sentential function
Noun: 1. A formal logical expression containing variables: A sentential function is a symbolic expression in logic that has the grammatical form of a statement (sentence) but contains one or more variables. It is not a complete, truth-evaluable sentence on its own. It becomes a specific, meaningful sentence (a "proposition") only when its variables are replaced by specific constants or values.
A sentential function is a template or schema. It represents the structure of a possible statement. The primary use is to analyze logical form, create generalizations, and define logical operations. * In the sentential function "x is capital of y," 'x' and 'y' are variables. * This function becomes a true sentence when we substitute 'x' with "London" and 'y' with "the UK." * It becomes a false sentence when we substitute 'x' with "Paris" and 'y' with "Germany."
- Simple Example: "p ∧ q" is a sentential function where 'p' and 'q' are propositional variables. It becomes a specific logical sentence when 'p' is replaced by "It is raining" and 'q' is replaced by "The ground is wet."
- Predicate Logic Example: "F(a)" is a sentential function where 'F' is a predicate variable (e.g., "...is tall") and 'a' is an individual variable (e.g., a person's name). Substituting constants yields "Socrates is mortal."
- Mathematical Example: "x + 2 = 5" is a sentential function. It becomes the true sentence "3 + 2 = 5" when x is replaced by 3, and a false one when x is replaced by 4.
- Quantification: Sentential functions are the objects over which quantifiers (∀ "for all", ∃ "there exists") operate. For example, in "∀x (x is mortal)," the part "(x is mortal)" is a sentential function.
- Open Sentence: In mathematics and philosophy, a sentential function is often called an "open sentence" because its truth value is "open" or undetermined until the variables are instantiated.
- Well-Formed Formula (WFF): In formal logic, a sentential function is a type of well-formed formula that is not closed (i.e., it contains free variables).
- Propositional Function: A near-synonym, often used interchangeably with "sentential function," particularly in the works of philosophers like Bertrand Russell.
- Open Formula: A synonymous term commonly used in mathematical logic.
- Statement Form: A term emphasizing the structural template aspect of a sentential function.
- Sentence (or Proposition): The completed, truth-evaluable result of substituting constants into a sentential function.
- Open sentence
- Propositional function
- Open formula
- Statement form
- Variable: A symbol (like x, y, p, q) in a sentential function that stands for an unspecified constant.
- Constant: A symbol with a fixed, specific meaning that replaces a variable to create a sentence.
- Quantifier: A logical operator (e.g., "for all," "there exists") that binds variables in a sentential function to create a closed statement.
- Truth Value: The property (true or false) of a sentence resulting from a completed sentential function.
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formal expression containing variables; becomes a sentence when variables are replaced by constants
