20th WCP: ‘Probabilist’ Deductive Inference in Gassendi's Logic

In Book IV (On method) of his Institutio Logica, Pierre Gassendi proposes an unusual venue for probable and nondeductive inference in empirical reasoning: demonstrative syllogism. Thus, in 'resolution'—the seeking of a thing's causes given the evidence of its effects—he recognizes the critical role of inferring general claims from the particulars of empirical data, not least from what he calls 'the evidence of signs'. And intriguingly, he construes resolution-based claims as merely probable, though we attain them through classically deductive syllogism, because they represent merely possible claims among a field of alternatives—perhaps in the manner of Descartes's method. In the preceding book (On the syllogism), Gassendi presents a relatively traditional analysis of syllogisms which model our inductive and probabilistic reasoning, in which he anticipates Hume's critique of induction. All these views bear the mark of his distinctively strong empiricism. He proposes (quite reasonably) that our inductive inferences lack the information we would need to be certain of claims they suggest, and (a bit more surprisingly) that not even deductivist inference can insure our certainty about empirical claims because the experientially attained premises we adduce in support of such claims are no greater than probable.

We might think, on the basis of this last notion, that Gassendi has a good enough seventeenth century grasp of inductivist logic, and that it's rather deductivist logic he doesn't fully understand. Yet, while something is surely amiss in calling deductivist inference 'probabilistic', it seems Gassendi has hit upon a now-familiar, sensible point—that the use of deductive reasoning in empirical contexts, while providing certain formal guarantees, does not insulate empirical arguments from judgement by the measure of belief we invest in their premises. Such a view is possible for Gassendi to begin with because he is among those early Moderns who allow that we may have warrant for claims though we are not certain of them; this is the 'degrees of belief' concept which figures prominently in the development of modern probability theory. The more general point, which distinguishes Gassendi among his contemporaries, is that the strength all empirical claims share, irrespective of the way we infer them, consists in the warrant from experience for those claims we introduce in their support.

Thus, Gassendi's account of inductive reasoning also suggests we may infer claims about which we are less than fully certain. Yet he doesn't think of induction as a probabilist enterprise per se, because he reserves 'probabilist' for characterizing deductivist reasoning about empirical matters. His main concern about the less than certain nature of such reasoning is not with the conditional probability of some claim given our assumptions but with the probability that we are wrong about those assumptions to begin with. And just in case induction and this sort of probabilistic reasoning seem to be linked—as when we embed background inductive inference in assumptions of demonstrative syllogism—Gassendi thinks our inferences count rather as deductivist reasoning. In short, to call a piece of reasoning 'probabilistic', it is not enough that we can evaluate the merits of the conclusion given our degree of belief in its less-than-certain supporting premises; that would, of course, make inductive reasoning 'probabilistic' as well. Rather, a piece of reasoning is 'probabilistic' for Gassendi only in the sense that we draw upon that degree of belief to suggest the fallibility of the conclusion though we recognize that our argument is a valid one—and, in the context of his logic, this recognition occurs only when the argument is an instance of viable deductive syllogism. Whatever the flaws of this view (and, as I suggest below, they are several), it is noteworthy that Gassendi attempts to fit a suitable ('modern') probability concept into a general theory of what makes for adequate reasoning patterns. It is further noteworthy that he focuses here more on the content of our inferences and less on their formal character, and that he suggests we make room for probabilist inference without leaving behind the demonstrative force of deductive syllogism. This deductivist persistence might be a consequence of his recognition of induction's insecurities. His attempt to characterize a certain class of deductive syllogism as 'probabilist' (however quixotic) is surely an outcome of his global empiricism.

