# Does a biconditional necessarily imply a causal relationship?

Supposing A if and only if B, is it necessarily true that either A causes B or B causes A?

I'm considering this question where the truth values of A and B are both True, not both false.

In theory, I'm thinking no, it doesn't have to be a causal relationship. All that's necessary is that A and B always coexist. There could be some third variable causing both of them - C causes A and B. Or they could even be independently caused - C causes A, D causes B, but C and D coexist so A and B coexist...

However, in trying to find a concrete example of these alternative cases, I'm a little stumped. The tricky part is we need to ensure that A can never be without B and B can never be without A... So in the third variable alternative, we need to ensure that 1) C always causes both A and B, and 2) if something else besides C can cause either A or B, it also always causes both of them .... In the independent causes alternative, we need to ensure 1) that C and D always coexist, and 2) that if there are any other 2 things that independently cause A and B, these 2 things also always coexist...

First of all, please correct me if any of the above reasoning is off.

Second of all, can anyone think of a concrete example to illustrate that an alternative causal relationship is possible?

The closest I could get was this: "If there is a parent there is a child."

I was thinking: A parent and a child necessarily co-exist, but they don't really cause each other.... Technically the event of conception would cause both. But I could see someone objecting and saying that the event of conception is equivalent to the statement "There is now a child".... In which case the causal relationship would simply be There is a child causes there is a parent... Which is what we wanted to avoid...

• Not the classical material biconditional. First, there is no causation in mathematics, but plenty of biconditionals (e.g. a triangle has equal sides if and only if it has equal angles). Second, a biconditional may be due to common cause, neither the thunder nor the lightning cause each other, or verbal convention, A is a brother of B if and only if B is a brother of A. Sep 21, 2019 at 23:06

No. If and only if [time passes] then [Uranium238 decays]. Time does not cause the decay, but it can't happen unless time passes.

Another example is quantum entanglement. Given two entangled photons A and B, then: If and only If [A changes phase 45°] then [B changes phase 45°]. Because these changes happen at the exact same instant then by definition neither one caused the other (no cause can create an effect faster than the speed of light).

Biconditional does not assume causality, though we tend to only think of things in that way.

There exists relationships as you said such that C ⇒ ( A ∧ B ) ∴ A ⇔ B ; however

Example: CHILD if and only if ( MOTHER and FATHER) therefore MOTHER and FATHER are a biconditional however neither causes the other; a CHILD existing causes both conditions in the parents.

• Ahhhhh!! thank you so much!! :D
– Lily
Sep 21, 2019 at 16:28

A iff B, given A,B both true, simply means occurance of A implies occurance of B, and existence(/occurance) of B implies existence(/occurance) of A and nothing more.

Now for a technical interpretation: We can take A,B to be two sets of sentences, then A⇔B is equivalent to (A→B)&(B→A). Which further means (A⊆B) & (B⊆A). That is, every sentence in the set A is also in the set B and vise versa. Which means the two sets A and B are one and the same.

Suppose, now, we have to events x,y such that x causes y, but y does not cause x: or C(x,y)&~C(y,x), then by necessity x must precede y. However, if we suppose biconditional to be causation, then we get: x⇔y which is equivalent to (x→y)&(y→x) which is equivalent to (y→x)&(x→y) which is equivalent to y⇔x which is, in turn, equivalent to C(y,x)&~C(x,y). Therefore, by contradiction, biconditional is not causation.

Tl;dr,

A iff B means A and B paraphrases of each other; they are logically the same as far as logicians/set-theorists/mathematicians/metaphysicians are concerned. Therefore, A causing B or B causing A would be ridiculous since nothing can cause itself.

Supposing A if and only if B, is it necessarily true that either A causes B or B causes A?

Suppose A is the cause of B. Then, if B happens at an xyz location and time t, then A happened at an x'y'z' location and time t'. No quite what the question suggested but nonetheless a logical implication.

However, the notion of cause is much larger than that of material cause, to include in particular the notion of "formal cause":

Cause 7. (Philosophy) (in the philosophy of Aristotle) any of four requirements for a thing's coming to be, namely material (material cause), its nature (formal cause), an agent (efficient cause), and a purpose (final cause)

What is the nature of human intelligence? Presumably, most reasonable people today would say that human intelligence is caused by the human brain. Caused, or produced, or the result of, or the end-product of etc. What Aristotle called "formal cause".

The point is that there is a clear distinction to be made with the notion of material cause. In a formal cause, the effect and the cause are coincident. The nature of human intelligence is that it is ("formally") caused by a human brain. Some would go so far as saying that intelligence is a property of an operational, i.e. normally functioning, brain. Some will go further and specify sometimes the relation of formal cause as what is called emergence: human intelligence is an emergent property of operational brains.

Thus, when we have a formal cause, we also have an implication: Human intelligence implies an operational brain.

It should be noted, however, that strictly speaking, emergence is not a scientific notion. Rather, it is an abstraction which is part of our abstract representation of the world. If A is an emergent property of B, then A is an abstraction, the abstraction part of our abstract view of the world. If B is thought of as a material object, such as a brain or the material things that make up an organisation, then A will be thought of as more abstract, such as the mind, thought of essentially as an activity of the brain, or such as the activity of the organisation, something you couldn't measure the mass of.

Thus, the relation of formal cause shouldn't be thought of as a physical interaction, but as a formal relation, an abstraction we use to make sense of the world. As such, it is similar to a predicated property, such as "blue" in "The sky is blue".

In our example here, the predicated property is intelligence: The brain is intelligent. However, it works the other way around: Intelligence is a property of brains. Something we can also express by saying that a brain is the formal cause of intelligence.

• Ahhh that's interesting. Even though I'm a fan of Aristotle's 4 causes, I always kind of regarded them to be separate from what we generally mean when we talk about causality. Thank you though, this is an interesting answer to my question. :)
– Lily
Sep 23, 2019 at 3:16