A simplistic approach to causation and causal
reasoning
Michael Khalil
The German University in Cairo
Abstract. In this paper, we provide an approach to reason about causation and to identify causes using a simple logic. The aim is to capture
some ideas presented from past literature and use them but in a more
simple with avoiding complex notions and logics. The paper tries to
provide a basis or a building block for further advanced approaches by
being generic on the notion of causation with no hard specifications and
by avoiding complex notions or ideas without losing their benefits and
effectiveness. In such a way, the logic may be then improved in future
work to have a strong and powerful logic for reasoning about causation.
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Introduction
In past literature, there were different approaches to reason about causation:
Some were simple and others were more complex; Also some have focused on
specific ideas or specific aspects of causation, while others tried to be more
general. A good way to benefit from these efforts is to try and encompass some
of these ideas in a logic that is simple yet effective. We will try to express the
same concepts proposed in other papers such as [2], [3], and [4] and show that
it produces good results by testing on examples provided in past literature. We
fist show some important aspects that this paper is trying to cover and what
is it trying to achieve. Then the logic used is discussed. Then we show some
similarities with approaches from past literature and in what ways that this
approach is similar, specifically we show similarities with Shoham’s[2], Hobbs[3],
and Bochman’s[4]. Afterwards we define how causal relations are formed and
how to use the logic to do causal reasoning. Also we show how to deal with
counterfactuals, and background conditions and the difference between them
and causes. Some examples from past literature are then provided and we show
how to reason with them and compare the results with our intuition.
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Important ideas and benchmark
First, It is important to state clearly the ideas that this approach is based on
and what benchmarks is it trying to achieve.
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Michael Khalil
Differentiating between actual and general causality. General causation
defines the basic structure of causation: What generally causes what? Turning
the ignition key generally causes the car to work[2]. We do not need think about
if there were a banana in the tail pipe[2] or any other things to judge that idea.
General causation is not concerned with events or metaphorical situations, but
it is what we are taught or knew from experience. These are the things that
we base our causal thinking on and are the things that ”usually” happen. But
even that we know that turning the ignition key causes the car to work, if there
were a banana in the tail pipe, the car won’t work. In this situation turning the
ignition key does not cause the car to work but that does not negate the fact
that it generally does. Actual causation is concerned with what actually take
place and the events that occur. In the example by Shoham in [2], if the car key
is stuck in the ON position, disconnecting the battery is what actually causes
the car to stop working even if it generally does not.
Deriving wrong conclusions with insufficient information This idea is
related to nonmonotonicity. An agent is not always correct even if it was always
logical. If an agent does not know all the information it might never reach the
correct conclusion. If an agent does not know that a connected battery is related
to making the car works, it won’t know that connecting the battery if it was
disconnected causes the car to work. But an agent that knows that piece of
information, it would know that turning on the key and connecting the battery
cause the car to work.
Being able to derive counterfactuals A counterfactual statement, as described by Hobbs[3], is a conditional statement such that the antecedent of the
statement is counter to the truth, as in the following example borrowed from
Hobbs[3]: If the vase had fallen, it would have shattered. These statement should
be able to be driven from a logic that reason about causation. Some based their
ideas about causation on counterfactuals as in[1], but I believe counterfactuals
do not serve as basis for causation and are only relations that could be formed
through causal reasoning. Either way, counterfactuals play an important rule in
the literature and a logic should account for this aspect.
Dealing with preemptions (Early and Late) Preemption is when E would
cause C, but B causes C so E won’t anymore. There are two types of that case:
– Early: E would cause C, but B occurs and causes C, so observing that E
does not occur.
– Late: E would cause C. Both E and B occur but B is the one to cause C
preempting E from causing C
The difference is that, in early preemption, B preempts E early so E does
not occur while in late preemption E does occur but does not get to cause C.
Causation: A simplistic approach
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Differentiating between causes and background conditions When turning on the ignition key, it causes the car to work. But is it the only cause? A
working battery contributed to the car working, but we do not think of the
working battery as a strong cause as of turning on the ignition key. A logic
should differentiate between what we consider causes and what we regard as
“not-so-much” of a cause.
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Logic
The syntax, semantics, and inference rules used are that of classical propositional
logic. With the addition of these three sets:
Set of Causal Beliefs The set B consists of formulas of the form of φ → ψ.
This set constitutes the basic beliefs of an agent.
Set of Events The set E consists of atomic propositions that are acted or
observed by an agent. The set E is updated with new observed information
after an action is performed or observed. E becomes Enew .
Set of Causal Complexes The set C consists of formulas of the form of φ ↔ ψ
A causal model M is a model of the following structure M = (B, E, C, A)
where E is a truth assignment function.
