The epistemology of causality
There are two epistemic approaches to causal theory. Under a hypothetico-deductive account, we hypothesize causal relationships and deduce predictions based on them. We test these hypotheses and predictions by comparing empirical phenomena and other knowledge and information on what actually happens to these theories. We may also take an inductive approach in which we make a large number of appropriate, justified observations (such as a set of data) from which we can induce causal relationships directly from them.
Hypothetico-Deductive discovery
The testing phase of this account of discovery and causality uses the views on the nature of causality to determine whether we support or refute the hypothesis. We search for physical processes underlying the causal relationships of the hypothesis. We can use statistics and probability to determine which consequences of hypotheses are verified, like comparing our data to a distribution such as a Gaussian or Dirichlet one. We can further probe these consequences on a probabilistic level and show that changing hypothesized causes can predict, determine, or guarantee effects.
Philosopher Karl Popper advocated this approach for causal explanations of events that consist of natural laws, which are universal statements about the world. He designated initial conditions, single-case statements, from which we may deduce outcomes and form predictions of various events. These case initial conditions call for effects that we can determine, such as whether a physical system will approach thermodynamic equilibrium or how a population might evolve under the influence of predators or external forces. Popper delineated the method of hypothesizing laws, deducing their consequences, and rejecting laws that aren’t supported as a cyclical process. This is the covering-law account of causal explanation.
Inductive learning
Philosopher Francis Bacon promoted the inductive account of scientific learning and reasoning. From a very high number of observations of some phenomenon or event with experimental, empirical evidence where it’s appropriate, we can compile a table of positive instances (in which a phenomenon occurs), negative instances (it doesn’t occur), and partial instances (it occurs to a certain degree). This gives a multidimensionality to phenomena that characterize causal relationships from both a priori and a posterior perspectives.
Inductivist artificial intelligence (AI) approaches have in common the feature that causal relationships can be determined from statistical relationships. We assume the Causal Markov condition holds of physical causality and physical probability. This Causal Markov Condition plays a significant deterministic role in the various features of the model and the events or phenomena it predicts. A causal net must have the Causal Markov Condition as an assumption or premise. For structural equation models (SEM), Causal Markov Conditions result from representations of each variable as a function of its direct causes and an associated error variable with it. We assume probabilistic independence of each error variable. We then find the class of causal models or a single best causal model with probabilistic independences that are justified by the Causal Markov Condition. They should be consistent with independences we can infer from the data, and we might also make further assumptions about the minimality (no submodel of the causal model also satisfied the Causal Markov Condition), faithfulness (all independences in the data are implied via the Causal Markov Condition), linearity (all variables are linear functions of their direct causes and uncorrelated error variables). We may also define causal sufficiency, whether all common causes of measured variables are measured, and context generality, every individual or node in the model has causal relations of the population. These two features let us describe models and methods of scientific reasoning as causal in nature and, from there, we may apply appropriate causal models such as Bayesian, frequentist, or similar methods of prediction. We may even illustrate a causal diagram or model elements under various conditions such as those given by independence or constraints on variables.
This way, in the intercorrelatedness of the graph or model, we can’t change the value of a variable without affecting the way it relates to other variables, but there may conditions in which we construct models that have autonomous nodes or variables. The way these features and claims of inductivist AI interact with another is subject to debate by the underlying assumptions, justification, and methods of reasoning behind these models.
Metaphysics of causality
We can pose questions about the mathematization of causality even with the research and methods that have dominated the work on probability and its consequences. We can speculate what causality is and the opinions on the nature of causality as they relate to the axioms and definitions that have remained stable in the theories of probability and statistics.
We can elaborate three types of causality approaches. The first is that causality is only a heuristic and has no role in scientific reasoning and discourse, as philosopher Bertrand Russel argued. Science depends upon functional relationships, not causal laws. The second position is that causality is a fundamental feature of the world, a universal principle. We should, therefore, treat it as a scientific primitive. This position evolved out of conflict with purported philosophical analyses that appealed to asymmetry of time (that it moves in one direction) to explain the asymmetry of causation (that they move in one direction and one direction only). This raises concerns of how to interpret time in terms of causality. The third is we can reduce causal relations to other concepts that don’t involve causal notions. Many philosophers support this position, and, as such, there are four divisions within this position.
The first schism we discuss is that causality is a relation between variables that are single-case or repeatable according to the interpretation of causality in question. We interpret causality as a mental in nature given that causality is a feature of an agent’s epistemic state and physical if it’s a feature of the external world. We interpret it as subjective if two agents with the same relevant knowledge can disagree on a conclusion of the relationships with both positions correct, as though they were a matter of arbitrary choice. Otherwise we interpret it as objective. The subjective-objective schism raises issues between how different positions would be regarded as correct and what determines the subjective element or role subjectivity plays in these two different positions.
The second partition is the mechanistic account of causality – that physical processes link cause and effect. We interpret causal statements as giving information about these processes. Philosophers Wesley Salmon and Phil Dowe advocate this position as they argue causal processes transmit or have a conserved physical quantity to them. We may describe the relation between energy and mass (E = mc²) as causal relations from start (cause) to a finish (effect). One may argue against this position on the grounds that these relations in science have no specific direction one way or another and are symmetrical and not subject to causality. It does, however, relate single cases linked by physical processes even if we can induce causal regularities or laws from these connections in an objective manner. If two people disagree on the causal connections, one or both are wrong.
This approach is difficult to apply. The physics of these quantities aren’t determined by the causal relations themselves. The conservation of these physical quantities may suggest causal links to physicists, they aren’t relevant in the fields that emerge from physics such as chemistry or engineering. This would lead one to believe the epistemology of the causal concepts are irrelevant to their metaphysics. If this were the case, the knowledge of a causal relationship would have little to do with the causal connection itself.
The third subdivision is probabilistic causality in which we treat causal connections with probabilistic relationships of variables. We can debate which probabilistic relationships among variables of probabilistic causality determine or create causal relationships. One might say the Principle of Common Cause (if two variables are probabilistically dependent, then one causes the other or they’re effects of common causes that make them independent from one another). Philosopher Hans Reichenbach applied this to causality to provide a probabilistic analysis of time in its single direction. More recent philosophers use the Causal Markov Condition as a necessary condition for causality with other less central conditions. We normally apply probabilistic causality to repeatable variables such that probability handles them, but critics may argue the Principle of the Common Cause and the Causal Markov Conditions have counterexamples showing they don’t hold in under all conditions.
Finally, the fourth subclass is the counterfactual account, as advocated by philosopher David Lewis. In this way, we reduce causal relations to subjunctive conditions such that an effect depends causally on a cause if and only iff (1) if the cause were to occur, then the effect would occur (or its chance to occur would raise significantly) and (2) if the cause didn’t occur then the effect wouldn’t occur. The transitive closure of the Causal Depedendence (that a cause will either increase the probability of a direct effect or, if it’s a preventative, make the effect less likely, as long as the effect’s other direct causes are held fixed) holds. The causal relationships are what goes on in possible worlds that are similar to our own. Lewis introduced counterfactual theory to account of the causal relationships between single-case events and causal relationships that are mind-independent and objective. We may still press this account by arguing that we have no physical contact with these possible worlds or that there isn’t an objective way to determine which worlds are closest to our own or which worlds we should follow and analyze in determining causality. The counterfactualist may respond that the worlds we choose are the ones in which the cause-and-effect relationship occurs as closer to our own world and, from there, determine which appropriate world is closest to our own.
Heuristic 