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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?

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  • I've searched and I haven't found a good model for causation. However, I have this intuitive proof of why they aren't the same. Suppose they are the same, "rain causes wet floor", "rain implies wet floor", by contrapositive, "no wet floor implies no rain", "no wet floor causes no rain", which is intuitively absurd. QED. Commented Jul 1, 2023 at 23:03

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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

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    Ok, so generally in the case of causality the predications have to be related somehow. Makes sense. Material implication is indeed a bit weird from the standpoint of common sense reasoning.
    – Atamiri
    Commented Jun 1, 2015 at 15:53
  • @Atamiri Even if we restrict both implication and causation to real events, and make implication semantic (e.g. undefined when the premise is false) there is still a difference. It is often said that even perfect correlation does not imply causation, and Romans had a quip post hoc, non propter hoc (after it, not because of it). Two events can be perfectly correlated without causing each other, e.g. if they have a common cause that always forces both. Relationship between correlation and causation is explored in the theory of scientific inference. www3.nd.edu/~rwilliam/stats2/l32.pdf
    – Conifold
    Commented Jun 2, 2015 at 4:14
  • @Conifold But then it's not implication either. Correlation means departure from independence, which raises an additional question: What does it mean that two eventualities are (in)dependent?
    – Atamiri
    Commented Jun 2, 2015 at 18:56
  • @FreeMind Facts are statements. And there are implications between facts that are not causes.
    – Atamiri
    Commented Jun 2, 2015 at 19:00
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    The best answer I heard so far by a member of a working group on evidence and causation in medical sciences was: Causation has at least two components, i.e. correlation and a reasonable mechanism that explains how one event implies the upcoming of the second one. Correlation without an explanation why they should be linked through laws of nature does not qualify for causation.
    – Philip Klöcking
    Commented Feb 25, 2018 at 2:56
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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).

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  • Propositions aren't terms.
    – Atamiri
    Commented Jun 1, 2015 at 17:02
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    @Atamiri Yes, they are. The thing is that I used here the word 'term' in a different sense than the one you seem to think about (the linguistic). One of the senses of 'term' is "end point", as in "bus terminal". There is an old philosophical sense of 'term' according to which the "terms" of a relation are the things that are related by the relation. A relation is compared to a line connecting two points. These two end-points, the terms, are what the relation relates. And since in my answer I talk of a (logical) relation between propositions, these propositions are its (the relation's) terms. Commented Jun 2, 2015 at 8:38
  • My context was formal logic, where propositions aren't terms.
    – Atamiri
    Commented Jun 2, 2015 at 11:43
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    @At Doesn't matter. Just understand what I meant by the word 'term'. Commented Jun 2, 2015 at 12:13
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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.

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  • I think I understand what you are getting at and it is related to this: en.m.wikipedia.org/wiki/Material_conditional . Commented Jan 13 at 18:35
  • I am thinking about the class of sentences about causality that can be transformed into conditional propositions. For example the transition from "the bodies have acceleration, because a force acts on them" to "if a force acts on a body, then it has acceleration"
    – SK_
    Commented Jan 14 at 20:33
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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")

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  • Can you add detail for what you consider a ‘real relation’? Can you add detail for your claim ‘implication iss (‘=‘) causation when two objects are identical’? Commented Jan 22 at 21:04
  • I feel like you can go deeper into the difference between ontological (atemporal) causation and implication. I feel like they actually have potentially deep overlap. If something is “atemporally caused”, we are asking for why logically it exists. This is similar to implication: why premises imply conclusions. Commented Jan 22 at 23:28
  • The difference in definition is shown by their definitions. Implication (in a logical sense) just refers to one proposition being true whenever other is true, or being false whenever other is false. That's all it means. It has no relation to the proposition explaining another. Basically, one case is linked to explanation, and the other isnt. In one case, the elements are "explanatorily" linked (necessarily), in the other case this isnt necessary. I think that suffices. And the counterexample shows how they differ in reality. Commented Jan 22 at 23:57
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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).

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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.

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