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?
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).
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
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).