Paradoxes and Their Resolutions

A Thematic Compilation by Avi Sion

4. Clarifying paradox


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 antinomic! 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 antinomic. It may be true or false, like any other proposition. Granting its truth, it is its antecedent thesis which is antinomic, 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 antinomic, 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.

3.    Systematization

Normally, we presume our information already free of self-evident or self-contradictory theses, whereas in abnormal situations, as with paradox, necessary or impossible theses are formally acceptable eventualities.

A hypothetical of the primary form ‘If P, then Q’ was defined as ‘P and nonQ are impossible together’. But there are several ways in which this situation might arise. Either (i) both the theses, P and nonQ, are individually contingent, and only their conjunction is impossible — this is the normal situation. Or (ii) the conjunction is impossible because one or the other of the theses is individually impossible, while the remaining one is individually possible, i.e. contingent or necessary; or because both are individually impossible — these situations engender paradox.

Likewise, a hypothetical of the contradictory primary form ‘If P, not-then Q’ was defined as ‘P and nonQ are possible together’. But there are several ways this situation might arise. Either (i) both the theses, P and nonQ, and also their conjunction, are all contingent — this is the normal situation. Or (ii) one or the other of them is individually not only possible but necessary, while the remaining one is individually contingent, so that their conjunction remains contingent; or both are individually necessary, so that their conjunction is also not only possible but necessary — these situations engender paradox.

These alternatives are clarified by the following tables, for these primary forms, and also for their derivatives involving one or both antitheses. The term ‘possible’ of course means ‘contingent or necessary’, it is the common ground between the two. We will here use the symbols ‘N’ for necessary, ‘C’ for contingent (meaning possible but unnecessary), and ‘M’ for impossible. The combinations are numbered for ease of reference. The symmetries in these tables ensure their completeness.

Table 1            Modalities of Theses and Conjunctions.


The following table follows from the preceding. ‘Yes’ indicates that an implication and its contraposite are implicit in the form concerned, while ‘no’ indicates that they are excluded from it. ‘à‘ here means implies, and ‘ß‘ means is implied by.


Table 2            Corresponding Definite Hypotheticals.

Normal hypothetical logic thus assumes the theses of hypotheticals always both contingent, and so limits itself to cases Nos. 1 to 7 in the above tables. However, the abnormal cases Nos. 8 to 15, in which one or both theses are not contingent (that is, are self-evident or self-contradictory), should also be considered, to develop a complete logic of hypotheticals.

The definition of the primary positive form ‘If P, then Q’, while remaining unchanged as ‘P plus nonQ is not possible’, is now seen to more precisely comprise the following situations: Nos. 3, 6, 8, 11, 12, 14, or 15, that is, all the cases where ‘P and nonQ’ is impossible (‘M’), or ‘P implies Q’ is marked ‘yes’.

The definition of the primary negative form ‘If P, not-then Q’, while remaining unchanged as ‘P plus nonQ is not impossible’, is now seen to more precisely comprise the following situations: Nos. 1, 2, 4, 5, 7, 9, 10, or 13, that is, all the cases where ‘P and nonQ’ is contingent (C), or ‘P implies Q’ is marked ‘no’.

The other six hypothetical forms, involving the antitheses of P and/or Q, can likewise be given improved definitions, by reference to the above tables.

Notice the symmetries in these tables. In case No. 1, all conjunctions are ‘C’ and all implications are ‘no’. In cases Nos. 2-5, one conjunction is ‘M’, and one implication is ‘yes’. In cases 6-11, two conjunctions are ‘M’, and two implications are ‘yes’. In cases Nos. 12-15, three conjunctions are ‘M’, and three implications are ‘yes’. Note the corresponding statuses of individual theses in each case.

The process of contraposition is universally applicable to all hypotheticals, positive or negative, normal or abnormal, for it proceeds directly from the definitions. For this reason, in the above tables, each implication is firmly coupled with a contraposite. Likewise, the negation of any implication engenders the negation of its contraposite, so that the above tables also indirectly concern negative hypotheticals, note well.

