In formal logic, a **valuation** is a function that assigns truth values to propositional variables in a logical formula. It's used primarily in propositional logic and model theory to determine the truth of propositions under various interpretations or scenarios. The concept of valuation is fundamental when analyzing and understanding logical statements and their dependencies within a given logical system. ### Key Aspects of Valuation: 1. **Assignment of Truth Values**: A valuation function takes each propositional variable in a formula and assigns it a truth value, typically either true ($T$) or false ($F$). For example, in a formula with variables $p$ and $q$, a valuation might assign $p = \text{true}$ and $q = \text{false}$. 2. **Determination of Formula Truth**: Once the truth values are assigned to the variables, the truth of the entire formula can be evaluated based on these values. This process involves applying logical connectives (such as AND, OR, NOT, IMPLIES) according to their definitions. For instance, if the formula is $p \land q$ (p AND q), and the valuation is $p = \text{true}$, $q = \text{false}$, then the formula $p \land q$ evaluates to false. 3. **Systematic Exploration**: By changing the valuations, one can systematically explore all possible truth scenarios for a formula. This is especially useful in constructing truth tables, where each row corresponds to a different valuation, showing how the truth value of the formula depends on the truth values of its components. 4. **Use in Satisfiability Testing**: Valuations are crucial in testing whether a formula is satisfiable, which means there is at least one valuation that makes the formula true. If no such valuation exists, the formula is unsatisfiable. 5. **Relevance in Model Theory**: In model theory, a branch of mathematical logic, valuations extend to more complex structures where not only truth values but also elements of a domain are assigned to variables. This helps in interpreting formulas in different models to explore their properties across various mathematical structures. Valuations provide a concrete method by which the abstract symbols of formal logic can be connected to concrete truth conditions, enabling logicians and mathematicians to analyze and manipulate logical formulas effectively. This tool is fundamental in various applications across computer science, mathematics, and philosophical logic, where formal reasoning about truth and falsehood is crucial. # References ```dataview Table title as Title, authors as Authors where contains(subject, "valuation") sort title, authors, modified, desc ```