of universals, since it appeals to some itself, namely, ‘universals’, ‘do not’, and ‘exist’.

More details on these and other paradoxes, may be found scattered throughout the text. Thus, the uncovering of paradox is an oft-used and important logical technique. The writer Ayn Rand laid great emphasis on this method of rejecting skeptical philosophies, by showing that they implicitly appeal to concepts which they try to explicitly deny; she called this ‘the fallacy of the Stolen Concept’.

A way to understand the workings of paradox, is to view it in the context of dilemma. A self-evident proposition P could be stated as ‘Whether P is affirmed or denied, it is true’; an absolute truth is something which turns out to be true whatever our initial assumptions.

This can be written as a constructive argument whose left horn is the axiomatic proposition of P’s identity with itself, and whose right horn is the paradox of nonP’s self-contradiction; the minor premise is the axiom of thorough contradiction between the antecedents P and nonP; and the conclusion, the consequent P’s absolute truth.

If P, then P — and — if nonP, then P

but either P or nonP

hence, P.

A destructive version can equally well be formulated, using the contraposite form of identity, ‘If nonP, then nonP’, as left horn, with the same result.

If nonP, then nonP — and — if nonP, then P

but either not-nonP or nonP

hence, not-nonP, that is, P.

The conclusion ‘P’ here, signifies that P is logically necessary, not merely that P is true, note well; this follows from the formal necessity of the minor premise, the disjunction of P and nonP, assuming the right horn to be well established.

Another way to understand paradox is to view it in terms of knowledge contexts. Reading the paradox ‘if nonP, then P’ as ‘all contexts with nonP are contexts with P’, and the identity ‘if P, then P’ as ‘all contexts with P are contexts with P’, we can infer that ‘all contexts are with P’, meaning that P is logically necessary.

We can in similar ways deal with the paradox ‘if P, then nonP’, to obtain the conclusion ‘nonP’, or better still: P is impossible. The process of resolving a paradox, by drawing out its implicit categorical conclusions, may be called dialectic.

Note in passing that the abridged expression of simple dilemma, in a single proposition, now becomes more comprehensible. The compound proposition ‘If P, then {Q and nonQ}’ simply means ‘nonP’; ‘If nonP, then {Q and nonQ}’ means ‘P’; ‘If (or whether) P or nonP, then Q’ means ‘Q’; and ‘If (or whether) P or nonP, then nonQ’ means ‘nonQ’. Such propositions could also be categorized as paradoxical, even though the contradiction generated concerns another thesis.

However, remember, the above two forms should not be confused with the lesser, negative hypothetical, relations ‘Whether P or nonP, (not-then not) Q’ or ‘Whether P or nonP, (not-then not) nonQ’, respectively, which are not paradoxical, unless there are conditions under which they rise to the level of positive hypotheticals.

Drawn from Future Logic (1990), Chapter 31 (sections 1 & 2).