What is formal theory?

A formal theory can be understood in two main contexts: within psychology and within mathematical logic.

In Psychology

In the context of psychology, a formal theory is a structured and often mathematical approach to understanding and explaining psychological phenomena. Key characteristics include:

• Mathematical Models: Formal theories in psychology frequently use equations or models to describe relationships between variables in quantitative terms. Examples include decision-making theories like Signal Detection Theory and mathematical models in cognitive psychology1.
• Logical Rigor: These theories require a high level of logical consistency, with clear definitions and logical relationships between different parts of the theory1.
• Predictive Power: Formal theories aim to predict future behaviors or outcomes based on specific variables. For instance, they might predict how individuals will behave in certain social contexts1.
• Empirical Testing: These theories are designed to be testable and falsifiable, allowing researchers to validate or refute parts of the theory through empirical research1.

In Mathematical Logic

In mathematical logic, a formal theory is defined more broadly:

• Definition: A formal theory is a set of sentences in a formal language, often within a deductive system. It is based on a set of axioms and inference rules that generate theorems3.
• Axiomatic System: A formal theory typically includes a subset of sentences called axioms, from which other theorems are derived using logical inference rules. Every axiom is automatically a theorem, and any sentence that is a logical consequence of the axioms is also a theorem of the theory3.
• Consistency and Completeness: A formal theory is considered consistent if it does not prove both a sentence and its negation. It is complete if for every sentence in its language, either the sentence or its negation is provable from the theory. However, according to Gödel's Incompleteness Theorems, any consistent formal theory that includes the arithmetic of natural numbers is incomplete23.
• Interpretation: A formal theory can be interpreted by relating its elementary statements to some subject matter. This interpretation can be full or partial, depending on whether every elementary statement has a correspondent in the subject matter3.

In both contexts, formal theories emphasize rigor, logical consistency, and the ability to make precise predictions or derive theorems from a set of axioms.