The Logic of Transcendental Logic

Immanuel Kant employs transcendental logic to show that the synthesis involved in judging that the conceptual "presentation" (Vorstellung) P applying to the conceptual "presentation" (Vorstellung) S also applies to intuitional presentations (Vorstellungen). In other words, the syntheses involved in the act of judgment in general ultimately make possible the world of our experience, a world in which we know objects. 

According to Kant, while general logic abstracts from the particular content of concepts related, concentrating instead on the formal features involved in relating the concepts, transcendental logic deals with the most general features of our experience of objects in space and time.  Unlike general logic, transcendental logic is not about the capacity for thinking as such, but concerns itself with our thinking in relation to our experience of objects as such.  Accordingly, transcendental logic deals with rules of synthesis in so far as this synthesis applies to intuitions as well as to concepts.  It is left to Kant's transcendental deduction to show that the necessary condition for the possibility of experience as such is that there exists a transcendental unity of apperception, an "I think" that is presupposed in all activity of knowing objects.  

Kant famously offers a transcendental deduction in the first edition of the Critique -- the "A deduction" -- which he completely rewrites in the second edition six years later -- the "B deduction." There is a pronounced difference in emphasis between the two deductions with the first being predominantly a "subjective deduction" while the second attempts an "objective deduction."  

The precise contour of the transcendental arguments are a matter of considerable debate, but one might broadly paint the  "B deduction" as follows: 

• Our experience is one of a succession of awarenesses, that is, a succession of contents of consciousness.

• The condition for a succession of awarenesses, however, is an awareness of the succession itself, that is, the successive contents of consciousness must be combined and held together in a unity of consciousness. Such a unity is a necessary condition for an experience of succession. 

• For this synthesis to be presented (represented), I must think it. 

• But this analytic unity of the self thinking its objects presupposes a synthetic unity of the manifold.  In other words, presupposed is a transcendental unity of apperception, a unity of the "I think" that is neither the empirical "self" of psychology, nor a metaphysical thinking substance a la Descartes. (The "I" could never know itself if it were not possible to unify the manifold through synthesis.)

• The transcendental unity of apperception is an objective, not a subjective, unity.  The conditions for this unity are the conditions by which we have consciousness of objects in general. 

• An object is that under the concept of which the manifold is united.  The necessary conditions for uniting the manifold is a unity of consciousness, a unity that bestows objective affinity to transcendental apperception.  

• Since there is an objective unity in the transcendental unity of apperception, the synthesis must proceed according to the categories and the rules required for experience as such.  

My aim in this brief blog post is not, however, to discuss the differences between the deductions, nor to talk about the differing views on the structure of the deductions within the voluminous secondary literature seeking to understand them. Rather I want to highlight the general modal features of transcendental arguments. I am not the first to do this, of course, but sometimes people reading Kant miss the forest for the trees. Sometimes people simply forget to mention that Kant is engaged in a modal argument of a particular kind.  Let us look at the logical structure of Kant's transcendental argument. 

Kant is interested in the necessary conditions for the possibility of experience as such.  Clearly, the argument is difficult to state if we do not include its modal features.  So what is the argument structure, when these are included? 

• Premise I:  There is the possibility of experience as such.  Using Polish notation of L for the necessity operation and M for the possibility operator, we might say 'Me', experience is possible.  

• Premise II:  It is necessary that, if experience is possible, then there exist conditions C for that experience.   We might express this as 'L, if Me, then o'.  (I am using 'o' for 'conditions'.) 

• Conclusion: Lo. 

Kant is claiming that from the mere possibility of experience we can conclude to some necessary features making possible that experience.  He is not arguing that as a matter of contingent fact some conditions (or other) obtain -- that is, empirical conditions -- that would account for that experience.  He is saying that in each and every possible world, the same conditions C must obtain, if there is a possible world where experience E is had.   

Those familiar with modal logic will understand that Kant is presupposing Lewis' S5 in order to conclude to the necessity of C.  Let us review basic modal systems briefly. 

