Knowledge of Entailment







Knowledge of Entailment



How do we know that one proposition entails another proposition? If we think of entailment as logical necessitation, how do we know that one proposition logically necessitates another? I suggest that we consider this question by analogy with Hume’s treatment of causation. According to Hume, we do not perceive necessary connection in causally related objects (though it exists in them objectively), so our knowledge cannot be based on any such perception. Instead we perceive constant conjunction and the mind develops the habit of moving from cause to effect as a kind of animal instinct. No matter how much we may scrutinize a cause, such as a billiard ball about to collide with another billiard ball, we never discern any power or necessity in the object: thus necessary connection is invisible (though real). That is, nothing about the nature of the effect can be inferred from the cause considered alone and by itself; we can only gain knowledge of causal relations by observing the conjunction of cause with effect in repeated cases. We believe in causal relations because of these observations, not because we can detect causal power in the individual instance. Causal powers cannot be extracted by the mind from knowledge of the cause considered in itself—though such powers are present in the object considered in itself.

            The analogous view for logical consequence is that we cannot discern in propositions anything that would lead us to deduce their entailments. We cannot examine a proposition and extract from that proposition considered in itself what it entails. The proposition as it appears to us is an entity complete in itself and bounded by its own identity—it makes no reference to other propositions. Its logical powers are not evident from its manner of presentation to the mind. Instead we believe that it entails other propositions because its truth is constantly conjoined with the truth of other propositions: whenever the premise proposition is true the conclusion proposition is true. We perceive no connection between the two propositions—such as logical necessitation—but we form the habit of thinking of one when the other is presented to us by dint of instinct and habit. We take one proposition to entail the other because of the observed conjunction of truth-value, but we detect nothing in the one proposition that could ground the belief that the other proposition must be true. We don’t perceive any logically necessitating properties in the premise that could warrant inferring the conclusion. We don’t apprehend the logical powers of the premise, only a coincidence of truth-value. Our knowledge of entailment is thus general not particular—it comes from observing a totality of instances. We do not and could not perceive in the premise proposition any features that could lead us to infer the conclusion proposition. We perceive no necessary connection between the two propositions.

            This position sounds utterly incredible: surely we can tell just by looking at a proposition what it entails! Propositions have their entailment relations built into them, transparently so.  For instance, once we grasp a conjunctive proposition we see that it entails each of its conjuncts. But the “Humean” skeptic about our knowledge of entailment is precisely questioning this widespread assumption, as Hume questioned a similar assumption about causation: for the entailment skeptic cannot see how we could have access to any such knowledge—as Hume could not see how we could have access to necessitating causal powers in objects. What we have done, according to the skeptic, is to construct a mythology of propositions that bamboozles us into accepting that we can read entailments off propositions. We think that somehow propositions contain their logical consequences, like marbles contained in a drawer. But this metaphysics of containment is a myth, a chimera. Consider sentences, sequences of symbols: no sentence literally contains the sentences that it entails (except itself), since these are distinct sentences.  [1] The sentence “snow is white” does not contain the sentence “snow is white or grass is green”; nor does “Socrates is a man” contain “Someone is a man”. And it is not that such containment is implicit rather than explicit: there is simply no containment of any kind. Nor does a supposed language of thought exhibit any such containment: sentences in any medium cannot contain sentences they manifestly don’t contain. But it must be the same with propositions, whatever exactly they are: they must be made up of constituents that are self-enclosed and bounded, and knowledge of these constituents cannot ground knowledge of entailments. Consider propositions as ordered pairs of objects and properties: nothing in these constituents alone can lead the mind to other propositions—examine them from any angle and you will not be led to grasp logical necessitations. What could a proposition be such that we could detect in it the entailments that it has? Propositions are like billiard balls: we cannot read logical consequence or causal consequence off them by examining their isolated and intrinsic nature. So our beliefs about causation and entailment must derive from some other source, such as constant conjunction. In Hume’s terms, we cannot have any impression of necessary connection, either in causal relations or logical relations; so we cannot gain our knowledge of these relations by perceiving necessary connection. Logical powers and causal powers are both inscrutable in the entities that possess them.

            There seems to be only one way to avert this line of argument: adopt a radical kind of holism about causality and entailment. If we say that causal relations between particulars essentially involve other particulars and their constant conjunction, then we can maintain that causal relations actually embed facts about other particulars, notably effects of similar causes. Likewise, if we say that propositions are constituted by other propositions—all those that they entail and are entailed by—then we can maintain that knowledge of their identity actually embeds knowledge of those other propositions. That is, we hold that a given proposition is identical to the totality consisting of all the propositions logically related to the given proposition. But this is a completely incredible position: it is tantamount to claiming that a given proposition is a set of propositions, possibly infinite, that stands in logical relations with the given proposition. The individuation of propositions collapses under such a theory. A proposition is not the set of propositions it entails and is entailed by; these are distinct propositions. This kind of holism is a desperate attempt to make sense of the metaphysics of propositions suggested by the mythological idea of containment: it takes that idea literally and interprets it as something like set membership. What the mythology really needs is the idea of implicit containment and a corresponding epistemology; but that is precisely what the Humean critique is calling into question. It is the analogue of the idea that a billiard ball may not present its causal powers explicitly but it presents them implicitly, so that we still have access to them in some kind of attenuated form. But this is to enter fairyland (to use another term of Hume’s)—as if propositions contain other propositions in a ghostly form (as causes may be thought to contain their effects in a ghostly form).