Let B ψ C is the set resulting from replacing φ1 in φ1 → ψ ∈ B with φ2
such that ψ ↔ Φ ∈ C and φ2 is some conjunction that occurs in Φ. φ2 is called
a replacement on ψ.
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Similarities with past literature
The set B is similar to the structural equations in [4]. This set denote general
causation to an agent.
The set C is similar to the notion of causal complex in [3]. However, in C, not
all propositions needed for an effect are there, only the ones that are received
by the agent and they are subject to change or addition. While in [3], causal
complex denotes all things that have to hold for an effect to happen, where such
a set is hard to construct compared to C.
The operator and the update on E resemble in a way advances in time without any focus on time itself. They resemble the advances in reasoning steps and
the changes upon actions and observations. They also achieve nonmonotonicity
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Michael Khalil
since it allows for the agent to arrive to some conclusions and change the conclusions upon new observations in the world. In that way the logic have some
similarities with Shoham’s[2] but with avoiding dealing with how time progresses
and having no notion of any ordering that is preferred over another.
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Causation
The set of general causes GC is the set containing any φ such that B |= φ. φ is
a general cause of ψ iff Φ → ψ ∈ GC and φ occurs in Φ
The set of actual causes AC is the set constructed as follows:
– if φ ∈ E and E ∪ B |= ψ using general causal rule φ → ψ then φ → ψ ∈ AC
and ψ ∈ Enew
V
– if φ ∈ E and φ → x1 , x2 → x3 , ..., xi → ψ then φ ∧ V
xi → ψ ∈ AC and any
conjunction φx that is subset of the conjunction φ ∧ xi is an actual cause
of ψ (φx → ψ ∈ AC) and ψ ∈ Enew
– if φ1 , φ2 ∈ E and Φ1 → ψ and Φ2 → ψ where Φ1 is a conjunction that
contains φ1 and Φ2 is a conjunction that contains φ2 , then Φ1 ∨Φ2 → ψ ∈ AC
and ψ ∈ Enew
– If B ψ C ∪ (E ∪ {φ}) |= ψ such that φ is the smallest replacement on ψ,
then actual causes of ψ of that model are in AC and ψ ∈ Enew .
φ is an actual cause of ψ iff Φ → ψ ∈ AC and φ occurs in Φ
The set of counterfactuals is the set F constructed as follows:
– If φ → ψ ∈ AC and adding ¬φ to E does not result in ψ being a model,
then ¬φ → ¬ψ ∈ F
– If Φ → ψ ∈ AC and Φ is a dis-junction, for any dis-junction s1 ⊂ Φ and
dis-junction s2 ⊂ Φ and s1 ∩ s2 = {}, ¬s1 → (s2 → ψ ∈ F )
– If φ → ψ ∈ AC then formulas in the form of ¬φ → ac where ac is the set of
actual causes of ¬φ in a model with E − {φ} ∪ {¬φ} are in F
– Formulas in the form of ¬φ → ac where ac is the set of actual causes of ¬φ
in a model with E − {φ} ∪ {¬φ} are in F
We assign degree of importance for counterfactuals depending on in which form
does it occur. Where the first form has an importance degree of 1 while last form
has an importance degree of 4. One formula might satisfy different forms and
in such a way it would have different degrees. The importance degrees of these
formulas are the average of these degrees. These degrees resemble a measure for
closest worlds in a way.
φ is a background condition of ψ iff ψ ↔ Φ ∈ C and φ occurs in Φ
Causation: A simplistic approach
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Examples
Example 1. Consider Shoham’s example from [2]. And first Consider the case
where turning on the ignition key is enough for the car to work. We construct
B = {key → car} and an agent takes an action of turning on the key E = {key}.
We now have B ∪ E |= car and since key ∈ E and we have key → car ∈ B. That
results in making turning on the key is an actual cause of the car working. We
also have the following counterfactual ¬key → ¬car since key → car ∈ AC.
Now consider that the battery is disconnected and the agent turns on the key
with no clue about how cars work except that key is a factor (C = {car ↔ key}).
The car does not work which would make ¬car ∈ E so there are no actual causes
of the car working because it did not happen. Now consider the case that the
agent has a knowledge that a connected battery is important for the car working
(C = {car ↔ key∧battery}). Then, now we have Bcar C∪(E∪{key, battery}) |=
car and car ∈ Enew . Now we have key and battery and key ∧ battery are actual
causes for car since they are both in E and we have key∧battery → car ∈ Bcar C
with {key, car} is the smallest set resulting for making the car work. And now
we have the counterfacatuals:
– ¬(key ∧ battery) → car
– ¬key → ¬car
– ¬battery → ¬car
Example 2. Consider the “Loader” example from [4] where shooters B and C
are going to shoot a prisoner. E’s job is to load B’s gun while C loads it’s own
gun. On the shooting day, E does load B’s gun and C loads his own gun. C
shoots the prisoner. It is clear that C is responsible of prisoner’s death (D)
First, we construct the set B = {(A ∧ B) → D, C → D} The set E = {A, C}
From the third causal rule and C we get D. So C actually caused D (C → D)
while B only generally causes D but actually didn’t in that situation.
