# Paper Review: The Leibniz-Carnap Program for Inductive Logic

Hacking’s broad claim in this paper is that there are deep connections between Leibniz’s program for a “new kind of logic” and Carnap’s inductive logic program. His more specific thesis is that inductive logic as Carnap imagined it would only be possible if some theory like Leibniz’s were true.

***You can find the original paper here.***

Hacking thinks the problem arises from a tension between the goals of inductive logic and the limitations of its two main ingredients: the doctrine of chances and bet-hedging. In Hacking’s words “the doctrine of chances is objective but local, whereas bet-hedging is global but subjective” (p. 599). Chances are objective because they occur in contexts in which there is some stable frequency or propensity of outcomes (think dice). However they are local, because knowing the chances in one context only allows one to reason about probability in that specific context. For example, if we know the chances of a die, this does not impose any constraints on how likely we think it is that Jupiter has more than 65 moons. Thinking in terms of bet-hedging, on the other hand, is very general: in principle we can bet on any set propositions, and bet-hedging tells us how our degrees of beliefs across all propositions should cohere. However, it is not objective because beyond consistency constraints, we can have whatever probabilities we want. However, on Hacking’s view, inductive logic is meant to be both global and objective.

Hacking thinks this goal cannot be met by a program like Carnap’s unless it has theoretical underpinnings that justify the choices the inductive logician makes. In particular, this is important for the choice of language and the choice of symmetry principle. We will see how a theory like Leibniz’s would provide such an objective justification.

On Carnap’s view, one’s inductive inferences are based in no small part on the choice of a language. The general observation is that having different natural kinds in a logic can dramatically change the inferences one can carry out (think, for example, of Goodman’s new riddle of induction). Hacking notes that Leibniz and his contemporaries were aware of this problem, thinking of it as a problem of distinguishing good ideas from bad ones (indeed, it is discussed in the Port Royal Logic).

Thus, if you and I adopt different choices for our base language, then we might make radically different inductive inferences based on the same observations. The claimed objectivity of inductive logic is threatened.

One obvious way out of this problem is for there to be a single correct language. If this were the case, then this particular form of subjectivity would evaporate. This is precisely what Leibniz’s background theory makes available. Leibniz takes there to be a single true language of thought. Hacking writes:

His plans for academies and scientific journals intend to coordinate knowledge so that we can discover what are the true underlying ideas. Many of his predecessors hoped to uncover an original language preceding Babel. It would encode the true ideas. Leibniz’s better plans did not believe in lost innocence but rather in a science and a language that more and more closely correspond to the structure of the universe.

p. 605

Following the Lebinzian programme we would break our ideas apart until we reach the simple, basic ideas. These are the basic components of our language. We can then apply combinatorial principles to find a description for every possible world. If Carnap had such a justification for a choice of language then this would prevent the choice of language from leading to subjectivity of the inferences.

The second way in which subjectivity appears in Carnap’s system is (in the language we use) the choice of which symmetry principle to apply (if any). Hacking writes:

How are we entitled to apply probability theory to the set of alternatives offered by some favored set of signs? We can, of course, hedge bets. We can say that if we place bets on some possibilities we are bound, in coherence, to make other bets on other possibilities. But this gives us no objective measure of the probability of propositions in the light of the data…

(p. 608)

This is a version of the problem of the priors: on what basis do we choose our prior probability? The standard Carnapian approach is to apply some symmetry principle over some set (for example, over state descriptions). But how is this justified?

Once again, if we were to accept a system like Leibniz proposes then we would have a justification. For Leibniz every possible world has some propensity to occur (this is Hacking’s language — the exact details of how Leibniz thinks that [com]possible words are equally likely/possible is subtle). This would then give a justification for adopting a specific prior probability. In conjunction with Leibniz’s true language of thought, we would have an objective uniform distribution over possible worlds. (This approach is vaguely reminiscent of Wittgenstein’s views on probability and the independence of atomic propositions his Tractatus Logico-Philosophicus — see propositions 5.1. – 5.156).

We see how a theory like Leibniz’s — one that gave us justification to choose one true language and one true symmetry principle — would allow induction logic to be global and inductive. However, Hacking thinks that no such theory will be true. Thus, he concludes, inductive logic along the Leibniz-Carnap lines is impossible.