            Here is another way to put the point, recalling Wittgenstein according to Kripke. As we “grasp a proposition” we may have certain experiences, such as the formation of a mental image, and it is tempting to suppose that these experiences play a role in producing our knowledge of both the identity of the proposition and its entailments. But no such experience could ever constitute grasping a particular proposition or fixing the entailments of a proposition—for the experience can be interpreted in multiple ways and may not even occur. Nor can our knowledge of entailments arise from access to the dispositions we have to make inferences: for such dispositions fail to track logical entailments, for the reasons Kripke spells out (entailments are normative). So these kinds of facts cannot the basis of our knowledge of entailment; the facts exist, unlike the mythology of containment, but they are of the wrong type to deliver entailment. Invoking them is like supposing that mere contiguity will add up to causation. We need logical necessitation to get entailment, as we need necessary connection to get causality; but we search in vain for experiences or impressions of such necessitation—and dispositions are the wrong kind of thing to ground our knowledge of entailment. The epistemology of entailment is thus as problematic as the epistemology of causation, as Hume diagnoses the latter. It is even difficult to see how we get the idea of entailment, as Hume argued in the case of causation (hence the need for constant conjunction and habit to account for how the idea arises in us).

            Why didn’t Hume extend his critique to logical entailment? I think it is because he had an overly simple theory of ideas, to the effect that all entailment results from the containment of simple ideas in complex ideas—the mosaic theory of ideas. It is a matter of distinguishing the parts of a complex whole. But this model doesn’t work for most logical entailment, such as disjunction introduction or existential generalization. It is as hard to see how ideas contain logical consequences, as it is to see how objects contain causal consequences: for how can other things be present in a given thing? How is the effect present in the cause and deducible from it, and how is an entailment present in a proposition and deducible from it? What is this idea of containment? How can what is absent be present? What are causal and logical powers? How do we manage to see extrinsic consequences of things in those things? Don’t we just see the thing? As Wittgenstein says, it is “as if everything is already present in advance”: but we have no model for such anticipation, and no real idea of what those words mean.

            It might be replied that we can just appeal to the notion of “grasping a concept” as primitive. When we grasp a concept we know in some primitive way what its entailments are—these are evident to us in the grasping. The trouble with this reply is that it is not explanatory and does nothing to address our unease about how such knowledge is possible. It is like agreeing with Hume that nothing perceptible can constitute necessary causal connection and then insisting that we have a primitive apprehension of necessity in objects. Worse, it is like agreeing that nothing could conceivably amount to an impression of necessary connection and then asserting that we just “intuit” the connection by some unknown faculty. The problem is that the Humean has a strong negative argument to the effect that such necessities cannot be discerned in objects or propositions, so the primitive grasp response looks like pure hand-waving. We have no account of how we could extract knowledge of effects from causes or knowledge of entailments from propositions—by what means or mechanism. The very idea is baffling. Hence the threat of skepticism about whether we have knowledge of entailment at all, or the possibility of having to accept a skeptical solution along the lines of habit and constant conjunction. Nothing seemed clearer than that we can know the basic entailments of logic, but upon examination this looks to be deeply problematic for Humean reasons. The root of the problem is explaining how we can know that one thing necessitates another thing. After all, propositions and their entailments are “distinct existences”, as causes and their effects are.

            In response to this it might be suggested that the whole idea of necessitation should be rejected. There is simply no such thing. In the case of causation we replace necessitation between particulars with regularity between types of particulars—the regularity theory of causation. Thus there is nothing inscrutable to begin with; Hume was wrong to believe in necessity in the objects. Knowledge of causal relations is just knowledge of constant conjunctions—there is no “tie” (Hume’s word) between cause and effect, just regular association. The analogue for entailment would be the idea that propositions do not stand in necessitation relations–that would indeed be surpassingly difficult to discern; instead different propositions have regularly associated truth-values—whenever one is true another is. There is no dependence here, no connection, no tie; there is just brute coincidence of truth-value (“just one damn truth after another”). This kind of approach can take different forms, depending on what we generalize over—places, times, substitution instances, possible worlds. The first two are appropriate for indexical entailment, since indexical sentences can vary in truth-value across space and time. The second two are applicable to eternal sentences, which do not vary in truth-value over space and time. Thus we say that entailment consists in the fact that the entailed proposition is true in all the same worlds that the entailing proposition is true, or that pairs of schematic sentences are true under the same substitutions (whenever anything of the form “p and q” is true something of the form “p” is true). Thus we capture entailment by moving from the properties of particular propositions to the properties of collections—as the regularity theorist of causation locates causality in collections of events. We interpret entailment as something like constant conjunction: the truth-values of sentences or propositions are constantly conjoined across space, time, substitution instances, and worlds. We need merely to observe such conjunctions to know that we have an entailment; we don’t need insight into the necessitation relations between individual propositions.

One sees the attraction of such theories once Humean skepticism has taken hold, but it is hard to believe that entailment is merely constant conjunction of truth-value, just as it is hard to believe that individual causation exists in virtue of general conjunction. At any rate, the two issues are similar in form, with the same array of theoretical options and the same difficulty of selecting one option as clearly correct. As Hume called his problem (or one aspect of it) “the problem of induction”, so we can call the analogous problem for entailment “the problem of deduction”: how do we know what is deducible from a given proposition? In virtue of what does one proposition deductively entail another? That problem is even more radical than Hume’s original problem about causation.


  [1] Except in the case of conjunction: but the syntactic containment of conjuncts in a conjunction is not the ground of the entailment—that is the concept expressed by “and”. A sentence could contain a pair of sub-sentences without entailing either of them, as in “Snow is white or the sky is blue”.

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