For counterfactuals, since we have C → D then we have the following counterfactual ¬C → ¬D. Note that we also have the following counterfactual too
¬C → (B → D) but with less importance since we have (A ∧ B) → D as a
general cause for D but not fully fulfilled and we need to add B to E which has
an importance degree of 4.
Example 3. Consider the “Window example from [4] where S and B throw
rocks at a window and they both hit the window at the same time and the
window breaks W . In such case one could say that S broke the window or B
broke the window or S and B broke the window.
First, we construct the set B = {S → W, B → W } The set E = {S, B} From
the first causal rule and S we get W . From the second causal rule and B we get
W . So S ∨ B actually caused W which matches our intuition.
Since we have S ∨ B actually caused W , then we he have the following three
counterfactuals:
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Michael Khalil
– if ¬S then B → W would have happened (¬S → (B → W ))
– if ¬B then S → W would have happened (¬B → (S → W ))
– if ¬(B ∨ S) then W would have not happened (¬(B ∨ S) → ¬W ))
Example 4. (Early preemption) Consider the “Backup” example from [4]
where E poisons the coffee of a victim and dies after drinking it D. A backup B
would have poisoned the coffee if E did not which would also to D
First, we construct the set B = {A → D, (¬A ∧ B) → D} The set E = {A}
From the first causal rule and E we get D. So E actually did cause D.
Example 5. (Late preemption) Consider the “Bottle” example from [4]
where Suzy ST and Billy BT both throw rocks at a bottle and Suzy’s rock
hit the bottle SH and the bottle shatters BS. Billy’s rock hitting the bottle BH
does not occur.
First, we construct the set B = {ST → SH, ¬SH ∧ BT → BH, SH →
BS, BH → BS} The set E = {ST, BT } It is clear that B ∪ E |= BH. From
the first causal rule and ST we get SH, and with SH and the third rule we
get BH. Note that all of these would be in Enew . We would have AC contains
{ST ∧ SH → BS}. (Model M1 )
Now consider, in model M2 , if ¬SH. We would have BT and BH to be actual
causes for BS in M2 . So we would have ¬SH → (BH → BS) and ¬SH → BS
and ¬SH → BH as counterfactuals in M1 . So the agent believes that if Suzy
did not hit the bottle Billy would have and if Suzy did not hit the bottle and
if Billy dit hit the bottle then the bottle would shatter, and if Suzy did not hit
the bottle the bottle would have shattered anyway.
Now consider M3 where the second causal rule in B is changed to be BT →
BH. In that model we would have the counterfactual that ¬SH → ¬BS and we
would also have ¬SH → (BH → BS). Since in that model our belief dictates
that there is no relation between Suzy not hitting the bottle results in Billy
hitting the bottle. Our agent does not have that belief and would think that if
Suzy did not hit the bottle the bottle would have not shattered or maybe Billy
would have hit the bottle and then it would have shattered.
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Conclusion
As we have seen through the example, the logic is quite easy to use and that it
is flexible. An agent depends on the beliefs it has and would reason and derive
logical causal conclusions regarding his beliefs. We shown that the logic works
on examples of Shoham’s and Bochman’s and how it in some way capture the
same intuitions and concepts in a more simple way without tackling time or nonmonotonicity and without depending on hard-to-construct sets. We also avoided
the use of modal operators and first order logic for the aim of having a simple
logic without losing their benefits or effectiveness in terms of causation. Also,
a way to handle counterfactuals is shown and tested against famous examples
Causation: A simplistic approach
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in the literature and how it does match our intuition. A major point in this
paper is differentiating between general and actual causality and it is done in a
very similar way to Bochman’s[4]. The logic is quite general with the ability to
capture different logics with the hope that it can be further developed and be
more formally modeled in future work. With this, we hope that we provided a
strong and general basis for future approaches reasoning about causation.
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Michael Khalil
References
1. Causation. D. Lewis. (1973) The Journal of Philosophy, 70, pp. 556-567.
2. Nonmonotonic Reasoning and Causation. Yoav Shoham. (1990) Cognitive Science,
14, pp. 213-252.
3. Toward a useful concept of causality for lexical semantics. Jerry R. Hobbs. (2005)
Journal of Semantics, 22, pp. 181-209.
4. Actual causation in a logical setting. Alexander Bochman. (2018) Cognitive Science,
pp. 1730-1736.