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Degrees of Paradoxicality of Logical Constants
In this section, we first discuss whether tonk is an adjoint functor or not, or whether tonk counts as a genuine logical constant according to categorical harmony, and we finally lead to the concept of intensional degrees of paradoxicality.
Let L be a (non-trivial) logical system with a deductive relation L admitting identity and cut. And suppose L contains truth constants ⊥ and T, which are specified by adjunction-induced rules ⊥ ϕ and ϕ T, respectively. The first straightforward observation is that, if L has tonk, then tonk has both left and right adjoints, and thus
tonk is the left and right adjoint of two functors. Recall that the inferential rôle of
tonk is given by: ξ ϕ ξ ϕ tonk ψ ξ ϕ tonk ψ
ξ ψ (20)
which are equivalent to the following simpler rules in the presence of identity and cut:
ϕ ϕ tonk ψ ϕ tonk ψ ψ (21)
We can see tonk as a functor from L × L to L. Now, define a “truth diagonal” functor
ΔT : L → L × L by
ΔT(ϕ) := (T, T) (22)
and also define a “falsity diagonal” functor Δ⊥ : L → L × L by
Δ⊥(ϕ) := (⊥, ⊥). (23)
We can then prove that Δ⊥ is a left adjoint of tonk, and that ΔT is a right adjoint of tonk. In other words, tonk is a right adjoint of Δ⊥ and a left adjoint of ΔT; therefore, tonk is an adjoint functor in two senses (if L is already endowed with tonk).
At the same time, however, this does not mean that the principle of categorical harmony cannot exclude tonk, a pathological connective we ought not to have in a logical system. Indeed, it is a problem in the other way around: in order to define tonk in a logical system, the principle of categorical harmony requires us to add it as a right or left adjoint of some functor, or equivalently, via an adjunction-induced bi-directional rule. Thus, when one attempts to define tonk in a logical system L according to categorical harmony, the task is the following:
1. Specify a functor F : L → L × L that has a (right or left) adjoint.
2. Prove that tonk is a (left or right) adjoint of F , or that the rules for tonk are derivable in the system L extended with the bi-directional rule that corresponds to the adjunction.
As a matter of fact, however, this turns out to be impossible.
Let us give a brief proof. Suppose for contradiction that it is possible. Then we have a functor F : L → L × L, and its right or left adjoint is tonk. Assume that tonk is a left adjoint of F , which means that F is right adjoint to tonk. It then follows that F must be truth diagonal ΔT as defined above. The bi-directional rule that corresponds to the adjunction tonk -F is actually equivalent to the following (by the property of ΔT):
ϕ1 tonk ϕ2 L ψ (24)
But this condition is not sufficient to make the rules for tonk derivable, thus the right adjoint of F cannot be tonk, and hence a contradiction. Next, assume that tonk is a right adjoint of F , i.e., F is a left adjoint of tonk. Then, F must be falsity diagonal Δ⊥, and the rule of the adjunction F -tonk is equivalent to the following:
ϕ L ψ1 tonk ψ2 (25)
This is not enough to derive the rules for tonk, and hence a contradiction. This completes the proof.
It has thus been shown that:
• Tonk cannot be defined as an adjoint functor (of some functor) in a logical system without tonk, even though tonk is an adjoint functor in a logical system that is already equipped with tonk.
– This is a subtle phenomenon, and we have to be careful of what exactly the question “Is tonk an adjoint functor?” means. Due to this, naïvely formulating categorical harmony as “logical constants = adjoint functors” does not work.
• Consequently, tonk cannot be introduced in any way according to the principle of categorical harmony.
We may then conclude that tonk is a pseudo-logical constant, and the rules for tonk are not meaning-conferring, not because it is non-conservative (i.e., Belnap's harmony fails for tonk), but because it violates the principle of categorical harmony (which is able to allow for non-conservativity as discussed above). Still, it is immediate to see the following:
• Tonk can actually be defined as being right adjoint to falsity diagonal Δ⊥, and left adjoint to truth diagonal ΔT at once. We may say that tonk is a “doubly adjoint” functor.
• In categorical harmony, therefore, it is essential to allow for a single adjunction only rather than multiple adjunctions, which are harmful in certain cases.
We again emphasise that tonk cannot be defined in a system without tonk by a single adjunction (i.e., there is no functor F such that an adjoint of F is tonk); nevertheless tonk can be defined by two adjunctions: Δ⊥ -tonk -ΔT, i.e., Δ⊥ is left adjoint to tonk, and tonk is left adjoint to ΔT. Note that double adjointness itself is not necessarily paradoxical.
What is then the conceptual meaning of all this? After all, what is wrong with tonk? The right adjoint t of falsity diagonal Δ⊥ may be called the binary truth constant (the
ordinary truth constant T is nullary), because the double-line rule of this adjunction
boils down to ϕ L ψ1 t ψ2, which means that ψ1 t ψ2 is implied by any formula ϕ (for any ψ1, ψ2). Likewise, the left adjoint s of truth diagonal ΔT may be called the “binary falsity constant”, because the double-line rule of this adjunction boils down to ψ1 s ψ2 L ϕ, which means that ψ1 s ψ2 implies any ϕ. Now, the rôle of tonk is to make the two (binary) truth and falsity constants (t and s) collapse into the same one constant, thus leading the logical system to inconsistency (or triviality); obviously, truth and falsity cannot be the same. This confusion of truth and falsity is the problem of tonk.
