In the study of logic, the concepts of premises and conclusions are foundational, serving as the basic parts of any inference or argument. Logic is often defined as the study of the correctness of these arguments. An argument is essentially a collection of premises that lead to a conclusion, and an inference is the mental process of moving from these premises to the conclusion. The relationship between premises and conclusions is central to determining
whether an argument is considered correct or incorrect.
The Nature of Premises and Conclusions
For most forms of logic, it is widely accepted that premises and conclusions must be "truth-bearers." This means they possess a truth value, being either true or false. In contemporary philosophy, they are generally viewed either as propositions or as sentences. Propositions are abstract objects that represent the meaning of sentences. For example, the English sentence "the tree is green" and the German sentence "der Baum ist grün" express the same proposition, despite being different linguistic constructs. Sentences, on the other hand, are concrete linguistic objects, like the symbols printed on a page.Premises and conclusions exhibit an internal structure. They can be either simple or complex. A complex proposition is composed of other propositions, which are interconnected by propositional connectives such as "and" or "if...then." Simple propositions, however, do not contain other propositions as parts. Nevertheless, they too have an internal structure, being made up of subpropositional parts like singular terms and predicates. For instance, the simple proposition "Mars is red" is formed by applying the predicate "red" to the singular term "Mars." In contrast, "Mars is red and Venus is white" is a complex proposition combining two simple propositions with the connective "and."
Truth Values and Logical Connectives
The truth of a proposition depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth value is solely determined by the truth values of their component parts. However, this relationship becomes more intricate with simple propositions and their subpropositional elements. These subpropositional parts carry their own meanings, often referring to objects or classes of objects. The truth of the simple proposition they form then depends on their relationship to reality—that is, what the objects they refer to are actually like. This area of study falls under theories of reference.Truth tables are a crucial tool for illustrating how logical connectives function and how the truth values of complex propositions are derived from their parts. These tables feature a column for each input variable, with each row representing a possible combination of truth values these variables can take. Symbols like "T" and "F" or "1" and "0" are commonly used to denote "true" and "false." For example, the expression "p ∧ q" (p and q) is true only if both input variables, p and q, are true; otherwise, it is false. Other significant logical connectives include "¬" (not), "∨" (or), "→" (if...then), and "↑" (Sheffer stroke). Truth tables can also be constructed for more elaborate expressions involving multiple propositional connectives, providing a clear visual representation of their logical behavior.