• We might have a system that might allow us to move from necessity to possibility.  Using Polish notation, we have the distinguishing axiom 'CLpMp', if p is necessary, then p is possible.  (Read the 'C' as the conditional 'if, then', e.g., 'if Lp, then Mp'.) That is, if p obtains in all possible worlds, p obtains in some possible world.  (It is hard to conceive how something appearing in all possible world is not possible, for it is in every world that is, by definition, possible.)

• We could add to this first system another axiom this one from actuality to possibility: 'CpMp', if p obtains then p is possible.  That is to say, if p obtains in the actual world, then p obtains in a possible world.  (This seems plausible since the actual world is a possible world.)

• We can add to this second system another plausible theorem: 'CMMpMp'. We have now arrived at Lewis' system S4 holding that if something is possibly possible, then it is possible.  In other words, if p is possible in a possible world, then p is itself in a possible world.  (This seems plausible since all there are are possible worlds, and it would be strange were something possible in a possible world to somehow not simply be possible.) 

• Finally, we get to S5, sometimes assumed to be the "standard" system of model logic.  This system is generated from 'CLpMp', 'CpMp', CMMpMp' and the distinguishing assumptions of S5, 'CMLpLp', that is, if it is possible that something is necessary, then it is necessary.  Simply put, if there is possible world where some necessity holds, then, since for something to be necessary it obtains in all possible worlds, that which is necessary in that possible world is now ingredient in every possible world. (It is hard to see what being necessary in a possible world might be, if that necessity does not extend over all possible worlds.)  

Those familiar with ontological arguments for the existence of God should immediately recognize the importance of S5. Assume it is possible that God exists. Now reflect on the nature of God. Is God the kind of being that could exist contingently like a rat or an apple, or is God the king of being who, were God to exist, would exist necessarily?  If one's intuitions are of the latter, then God either exists in all possible worlds or in no possible worlds. But how do we know?  We know by checking whether or not God's existence involves a self-contradiction.  If God's existence is self-contradictory, then God does not exist in a single possible world. However, if God's existence is not self-contradictory and God's existence is not contingent, then the very possibility of God existing entails that God exists in all possible worlds including the actual world!  

So how do we apply S5 here?  Let us look at the argument again, and see if we can arrive at the conclusion. 

• Premise I: Me

• Premise II: LCMeo   (This says that necessarily, if possibly e then o.)

• S5 Assumption: CMLpLp

• But (2) is logically equivalent in all modal systems to 'CLMeLo'

•  From(3), 'C~Lp~MLp'. 

• (5) is equivalent to 'CM~pLM~p'. 

• Substituting 'e' for '~p' uniformly, we get, 'CMeLMe'. 

• Thus from (1), we derive 'LMe'. 

• Now by (4) through modus ponens we get 'Lo', and thus 'o' constituted necessary conditions for the possibility of 'e'.  QED. 

It is not immediately apparent what is wrong with this proof. Kant is engaged in critical or immanent metaphysics in the Critique. He is not talking about his believing or knowing primarily, but those states of affairs making true his believing and necessary for his knowing. The transcendental unity of apperception constitutes a necessary condition for any possible experience, that is to say, if there is a world in which there is experience 'e', then there can be no worlds in which transcendental unity fails to obtain.  The very possibility of 'e' entails the necessity of 'o'.  

Now the question of the claim: Is Kant really trying to say that 'o' obtains in all worlds, or simply that there is no world having 'e' that does not have 'o'? Are we saying that worlds in which 'e' does not obtain have 'o'?  In other words, are we asserting a necessity of consequence or a necessity of the thing consequent.  

In the medieval tradition God's foreknowledge was figured as a necessity of consequence, not a necessity of the thing consequent.  If God foreknows that S rejects God, does God's foreknowledge itself logically entail S cannot reject God? The solution was to discriminate the scope of the modal operator.  In worlds in which God foreknows S rejects God, S cannot not reject God.  However, in worlds where God does not have this foreknowledge, then S is presumably not logically determined to reject or not reject.  Are we saying that the transcendental argument is more like a necessity of consequence: In worlds were 'e' occurs, it cannot be that 'o' fails to obtain.  But how about those worlds in which 'e' does not transpire?  Must 'o' be ingredient in them as well?  And if 'o' is not ingredient, then how must we adjust the transcendental argument?  Clearly, these questions motivate a deeper investigation.