To put it differently, a right adjoint of Δ⊥ and a left adjoint of ΔT must be different, nevertheless tonk requires the two adjoints to be the same; the one functor that are
the two adjoints at once is tonk. The problem of tonk, therefore, lies in confusing two essentially different adjoints as if they represented the same one logical constant. We may thus conclude as follows:
• The problem of tonk is the problem of equivocation. The binary truth constant and the binary falsity constant are clearly different logical constants, yet tonk mixes them up, to be absurd.
This confusion of essentially different adjoints is at the root of the paradoxicality of tonk. There is no problem at all if we add to a logical system the right adjoint of Δ⊥ and the left adjoint of ΔT separately, any of which is completely harmless. Unpleasant phenomena only emerge if we add the two adjoints as just a single connective, that is, we make the fallacy of equivocation.
Let us think of a slightly different sort of equivocation. As explained above, ∧
is right adjoint to diagonal Δ, and ∨ is left adjoint to it. What if we confuse these
two adjoints? By way of experiment, let us define “disconjunction” as the functor that is right adjoint to diagonal, and left adjoint to it at the same time. Of course, a logical system with disconjunction leads to inconsistency (or triviality). Needless to say, the problem of disconjunction is the problem of equivocation: conjunction and disjunction are different, yet disconjunction mixes them up.
Then, is the problem of disconjunction precisely the same as the problem of tonk? This would be extensionally true, yet intensionally false. It is true in the sense that both pseudo-logical constants fall into the fallacy of equivocation. Nonetheless, it is false in the sense that the double adjointness condition of disconjunction is stronger than the double adjointness condition of tonk.
What precisely makes the difference between tonk and disconjunction? Tonk is a right adjoint of one functor, and at the same time a left adjoint of another functor. In contrast to this, disconjunction is a right and left adjoint of just a single functor. Disconjunction is, so to say, a uniformly doubly adjoint functor, as opposed to the fact that tonk is merely a doubly adjoint functor. The difference between tonk and disconjunction thus lies in uniformity. Hence:
• On the ground that uniform double adjointness is in general stronger than double adjointness, we could say that disconjunction is more paradoxical than tonk, endorsing a stronger sort of equivocation.
• We thereby lead to the concept of intensional degrees of paradoxicality of logical
constants. Degrees concerned here are degrees of uniformity of double adjointness or equivocation.
What is then the strongest degree of paradoxicality in terms of adjointness? It is self-adjointness, and it is at the source of Russell-type paradoxical constants. A selfadjoint functor is a functor that is right and left adjoint to itself. This is the strongest form of double adjointness. Now, let us think of a nullary paradoxical connective R defined by the following double-line rule (this sort of paradoxical connectives has been discussed in Schroeder-Heister [20, 22]):
Reformulating this, we obtain the following:
We may consider R as a unary constant connective R˜ : L → L defined by R˜ (ϕ) = R. Then, the double-line rule above shows that R is right and left adjoint to R, and therefore the Russell-type paradoxical constant R is a self-adjoint functor.
In order to express double adjointness, we need two functors (i.e., Δ⊥ and ΔT) in the case of tonk, one functor (i.e., Δ) in the case of disconjunction, and no functor at all in the case of paradox R. These exhibit differences in the uniformity of double adjointness. Tonk exemplifies the most general case of double adjointness and exhibits the lowest degree of uniformity. On the other hand, paradox instantiates the strongest double adjointness, and exhibits the highest degree of uniformity. Disconjunction exemplifies the only possibility in between the two.
We have thus led to three intensional degrees of paradoxicality (double adjointness
< uniform double adjointness < self-adjointness):
Right adjoint to Left adjoint to Genuine paradox R Itself R Itself R Disconjunction Diagonal Δ Diagonal Δ
Tonk Truth diagonal ΔT Fa lsity diagonal Δ⊥
The last two are caused by equivocation according to the categorical account of logical constants. In contrast, paradox R is not so for the reason that self-adjointness can be given by a single adjunction: if a functor is right (resp. left) adjoint to itself, it is left (resp. right) adjoint to itself. This is the reason why we call it “genuine paradox” in the table above. More conceptually speaking:
• Pseudo-paradoxes due to equivocation can be resolved by giving different names to right and left adjoints, respectively, which are indeed different logical constants, and it is natural to do so.
– The paradoxicality of such pseudo-paradoxes is just in mixing up actually different logical constants which are harmless on their own.
• On the other hand, we cannot resolve genuine paradox in such a way: there are no multiple meanings hidden in the Russell-type paradoxical constant, and there is nothing to be decomposed in genuine paradox.
– Genuine paradox is a truly single constant, and the paradoxicality of genuine paradox is not caused by equivocation, unlike tonk or disconjunction.
If we admit any sort of adjoint functors as logical constants, then we cannot really ban genuine paradox, which is surely an adjoint functor. A naïve formulation of Lawvere's idea of logical constants as adjoint functors, like “logical constants
= adjoint functors”, does not work here again (recall that we encountered another
case of this in the analysis of tonk). This is the reason why we have adopted the iterative conception of logic in our formulation of categorical harmony. In that view, logical constants must be constructed step by step, from old to new ones, via adjunctions. Since genuine paradox emerges via self-adjointness, however, there is no “old” operation that is able to give rise to genuine paradox via adjunction. In this way, categorical harmony based upon the iterative conception of logic allows us to avoid genuine paradox.
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