The Problem with Implication

Among the domains of philosophy, comprising, but not limited to the fields of epistemology, ethics, metaphysics and logic — the latter finds itself in a peculiar place as arguably the most sacrosanct, but at the same time fragile. In this essay we shall focus our attention on one of the scaffoldings of logic itself; the engine which keeps it running, what we call “it implies” also represented symbolically as such (→). I shall argue that the glue of logic which we take as a given, the implies, lies on much more fragile assumptions than what we would like to admit to ourselves. This essay is largely inspired by the seminal work Two Dogmas by Quine, while also taking inspiration from On Certainty by Wittgenstein.

Borrowing from Kant’s distinction of analytic and synthetic, for the scope of this essay I shall bifurcate implication into the following types: analytic and synthetic. Both of these suffer from a distinct type of failure which laymen, and even the great philosophers and logicians, choose to ignore by convention of the trade, rather than it being justified.

The Types

Analytic

Let us first focus our attention on the sacrosanct and fragile, the logical implication. In symbolic logic, implication may be written down as such: $\vdash_S (P \land Q) \to P$. For the sake of representation, I shall call this statement S. Translating into plain English, this means “P and Q implies P is a provable statement in system S”. If you may notice this statement is a tautology, now the natural move of any inquisitive mind would be to ask why? What makes this statement a tautology? What gave it the authority? How dare it?! These questions are largely answered from within the system, rather than by questioning the system itself. In this section my aim is to question the sacrosanctity of the system, and by extension logic itself.

Statement S as we saw above exists as a tautology, and what gives it the authority, is the “implies”. The point of contention here is not that we doubt the validity of it, but what it exists as ontologically. The analytic “implies”, is, as we can see, true — but here I shall draw a hard line between assumed truth and derived truth. The “implies” rests squarely in the former category. When we analyse the tautologicity of S, we assume the implication to hold true rather than having derived it from anywhere. Implication is true conventionally, and not because it was proven.

Some might raise an objection that I’m stating the obvious. Of course we are assuming some basic propositions to be true in order to derive more complex statements, that is literally what an axiomatic system is. I refute this objection by showing how implication is different from a general axiomatic statement.

Consider an axiom A in a system $S_1$. We can therefore say that A will be assumed as A is an axiom in system $S_1$, but A may not always be an axiom. Consider a system $S_2$. In $S_2$, A is a provable statement. Axiom and theorem status is system relative. The point being, an axiomatic statement is provable in a different, more general system. “Implication” on the other hand cannot be proven without circularity, as in order to construct a proof, we would need to employ the use of the very apparatus we are casting doubt on. Hence, we can see that “implication” is proven circularly. We assume it to be true, but we cannot prove it without first assuming its validity. The status of “implication” is system invariant.

Here it would be beneficial if we learned about the idea of a hinge proposition proposed by Wittgenstein in On Certainty. There he elucidates upon the idea of a hinge proposition, which is something one believes without it being justified or proven, as it forms the bedrock of existence. A hinge proposition just is. Analytic implication is an example of a hinge proposition as we believe it to hold despite it being unjustified to do so, as we have no reason to suspect the contrary. Notice what is happening. I am not casting doubt on the validity of the implication, but am expressing that its ontological basis is more fragile than people presuppose.

Synthetic

Having looked at the analytic, we may now focus on the synthetic. It, as the argument stands, is pertinent to first rigorously define what is meant by the word “synthetic”, as I use it in a marginally altered sense compared to the strictly Kantian. When I say a statement being synthetic, I mean that the meaning of the statement is not contained in it in its entirety. For example, “Cars made by Toyota are high quality.” In this sentence unless a person already knows what Toyota, car and high quality means, this sentence is nothing but gibberish to them. The sensibility of this sentence rests on a bet that a person has had prior exposure to the words mentioned. To a tribesman, Toyota and car may not mean anything as that person has never encountered a car before. Hence, a statement being synthetic means that the meaning of it is not contained within it in its entirety, but rather it rests on priors in order for it to be coherent. Naturally now the question arises on what even is meant by “prior exposure”, and in order to attack the ground of synthetic implication, I shall show using Quine’s arguments in Two Dogmas, why “prior exposure” is unreliable and shaky. The very “prior exposure” on which the coherence of synthetic statements lies.

Consider the statement “Cars made by Toyota are high quality.” once more. For a person to know what Toyota is, they must first know what a car is, as Toyota is described as “A car manufacturer”. And in order to know what a car is, a person must know what wheels, engines and “moving” are, as a car may be described as “A four wheeled apparatus equipped with an engine, which is used to move.” For the sake of generalizing, we can say, as I showed, that all words are defined in terms of other words. All things are experienced in terms of one another, in clusters. There is nothing definitionally which grounds the meaning of a word into something not a word. This is precisely Quine’s argument in Two Dogmas, specifically the first dogma. No word is defined without other words, which themselves have definitions. We can only ascertain the semantic meaning of “Toyota” within a web of belief. A word only makes sense within an experiential web. Having shown how the definition of the words we use is ungrounded, it is not difficult now, to formally describe the problem with synthetic implication.

Consider the following statements:

• P1: This apple is ripe as it’s sweet to taste
• P2: That big potato sack on the ground must be pretty heavy

On the surface these two sentences may look fairly different. In P1, the person has direct sense experience to infer from (sweetness implies ripeness), but in P2 the person infers the sack to be heavy based on prior experience with big potato sacks, without actually lifting it. It is trivial to see the problem with P2’s reasoning: the big sack may be filled with cotton, or hollow plastic potatoes. Size of the sack doesn’t tell you anything about its weight. My actual point of focus here isn’t P2, but rather P1.

In P1, the person infers ripeness from sweetness after having directly tasted the apple. According to the person the apple being sweet implies its ripeness. Here the problem is with defining sweet and ripe. As shown earlier, words have a grounding problem. As “sweet” is subjective, a person addicted to ultra processed food may only define Oreos to be the threshold of sweet. To them, that “ripe” apple may be sour and hence not ripe at all. A third person may completely abandon taste, to only focus on the colour of the apple, as in only a red apple is a ripe one. To them a sweet green apple may not be classified as ripe. As we can see “sweet” and “ripe” mean different things to different people, and hence the implication in P1, where sweetness implied ripeness, is dubious in nature.

The failure point of synthetic implication now, stated plainly, is that the words we use to imply one thing from another are not fixed. Who’s to say what’s in a word?

Conclusion

We saw in this article how implication, both analytic and synthetic, rests on shakier ground than what people think. How analytic implication is an assumption not proof, and how synthetic implication is betrayed by words. The implies smuggles with itself a dogmatic proclamation of its own certainty, while hiding behind the veil of an unexamined ground. It should be noted that in this article I did not concern myself with Hume’s problem of induction as it is not a problem of implication in general, but of induction specifically, which is a narrower case of implication. My arguments here do not concern themselves with Hume. My aim here was to show the unearned certainty the implies carries with itself, and to encourage the readers to use caution whenever using a strong word like “it implies”, because the word you think of being logically airtight, rests on nothing but a collective fiction which happens to hold true.