Inference

Inference is the process of carrying ideas forward to reach a conclusion. It involves using evidence and reasoning to move from known premises to logical consequences. The term comes from the idea of carrying something from one point to another. In many ways, inference is the engine of thought. It allows us to take what we already know and use it to discover new information. This process can be seen in how humans think, how scientists study data, and how computers solve complex problems.

There are several distinct ways to categorize how we infer new information. The most traditional division is between deduction and induction. This distinction dates back at least to Aristotle in the 300s BC. Deduction is a form of reasoning where you derive conclusions from premises that are assumed to be true. Induction is different because it moves from particular evidence to a universal conclusion. A third type, called abduction, was notably distinguished by Charles Sanders Peirce. Abduction differs from induction by focusing on finding the best explanation for an observation.

Deduction relies heavily on the concept of validity. In logic, validity does not refer to whether the facts are true. Instead, it refers to the form of the inference. A valid inference is one where, if the premises are true, the conclusion must also be true. For example, if all humans are mortal and all Greeks are humans, then it must follow that all Greeks are mortal. This is a three-part inference known as a syllogism. However, an inference can be valid even if the parts are false. If you use a valid form but start with a false premise, you might reach a false conclusion.

Inductive reasoning works by observing patterns to create general rules. You might look at many specific instances and conclude that a general law exists. For instance, if you see many white swans, you might induce that all swans are white. Unlike deduction, inductive conclusions are not always certain. They may be correct, incorrect, or only correct within certain degrees of accuracy. These conclusions can be tested by making additional observations. This makes induction a vital tool for learning about the world through experience.

In the modern era, inference has moved into the realm of mathematics and technology. Statistical inference uses mathematics to draw conclusions when there is uncertainty. It uses quantitative or qualitative data that may be subject to random variations. This generalizes deterministic reasoning, where there is no uncertainty. In artificial intelligence, researchers develop automated inference systems. These systems use inference engines to extend a knowledge base. A knowledge base is a set of propositions representing what the system knows about the world.

One specific application of logical inference is found in the programming language Prolog. Prolog, which stands for "Programming in Logic," uses an algorithm called backward chaining. It checks if a certain proposition can be inferred from a knowledge base. For example, if the knowledge base states that all men are mortal and that Socrates is a man, the system can deduce that Socrates is mortal. If the system does not have information about a specific person, it may default to a "closed world assumption," treating unknown facts as false.

Other advanced methods include Bayesian inference and fuzzy logic. Bayesian followers use the mathematical rules of probability to find the best explanation. They view probabilities as degrees of belief, where a probability of 1 is certain and 0 is impossible. This framework treats deductive logic as a special, certain subset of probability. Additionally, logicians study monotonic and non-monotonic reasoning. Deductive inference is monotonic because adding new premises does not change a reached conclusion. Everyday reasoning is often non-monotonic, meaning new information can undermine or change previous conclusions.