I begin my assessment of these views where Gassendi himself commences, with his discussion of induction, which he describes as reasoning that (typically, for his Logic) follows syllogistic form, the salient feature of which is that the concluding line is a generalization on enumerated members of a given class. Each such inference relies on the generally unexpressed but necessary premise that all members of the class so characterized have been enumerated. This much resembles the prevailing Aristotelian view, according to which we can get from instances to generalizing conclusions—via an intuition of the relevant universal. He steps away from that view, however, by suggesting that reliance on this unexpressed assumption produces the difficulty that we could never insure that we have given such an enumeration. If we could, then our reasoning would be demonstrative and thus of an altogether different sort.(1) Here he anticipates a Humean point, that justifying any generalization on particulars requires something that may lie beyond our cognitive powers, our empirical knowledge of all such particulars. Gassendi's proposed resolution also anticipates Hume: we might simply posit the conformity of all remaining unenumerated particulars. This proposal (which he attributes to Lucretius and Horace) nevertheless has only the force of a helpful supposition, and his only suggestion as to its warrant is its utility: "...because...it is exceedingly difficult, not to say impossible, for there to be a complete enumeration, when some have been enumerated...you suppose that apart from those enumerated there occurs none which is different."(2) Induction cannot take us from particulars to generalizations and so cannot be demonstrative, for lack of this supposition or the elusive generalization step.

By contrast, Gassendi does not think 'probabilistic reasoning' is missing any premises—nor does he think it is, in the main, nondeductive. He proposes that what makes a bit of reasoning probabilistic, or 'suasory', is that its premises are contingent and therefore persuade us of the conclusion's truth without making that truth necessary.(3) The measure of strength for such inferences is a function of the degree of clarity and certainty we attribute to the premises, rather the conditional probability of the conclusion given that the premises are true. By adopting this measure Gassendi classifies as 'probabilistic' all deductive inferences from empirical premises. His reason for this classification is that we base such claims on information from the senses which, however otherwise well-warranted, never warrants our certainty. As an example designed to illustrate how such uncertainty may arise, he cites the cases where being persuaded by some reasoning relies on trust in the person presenting the reasoning. In these cases, he points out, our awareness that the presenter may be unreliable or even deceitful should lead us to withhold certainty from the premises presented, hence to any claims thereby inferred.(4)

One problem here is that, if this is all it takes to render a piece of reasoning as 'probabilistic', then presumably any inference where it is merely possible that the premises are false, and not just those based on empirical claims, could count. Thus, despite Gassendi's apparent intentions, this view fails to pick out anything special about deductive inference in specifically empirical contexts. Moreover, it is a consequence of this view that there are no definitively false inferences any more than there are definitively true ones. An inference such as

Primes are divisible only by themselves and one.
Four is prime.
Four is divisible only by itself and one.

should be considered as merely 'probabilistic'—in this case, likely to be false—because the conclusion is based on a premise that we cannot be certain is false. This much is consistent with Gassendi's proto-Millian views regarding learning mathematical truths and our resulting certitude about them. In sum, we might well worry, that Gassendi understands 'probabilistic' reasoning to cover an extraordinary and not terribly useful range of cases.

But Gassendi actually conceives of 'probabilistic' reasoning more narrowly. For one, inductive reasoning cannot count, surprisingly enough, precisely because it is missing the all-important generalization step. Indeed, it turns out that a given piece of reasoning is persuasive and contingent in what he thinks is the right way only if it meets specific criteria which only deductive syllogisms can satisfy, such that any inference we call 'probabilistic' we must already classify as a species of deductive syllogism. So what does he think is the 'right way'? His criterion for determining that reasoning is probabilistic depends on our first identifying the standard relation of the middle term to other terms, which is only possible for viable deductive inferences in syllogistic form. Specifically, Gassendi understands a claim to be certain only if the inference by which we attain it is certain, as merely probable only if that inference is probable and, with this understanding, prescribes these guidelines: any claim we validly infer by conjunctive figure syllogism is certain just in case the middle term necessarily agrees with—that is, is the genus, property, or some other crucial feature of—the subject and the predicate; any claim we validly infer by discrete figure syllogism is certain just in case the middle term necessarily disagrees with—that is, lacks any such feature of—the predicate; and any other claim we validly infer by any other figure syllogism is merely probable. What counts as a claim about which we may be certain, then, is not simply that it follows a bit of reasoning in some canonical form of valid syllogism, but that some necessary relations additionally exist among the parts of the syllogism. Thus in the (conjunctive figure) syllogism:

