It is accepted, even by more sympathetic readers than me, that Locke’s theory of inference leaves something to be desired.
In the Essay Concerning Human Understanding (4.17.4) we have:
To infer is nothing but by virtue of one proposition laid down as true, to draw in another as true, i.e. to see or suppose such a connexion of the two ideas, of the inferred proposition v.g. let this be the proposition laid down, ‘men shall be punished in another world’, and from thence be inferred this other, ‘then men can determine themselves’.
It’s important that Locke sees the relation of inference as holding between two ideas. This rules out explaining inferences formally. The ideas can be complex, and inference can be explained in terms of relations among their components. But the relations Locke discusses aren’t standard logical relations: convertibility, subalternation, predication, etc. He goes on:
The question now is to know, whether the mind has made this inference right or no; if it has made it by finding out the intermediate ideas, and taking a view of the connexion of them, placed in a due order, it has proceeded rationally, and made a right inference. If it has done it without such a view, it has not so much made an inference that will hold, or an inference of right reason, as shown a willingness to have it be, or be taken for such.
Locke doesn’t so far as I know, justify his assumption that inference occurs ‘by finding out the intermediate ideas’. But he explains how it would work in his example:
In the instance above-mentioned, what is it shows the force of the inference, and consequently the reasonableness of it, but a view of the connexion of all the intermediate ideas that draw in the conclusion, or proposition inferred. v.g. Men shall be punished, — God the punisher, — just punishment, — the punished guilty — could have done otherwise — freedom — self-determination, by which chain of ideas thus visibly linked together in train, i.e. each intermediate idea agreeing on each side with those two it is immediately placed between, the ideas of men and self-determination appear to be connected, i.e. this proposition men can determine themselves is drawn in, or inferred from this that they shall be punished in the other world.
There is the well-known problem. If the relation of agreement is transitive, then the first item in the chain must agree with the last and the intermediate steps are superfluous. And if the relation isn’t transitive, it’s not clear why positing the intermediate steps should do anything to show why the original inference is justified: one could just as well show that seven is the successor of three by giving intermediate numbers, each of which is in a successor-relation to those on either side of it.
There is also a mystery. What does ‘agreement’ mean? I’ve said that it’s not a logical relation in the traditional sense.
If ‘punished by God’ agrees with ‘punished in another world’ this would seem to be because, in the context, somebody who says one means the other. Agreement here could mean sameness of meaning, at least in context.
But the agreement of ‘just punishment’ with ‘punishment of the guilty’ seems to depend on more than just agreement in meaning. It is a synthetic truth, I think, that just punishment is always of the guilty, though it might be an obvious point of agreement among all theories of justice. Worse still, agreement seems to be a sort of equivalence here — it is a symmetrical relation. And ‘punishment of the guilty’ is hardly equivalent to ‘just punishment’, at least it would be an awfully hard theory of justice that held that any punishment is just so long as the punished party is guilty.
‘Agreement’ becomes even woolier between ‘guilty’ and ‘could have done otherwise’. Locke himself would surely have realised that this is an equivalence from which we might demur. We might concede to it after a long argument about free will and moral responsibility, but not merely by perceiving an agreement of ideas.
Mary Astell (A Serious Proposal to the Ladies, 2.3.4), seems to draw out the puzzle here very eloquently:
And as the way to know the Worth of things is to Compare them one with another, so in like manner we come to the Knowledge of the Truth of ’em by an Equal Ballancing. But you will say, Tho I may learn the value of a Spanish Coin by Weighing, or Comparing it with some other Money whose Standard I know, and so discern what proportion it bears to those Goods I wou’d exchange; yet what Scales shall I find to weigh Ideas? What Hand so even as to poize them Justly? Or if that might be done, yet where shall I meet with an Equivalent Idea when I have occasion to use one?
Her answer is:
…as Light is always visible to us if we have an Organ to receive it, if we turn our Eyes towards it, and that nothing interpose between it and us; so is Truth, we are surrounded with it, and GOD has given us Faculties to receive it.
The assurance here is that we have the capacity to recognise agreement, insofar as we have the capacity to know Truth. We can know that just punishment is equivalent to punishment of the guilty (if that is true). We can know that the guilty could always have done otherwise (if that is true). Inferring, defined as perceiving the agreement of ideas, ends up amounting to nothing more than knowing many true theories and knowing their consequences.
But then we still don’t know what it is for one thing to be the consequence of another. Perhaps Locke and Astell weren’t interested in this question, though it is obviously circular to suggest that inferring A from B means perceiving that A agrees with B, which in turn means having a theory from which we can infer that A is equivalent to B.
What is interesting is that Astell goes on to list rules, derived in part from the Port-Royal Logic, for making inferences. Since making inferences on her view just means applying true theories to particular cases, the rules mostly amount to a recommendation not to judge about things you don’t understand properly, along with some very general recommendations about how to gain understanding.
I think this is a useful point in a contemporary context. Many people, especially online people, seem to think that there is some general skill, called ‘logic’ or ‘critical thinking’, the possession of which makes you good at seeing logical consequences, regardless of the domain of discourse. Astell’s suggestion, I think, is that there isn’t really any non-domain-specific skill like that.
The one piece of general advice for making inferences is not to judge of things you don’t know much about — not even what you perceive to be the mere logical consequences of various views expressed about them. ‘I don’t know much about climate science, but it follows from what you say that…’ Stop! If you don’t know the first, you don’t the second either. You can’t catch out people on their fallacies by being generically ‘logical’. Logic, taken generically, just means knowing what you’re talking about. And so it can’t compensate you in an argument for the handicap of not knowing what you’re talking about. Nothing can do that.
This is useful advice, I think, though it won’t be welcome on the internet.