We must be careful, in developing our theory of hypothetical propositions, to clearly formulate the breadth and limits of application of any process under consideration, and specify the exceptions if any to its rules. The validity or invalidity of logical processes often depends on whether we are focusing on normal or abnormal forms, though in some cases these two classes of proposition behave in the same way. If these distinctions are not kept in mind, we can easily become guilty of formal inconsistencies.

4.    Properties

Paradoxical propositions obey the laws of logic which happen to be applicable to all hypotheticals, that is, to hypotheticals of unspecified basis. But paradoxicals, being incontingency-based hypotheticals, have properties which normal hypotheticals lack, or lack properties which normal hypotheticals have. In such situations, where differences in logical properties occur, general hypothetical logic follows the weaker case.

The similarities and differences in formal behavior have already been dealt with in appropriate detail in the relevant chapters, but some are reviewed here in order to underscore the role played by paradox.

a.         Opposition.

In the doctrine of opposition, we claimed that ‘If P, then Q’ and ‘If P, then nonQ’ must be contrary, because if P was true, Q and nonQ would both be true, an absurdity. However, had we placed these propositions in a destructive dilemma, as below, we would have obtained a legitimate argument:

If P, then Q — and — if P, then nonQ

but either nonQ or Q

hence nonP

Likewise, ‘If P, then Q’ and ‘If nonP, then Q’ could be fitted in a valid simple constructive dilemma, yielding Q, instead of arguing as we did that they must be contrary because their contrapositions result in the absurdity of nonQ implying nonP and P.

It follows that these contrarieties are only valid conditionally, for contingency-based hypotheticals. There are exceptional circumstances in which they do not hold, namely relative to abnormal hypotheticals (including paradoxicals).

This is also independently clear from the observation of ‘yes’ marks standing parallel, in cases Nos. 8, 14, 15 (allowing for both ‘P implies nonQ’ and ‘P implies Q’, where P is impossible), and in cases Nos. 11, 12, 14 (allowing for both ‘P implies Q’ and ‘nonP implies Q’, where Q is necessary).

Similar restrictions follow automatically for the subcontrariety between ‘If P, not-then nonQ’ and ‘If P, not-then Q’, and likewise for the subalternation by the uppercase ‘If P, then Q’ of the lowercase ‘If P, not-then nonQ’ (which corresponds to obversion). These oppositions only hold true for normal hypotheticals; when dealing with abnormal hypotheticals (and therefore in general logic), we must for the sake of consistency regard the said propositions as neutral to each other.

b.         Eduction.

Similarly with the derivative eductions. The primary process of contraposition is unconditional, applicable to all hypotheticals, but the other processes can be criticized in the same way as above, by forming valid simple dilemmas, using the source proposition and the denial of the proposed target, or the contraposite(s) of one or the other or both, as horns.

Alternatively, these propositions can be combined in a syllogism, yielding a paradoxical conclusion. Thus:

In the case of obversion or obverted conversion (in the former, negate contraposite of target):

If Q, then nonP (negation of target)

if P, then Q (source)

so, if P, then nonP (paradox = nonP)

In the case of conversion by negation or obverted inversion (in the latter, negate contraposite of target):

If P, then Q (source)

if nonQ, then P (negation of target)

so, if nonQ, then Q (paradox = Q)

Thus, eductive processes other than contraposition are only good for contingency-based hypotheticals, and may not be imitated in the abnormal logic of paradoxes. This is made clear in the above tables, as follows.