Socrates is human.
All humans are animals.
Socrates is an animal.

the conclusion is certain because 'human' (the middle term) is the genus of Socrates and the species of 'animal', and so necessarily agrees with each. On the other hand, in the (conjunctive figure) syllogism:

Rhetoric is an art.
Every art is useful to life.
Rhetoric is useful to life.

of identical form, the conclusion is no more than probable because 'usefulness' is neither the genus of art (the middle term) nor a necessary or universal property of art, such that the last line cannot follow from the first two without some residual doubt. As Gassendi puts it, "...the mind is unable to assent to the conclusion without a degree of doubt; nor can the premises bestow upon the conclusion greater clearness and certainty than they themselves possess."(5)

But by standard lights, in the syllogism just cited the last line follows without doubt from the first two—and in this it does not differ any from the first syllogism—from which we might conclude that Gassendi simply doesn't understand what it means to call reasoning 'probabilistic'. Then we can easily write off this set of views as a peculiar, incoherent slip in the Institutio Logica. It is more reasonable, however, to suppose that he instead means to characterize any (deductivist) reasoning as 'probabilistic' where the actual truth-value of the premises is a contingent affair, and where he takes this very possibility that the premises may be false to entail that we cannot be certain of the claims we infer on their basis. In modern parlance, what he is really doing, then, is picking out the class of inferences which yield arguments we are unable to judge as sound because their premises have undetermined truth-values. If this is right, then calling a bit of reasoning 'probabilistic' for Gassendi reflects the limits on the conclusiveness of our epistemic judgements about such contingent matters as appear in the premises, more than it reflects any structural merits or deficits of the reasoning.

One oddity in this view should arise in cases where premises of what Gassendi takes to be 'probabilist' deductive inference incorporate or rely upon what we (and Gassendi) accept as inductive inference. Consider the following inference, based on enumerative induction, which Gassendi accepts as a viable instance of 'probabilist' reasoning:

A speech of Cicero has an elegant exordium, narration, confirmation, confutation, and peroration.
Whatever has these things persuades.
A speech of Cicero persuades.

Here, too, the second premise sometimes fails to be true, Gassendi warns, with the result that "...the conclusion is not a necessary one".(6) In this case, the inference is 'probabilistic' and, it seems, inductive. It is not, however, elliptical in the usual (offending) way, for the second premise provides the oft-missing generalization step. If cases of induction by this sort of "enumeration of parts" appear to us to count as what Gassendi takes to be induction and 'probabilist' inference, this is not to say there are 'probabilist' inferences he himself would concede are inductive. For though we would see such cases as canonical instances of inductive inference, Gassendi lights upon the fact that they supply the generalizing premise which he takes to be the mark of demonstrative (and thus deductive) syllogism. And so he understands those cases as 'probabilistic' deductive reasoning which incorporates complete enumerations—where it is perhaps the key 'probabilistic' feature of such cases that the enumeration is truly tendered as complete.

