Implication is said to be more general than causality since, for example, being a dog implies being a mammal, but it doesn't cause it.
Is there a formalization of the difference between implication and causality (in the field of metaphysics or philosophy of logic)?
What criteria could one use to examine these concepts?
Implication is a relation on statements. Causality is a relation among facts in a world (perhaps the real world, perhaps a possible or hypothetical one).
Causality is notoriously tricky to define and pin down. (For example, in a deterministic world, does it make sense to even talk about causality, since given an initial condition there is no other way any event could possibly have been?)
Implication, on the other hand, is really easy to define: statement A implies statement B in all cases except when A is true and B is false.
But this is emotionally unsatisfying because it doesn't match up with our intuitive sense of implication ("I want to know that B is true because A is true"). Well, I can't help you there. That's just what the word means. Totally unrelated statements imply each other. False statements imply everything. I'm really glad that false statements don't cause everything.
Source of this answer: https://www.quora.com/What-is-the-difference-between-implication-and-causality
Implication and causation are relations between different kinds of terms. Implication is a logical relation, holding between propositions, or declarative sentences. Fido is a dog (proposition), all dogs are mammals (proposition), therefore Fido is a mammal (proposition).
Causation is a real relation, holding in the world, outside language. Causation is a time-related relation, because it is relates changes. Causation is relevant only where there is change (so, for example, there is no causation in mathematics). I throw the switch, subsequently the light goes on. One change (switch off -> switch on) caused a subsequent change (light off -> light on).
Implication is explained by laws of logic (e.g. modus ponens). Causation is explained by laws of nature (e.g. laws of electricity).
The difference is rather simple: causation requires a real relation (i.e one element really depends on the other for something between the cause and the effect, implication does not.
A simple way to distinguish those concepts is to show an example in which they are distinguished. If they are identical, an implication is a causation, and vice versa. So, by showing an example in which there is implication but no causation, the difference is set. Here is the example: if the sky is blue, 1+1 = 2. The antecedent "the sky is blue" does not cause the consequent "1+1 = 2", nor vice versa, so there is no causation. In contrast, there is a logically valid implication here, even though those facts are pratically unrelated. The implication is true because the consequent (1+1 = 2) is true, and thus it doesnt matter what the antecedent may be.
This is the rule known as prefixation, or P → (Q → P). If a proposition P is true, then one can form an implication such that P is the consequent and Q is any antecedent. This follows because an implication is false only when the antecedent is true AND the consequent is false, and thus, when the consequent is true, the implication is not false, which entails that it is true.
But to be definitional, causation refers to the process by which one thing explains the being of another, be it temporally (as explaining the beginning of something's existence) or merely ontologically (explaining the existence of something in of itself, independently from time). The former would be "horizontal causation", the latter would be "vertical causation". Implication is just a logical statement between two things (be they propositions or whatever), such that, whenever x is true, y is true, or whenever y is false, x is false. (Both are forms to write "x → y")
The difference is that causality is a necessary connection whereas material implication is a contingent connection. In other words, the truth value of "p -> q" is a function of the truth values of p and q (it's a truth-functional connective) and the truth value of "necessarily (p -> q)" isn't. That leads to so-called paradoxes of material implication where in natural language we often want to express a necessary connection so the immediately obvious translation doesn't work.
Of course, causality is more complex than that but that's the basic reason why it isn't identical with material implication.
The following proposition:
A and B then C
Can be expressed (not always) as
A + B = C
In the second form, A is implicated in C. There's no sequence in time. The system A is part of C, or it has a relationship with C. That's the meaning of implication.
In the first form, A is causal of C, meaning that when A happens, C could follow. The word follow expresses a time sequence. Causality always is a time-based sequence. Time is essential to causality.
Consider that causality is only a mental construct. Most philosophers reject causality as an attribute of nature. It is our brain that works using a causal fashion. Probably because time is nothing more that a mental construct (Kant).
Of course implication is about propositions, though causality is about the external world. However, we can extend this; let P and Q be two phenomena, between which the relation of causality stands: P causes Q.
We have already enacted a sentential transformation, which corresponds to a non-sentential phenomenon.
The question is: Under which conditions can we transit from "P causes Q" to "if P, then Q"?
Τhere are many times that semantics betrays implication relations, which are set by e.g. causal conjunctions.