Consider the paradigmatic form ‘If P, then Q’. If we limit our attention to cases Nos. 1-7, then it occurs in only two situations, subalternating (3) or implicance (6). In these two situations, ‘P implies nonQ’ is uniformly ‘no’, so the obverse, ‘If P, not-then nonQ’ is true; and the contraposite ‘Q implies nonP’ is also ‘no’, so the obverted converse, ‘If Q, not-then nonP’ is true; ‘nonP implies Q’ is uniformly ‘no’, so the obverted inverse ‘If nonP, not-then Q’ is true; and the contraposite ‘nonQ implies P’ is also ‘no’, so the converse by negation ‘If nonQ, not-then P’ is true. With regard to inversion and conversion, they are not applicable, because ‘nonP implies nonQ’ and ‘Q implies P’ are ‘no’ in one case, but ‘yes’ in the other.

However, if now we expand our attention to include cases Nos. 8-15, we see that ‘If P, then Q’ occurs additionally if P is self-contradictory and Q is contingent (8) or P is contingent and Q is self-evident (11) or P,Q are each self-evident (12) or P is self-contradictory and Q is self-evident (14) or P,Q are each self-contradictory (15). The above mentioned uniformities, which made the stated eductions feasible, now no longer hold. There is a mix of ‘no’ and ‘yes’ in the available alternatives which inhibits such eductions.

c.         Deduction.

With regard to syllogism, the nonsubaltern moods, validated by reductio ad absurdum, remain universally valid, since such indirect reduction is essentially contraposition, and no other eductive process was assumed. But the subaltern moods in all three figures, are only valid for normal hypotheticals. Since these moods presuppose subalternations for their validation, i.e. depend on direct reductions through obversion or obverted inversion, they are not valid for abnormal hypotheticals.

With regard to apodosis, the moods with a modal minor premise provide us with the entry-point into abnormal logic. As for dilemma, it is the instrument par excellence for unearthing paradoxes in the course of everyday reasoning. If we put any simple dilemma, constructive (as below) or destructive (mutatis mutandis), in syllogistic form, we obtain a paradoxical conclusion:

If P, then R — and — if Q, then R

but P and/or Q

hence, R

This implies the sorites:

If nonR, then nonP (contrapose left horn)

if nonP, then Q (minor)

if Q, then R (right horn)

hence, if nonR, then R (paradoxical conclusion = R)

Thus, paradoxical propositions are an integral part of general hypothetical logic, not some weird appendix. They highlight the essential continuity between syllogism and simple dilemma, the latter being reducible to the former.

It follows incidentally that, since (as earlier seen) apodosis may be viewed as a special, limiting case of simple dilemma, and simple dilemma as a special, limiting case of complex dilemma — all the inferential processes relating to hypotheticals are closely related.

The paradox generated by simple dilemma of course depends for its truth on the truth of the premises. We should not hurriedly infer, from the paradox inherent in every simple dilemma, that all truths are ultimately self-evident, and all falsehoods ultimately self-contradictory. Knowledge is not a purely rational enterprise, but depends largely on empirical findings.

As already pointed out, simple dilemma yields a categorical necessity or impossibility as its conclusion, only if all its premises are themselves indubitably incontingent. Should there be tacit conditions for, or any doubt regarding the unconditionality of, the hypotheticals (the horns) and/or the disjunction (the minor premise), then the conclusion would be proportionately weakened with regard to its logical modality.

Thus, with reference to the foregoing example, granting the horns of the major premise: in the specific case where our minor premise is a formally given disjunction — if, say, P and Q are contradictory to each other (P = nonQ, Q = nonP) — then the R conclusion is indeed necessary. But usually, the listed alternatives P and Q are only contextually exhaustive, so that the R conclusion is only factually true.

So, although every logical necessity is self-evident, and every logical impossibility is self-contradictory, formally speaking, according to our definitions, we might be wise to say that these predications are not in practice reciprocal, and make a distinction between apodictic and factual paradox. The former is independently obvious; the latter derives from more empirical data, and therefore, though contextually trustworthy, has a bit less weight and finality.

Note lastly, the inconsistency of two ‘equally cogent’ simple dilemmas can now be better understood, as due to their implying contrary paradoxes.


From Future Logic 31.


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