The suggestion that a bit of reasoning is 'probabilistic' if it is deductivist in form and its premises have contingent truth-values has a curious result for Gassendi's picture of scientific method, namely, that all our empirical claims—even those he believes we reach by deductive inference—are probable at best. After all, we base reasoning to empirical claims on contingent premises given that these are at least partly derived from experience. Now, we might find this result curious because it seems to be inconsistent with the goal of lending a sense of surety to scientific reasoning—his obstensible motivation for adopting a regressus demonstrativa method. But there is nothing troubling about this result for Gassendi, whose general theory of empirical knowledge includes the notion that there is nothing we know about the world with certainty. There is also the echo here, in typical Gassendist fashion, of the ancient sceptical worry that we know nothing from syllogisms since we have no viable reason for accepting our initial premises as true. Of course, Gassendi rejects the Sceptics's view that warranted belief entails certainty about those beliefs, and so accepts that such probabilistic empirical judgements may be warranted. Yet in the end, like Glanvill, Mersenne, and other early moderns influenced by skeptical thought, he too tailors his scientific method to ancient caveats regarding knowledge from the senses. It's not surprising, then, that Gassendi insists reasoning employed in scientific method is 'probabilistic' (in his sense of the term) because it is based on contingent premises. It is also not controversial, since such reasoning generally is based on contingent premises. What is controversial or, minimally, quite unusual, is holding this view and all the while taking scientific method to be, in the main, a deductivist enterprise.

Gassendi's chief insight in this context, it seems to me, is that nothing guarantees that our empirical claims are certain—not even the inferential structure of valid syllogism—because the aim of scientific reasoning is not preservation of truth (across the premises and conclusion of a syllogism) but establishing credible claims about the world on the basis of other such claims we already believe, though we are not certain of them. And here Gassendi draws on his 'modern' notion that we judge claims by evidence in their support which is direct or without the benefit of intermediary testimony, such that the strength of our beliefs in those claims is a measure of the degree of our confidence, not in another person (as per their testimony), but in the evidence itself. What it means to say that our degree of belief in a given claim is less than whole is just that the direct evidence in its behalf is less than fully compelling and so fails to make us certain of that claim. So it is with empirical claims we attain by valid syllogism: the evidence for the supporting premises is never fully compelling so we can never be certain of them—nor, as a result, of the claims they support.

This insight leads Gassendi to propose, in effect, a familiar symmetry between inductive and deductive reasoning about empirical matters: in the former we lack certainty as to whether the conclusion follows from the premises; in the latter we lack certainty as to the truth of the premises themselves—but in either case, it is a consequence of our imperfect access to the way the world is, that the claims we infer by such reasoning cannot be known with certainty. I say 'in effect', for although Gassendi's actual, quirky view of 'probabilistic' inference suits only deductive reasoning, still his picture of inductive reasoning (quite correctly) allows that we have insufficient information from the premises to be certain of our conclusions. That he does not formally recognize (and perhaps doesn't even recognize at all) this symmetry makes all the more impressive his recommendation that one way we judge both sorts of empirical reasoning relies on our recognizing this lack of certainty: we gauge the content-wise strength of a given deductive argument by our degree of belief in its constituent premises, just as we do in inductive argument. The difference—and it's substantial, of course—is that in a given inductive inference we automatically get with this content-wise strength some measure as well of its structure-wise strength, the conditional probability of the conclusion given the premises.

While it seems Gassendi has all the elements of this familiar story, he falls short because he is convinced it is a syllogistic form special to deductive argument that determines when we are looking at a piece of reasoning we can call 'probabilistic'. Yet he makes an important contribution here, obviously not to the history of informal (let alone formal) logic, but to the history of the study of empirical reasoning. It is to at least recognize what it means to have and rely upon contingent propositions at the base of one's inferences about empirical matters. For although it is a bit awkward that he limits his characterization of 'probabilistic' argument to deductive inference, to his credit he sees it is not a formal feature of valid syllogism that we cannot be sure of empirical or other contingent claims we might so infer. It is simply a fact of life, so to speak, about such deductive inference. This may be indeed shaky ground upon which to assert our beliefs about the world, just as the ancient sceptics warn. But it is also the only grounds open to deductivists (including Gassendi in his regressus methodological mode) to build empirical claims upon. And so Gassendi accepts 'probabilist' deductive inference for the same reason he accepts inductive inference without the crucial generalization step: each regularly produces results—that is, viable empirical beliefs—even in the absence of the relevant necessitating beliefs. Utility emerges as a proper motivation in his analysis of inference patterns.