Logic studies reasoning not to explain actual processes of reasoning but instead to describe the norms of good reasoning.

The central focus of our study of logic will be inference. We will refer to the starting points of inference as assumptions or premises and its end as a conclusion. These two aspects of a stretch of reasoning can be referred to jointly as an argument. We will separate them by a horizontal line when they are listed vertically and by the sign / when they are listed horizontally.

We use the lower case Greek φ, ψ, and χ to stand for individual sentences and upper case Greek Γ, Σ, and Δ to stand for sets of sentences. Our notation for arguments will not distinguish sets from lists of their members; and, in considering the norms of inference, we will not distinguish between lists of sentences that determine the same set.

Inference that merely extracts information from premises or assumptions and thus brings no risk of new error is deductive. Inference that goes beyond the content of the premises to, for example, generalize or explain is then non-deductive. Deductive inference may be distinguished as risk free in the sense that it adds no further chance of error to the data. The study of the norms of deductive inference is deductive logic, and it is topic of this course.

Since deductive inferences are risk free, they provide a lower bound on the inferences that are good. Deductive reasoning also sets an upper bound on good inference by rejecting certain conclusions as absolutely incompatible with given premises.

The relation between premises and a conclusion that can be deductively inferred from them is entailment. When the premises and conclusion of an argument are related in this way, the argument is said to be valid. Our symbolic notation for this relation are the signs ⇒ and ⇐, where Γ ⇒ φ says that the premises Γ entail the conclusion φ and φ ⇐ Γ says that φ is entailed by Γ. A set of sentences is inconsistent when its members are mutually incompatible, and a sentence φ is excluded by a set Γ when φ and the members of Γ are mutually incompatible.

We will be interested in the deductive inferences whose validity is a result of the logical form of their premises and conclusions; so our study will be an example of formal logic. The norms of deductive reasoning based on logical form are analogous to some laws of mathematics. The recognition of these analogies (especially by Boole and Frege) has influenced the development of notation for formal deductive logic over the last two centuries, and logic studied from this perspective is often referred to as symbolic logic.