Formal versus informal implication

I want to compare and contrast two sorts of implication — and I want to suggest that our understanding of beliefs and logic is badly affected when we confuse them, as we often do. In the hope of making things a little clearer, I propose to use the following symbolism:  (written in Mistral font) stands for the belief that P, P (in italics) stands for the linguistic sentence expressing the same content P, and stands for the fact that P, which of course only exists if P is true. (I gave “earthy” colours because it’s “in the world”, geddit? Also the looks a bit like an octopus, i.e. a real thing in the world.)

For illustration, if P is the sentence ‘Snow is white’, is the belief that snow is white, and is the fact of snow’s being white — a very simple sort of fact that might be represented by a Venn diagram like this:

That silly diagram is intended as no more than a reminder that although we are using a letter for a mental state (belief ) which is true or false, and a letter for a linguistic utterance (sentence P) which is true or false in the same circumstances, in the third usage (fact ) a letter stands for those circumstances themselves — something that is neither true nor false. Now it may sound strange to say that a fact isn’t true — facts are “true by definition”, aren’t they? Well, a fact is what makes a true sentence or true belief true, so wherever there’s a fact there’s a truth. In a loose colloquial sense we might refer to truths as facts. But in the current philosophical sense, a fact is strictly a state of affairs corresponding to a truth.

So understood, facts cannot imply anything, being themselves neither true nor false. But their linguistic or mental counterparts can, and this is what I want to examine here. It seems to me that confusion between facts, sentences and beliefs has generated much misunderstanding about the nature of thought itself. I hope to disentangle a little of this confusion here, and in doing so I hope to persuade you that formal logic is much less useful than is widely supposed as a tool of critical thinking.

Although facts can’t imply one another, linguistic sentences often do. For example, what are we to make of the claim that P implies Q?

If it is true, it describes a fact of some sort of lawlike connection — formal, causal, categorical, or whatever — between two possible facts and . I say “possible” facts because the implication can hold at the same time as the individual sentences P and Q it connects are not true. What matters is the connection between the sentences rather than their truth-values. For that reason, material conditionals of elementary logic (whose truth-value depends simply on the truth-values of what they connect) don’t capture this sort of implication. The conditionals we use for that purpose have to be understood as counterfactual conditionals, or as having some sort of subjunctive mood, so that they can be true or false regardless of the truth or falsity of their component parts.

Just as the sentence P can both describe a purported fact and stand for the belief , the claim that P implies Q can both describe a purported fact and stand for a belief. The nature of this fact and of this belief have seemed a bit of a mystery, to me at any rate in the past. I now think that mystery is largely the product of confusion between formal and informal implication. Apologies if this is no mystery to you.

Formal implication

As a model of implication, most of us take the case we are most familiar with: implication in formal logic, where the premises of a valid deductive argument imply the conclusion. When I say the implication here is formal, I mean that the work is done by language, and thought follows. That is, relations between sentences guide the formation of beliefs.

When conditionals that express such implications are true, they are true by virtue of the fact that one sentence can indeed be derived from another sentence via rules of inference that enable the derivation.

Deriving one sentence from another is a bit like building a structure out of Lego bricks. In this analogy, our rule of inference might be “every new brick must engage at least half of the interlocking pins of the bricks underneath”. When we begin, we might have no clear idea whether a given point in space can be reached given our starting-point. But once we do reach it (if we do), we can believe that it is legitimately reachable, given that starting-point and the rules of inference. Or at least, we can “accept” it as true, because we “accept” the rules of inference simply by using them.

With formal implication, the fact that corresponds to a true claim that P implies Q is a “linguistic” fact, embodied by the actual derivability of Q from P. The belief that corresponds to a claim that P implies Q (or sort-of belief, if all we do is “accept” it as true) is about derivability in language.

Informal implication

With formal implication, the work is done by language and thought follows. But with informal implication it’s the other way around: the work is done by thought and language follows. Actually, if thought is working as it should, this one-thing-following-another goes deeper, all the way to facts. The world has some lawlike features, and the thoughts of animals reflect them — in other words, animals have true beliefs about lawlike facts. Later, we human animals try to express those thoughts using language. Here real-world relations guide the formation of beliefs, which in turn guide the formation of sentences.

These sentences can be misleadingly ambiguous. A sentence like ‘P implies Q’ can be read in three distinct ways. It can say something about the lawlike connections in the world, i.e. facts about how and are related; or it can say something about the way sentences P and Q are related; or it can say something about how beliefs and are related. This ambiguity is compounded by the fact that a sort of meta-level “conditional” corresponds to each of these types of relation, and the situation is made still worse by our inclination to take formal implication as our model of implication in general.

It seems to me that the way to avoid getting lost here is to constantly remind ourselves that the primary link is between things in the world where lawlike connenctions exist: “what goes up must come down”, “if it has feathers, it’s a bird”, etc. Thought captures these lawlike connections by forming belefs that stand or fall together in a systematic way. If and are related in a lawlike way, a mind captures that meta-level fact by being disposed to adopt the belief whenever it adopts the belief , and to abandon the belief  whenever it abandons the belief . Given the larger belief system to which the pair may or may not belong, they’re “stuck together” like the ends of a Band-Aid:

The system as a whole has the property that whenever gets added to it, gets added too, and whenever gets stripped away from the system, gets stripped away too, like a Band-Aid whose adhesive parts are put on or taken off (in reverse order).

