The Principle of Sufficient Reason

Causality is known to philosophers as the principle of sufficient reason, that everything has a reason or cause for its existence, from nothing nothing comes. A reason is sufficient as opposed to necessary if no other cause is needed for explanation (see necessary vs sufficient conditions logic). The only sufficient reason for all things is that they must have a reason, hence the name. The PSR is a law of thought along with identity, noncontradiction and the excluded middle. 

The PSR has several classes of causes such as Aristotle's four causes: 

Material- what the cause is made of. "Wood." 

Final- the end or purpose to which a cause occurs. "Write on"

Formal- what is the essence, nature, of the cause. "Rectangle."

Efficient- what brought the cause into existence. "Carpenter." 

Together we get the causes of "table."

There are also four different roots of the PSR which relate to Aristotle's causes from by Arthur Schopenhauer which distinguish valid uses for phenomena as opposed to objects (Kant): 

Becoming (material) - physical cause and effect

Knowing (final) - representation of causal relationships

Being (formal) - relation of representations

Acting (efficient) - internal motivation of causal agent

Schopenhauer's roots, I think, also relate to the four basic laws of thought.

1) Law of Identity- Becoming- Material

2) Noncontradiction- Knowing- Final

3) Law of the Excluded Middle- Being- Formal 

4) Principle of Sufficient Reason- Acting- Efficient 

Gottfried Leibniz came up with the distinction between intrinsic and extrinsic relations from his pioneering predicate logic, as opposed to Aristotle's syllogisms. A subject either contains its predicate or is predicated of another subject. If a predicate is contained in the subject, its existence is necessary, if the predicate is contained in another subject, its existence is contingent, dependent on the existence of something else. A necessary being is self-caused, causa sui, and a contingent being is caused by something else. Thus there are two fundamental sufficient reasons. Reason is different from cause, the former intrinsic to something's nature and the latter intrinsic. Leibniz used this distinction to criticize the mechanical philosophy 

For the purpose of conducting science, efficient causality applied to the phenomena of becoming (being individuated in time and space) is the most appropriate.  

Nowadays in science causal language is used more carefully. The mantra is correlation does not equal causation, thanks to Hume's skeptical doubts about induction. Instead of cause and effect we have an independent variable and a dependent variable. The DV is supposed to be changed in the presence of the IV, and the IV isn't. A causal relationship is a "change in the IV means a change in the DV." Causality as distinguished from correlation is a logical argument, giving some reason for a relationship between the variables. 

“[A]ll causes are of the same kind, and that in particular there is no foundation for that distinction, which we sometimes make betwixt efficient causes, and formal, and material … and final causes” (Treatise of Human Nature, I.iii.14)