If we can be said to have a “conditional belief” corresponding to this sort of implication, it amounts to little more than belief that a lawlike connection exists between and . This meta-level “conditional belief” is embodied in the way and stand together or fall together in the system. Even if such a belief is false — as it would if there were in fact no lawlike connection between and — that distinctive linkage of beliefs and in the system is all it amounts to. When we come to capture it in language, we may use arrows or similar symbols to indicate a non-symmetrical linkage of P and Q, but let’s be careful not to think of such informal links as perfectly mirroring formal links.

I hope you agree that the Band-Aid analogy goes too far in that it contains one unnecessary detail that ought to be omitted from our understanding of informal implication. That detail is the “bridge” between the adhesive parts, with its supposed “hidden mechanism” enabling an inference from P to Q. I think we are inclined to imagine such a mechanism exists because we are so used to taking formal implication as our model, and we have a tendency to assume something akin to interlocking Lego bricks are needed to “bridge the gap” between and . A better analogy perhaps would be a Band-Aid with the non-adhesive part removed:

What does it all mean?

The assumption that formal and informal implication are closely parallel misleads us about the nature of thought. It promotes the idea that thinking is a matter of “cogwheels and logic” rather than many direct acts of recognition by a richly-interconnected belief system, often of quite abstract things and states of affairs.

People who praise or actively promote logic as an aid to critical thinking routinely assume that beliefs work like discrete sentences in formal implication. That is, they assume beliefs have clear contents with logical consequences which are waiting to be explored. Well, as I’ve said several times now, in formal implication, language does guide thought. Beliefs correspond to sentences which are discrete because of their distinct form. One sentence leads to another thanks to the rules of inference, and beliefs follow their linguistic counterparts. The beliefs that are so led are themselves discrete because they are so closely associated with discrete sentences. Their contents determine the inferential connections between them. But most beliefs aren’t like that at all. Their content isn’t determined by prior association with discrete sentences whose form precisely determines their content. Rather, their content is attributed via interpretation, which is an ongoing affair and, well, a matter of interpretation. That interpretation involves “working our way into the system as a whole”, taking account of the inferences an agent draws and attributing whichever mental content best reflects his inferential behaviour. If someone behaves as if he is committed to lawlike connections in the real world, we attribute beliefs whose contents are appropriate to commitment to those lawlike connections. Here, inferential connections between beliefs determine their content rather than vice versa.

As far as I can see, this limits the usefulness and scope of logic. It’s useful in the academic study of logic, obviously, but outside of that field, only the most elementary applications are of much use, even in formal disciplines like computer science and mathematics. I agree that it’s useful to be aware of informal fallacies and to try to avoid them. But beyond that, the power of logic has been over-inflated by the assumption that beliefs are like “slips of paper in the head with sentences written on them”, and the assumption that thinking proceeds by drawing out their consequences — by examining what they formally imply.

We are not culpable for “wrong opinions”

When we act, our bodily movements are caused by mental states. These mental states consist of a desire to achieve a particular goal, and some relevant beliefs which help us “steer a course through the world” towards achieving the goal.

It all means a human agent is a bit like a sophisticated version of a cruise missile, which is programmed to reach a target, and to do something (usually explode) when it gets there. It steers a course towards its target by comparing the terrain it flies over with its onboard computer map.

Although both the map and the targeting are necessary for it to reach its goal, the map is “neutral” in the sense that it only contains information about the outside world. It is compatible with the missile hitting any other target within the mapped area, and with its doing good things like delivering medicine or food aid when it reaches its target (not just doing something bad like exploding).

If the “act” of a cruise missile is to be praised or condemned, we judge what it is programmed to do, and where. We do not judge its map, whose greater or lesser accuracy simply results in greater or lesser efficiency in fulfilling the aim of the programming.

It should be the same with human agents. If we praise or condemn what they do, it should be with reference to the good or evil they intend to do, or are willing to do, and to whom. We should suspend judgement of an agent’s beliefs when we judge his actions, as beliefs are “neutral” with respect to the good or evil of what they help to achieve, just like the cruise missile’s onboard map. Like the accuracy of the missile’s map, the truth or falsity of an agent’s beliefs affect his success or efficiency in achieving gaols, but the beliefs do not set any goals. A belief can be true or false, but it can’t be good or bad. The worst an opinion can be is false, rather than “aimed at an evil goal”.

Despite the neutral role of beliefs, some people blame others for having the “wrong” opinions, or in other words for not believing what they “should believe”. For example, many Muslims think “apostasy” should be punished by death. Many Westerners think “denialism” should be ostracised or worse.

Those are remarkably similar views, and both are primitive, in the worst sense of the word. They belong to a backward state of society. They are inspired by confused understandings of agency, and we should reject them. If someone has false beliefs, he has either had bad luck (by being exposed to unreliable sources of knowledge) or he is epistemically ill-equipped. In neither case is he culpable.