In the language of mathematics and logic there are essentially two types of statements. There are basic propositions (abbreviated here by lower-case, italicized letters, e.g., p, q, etc.), like "the triangle ABC is equilateral," or "the integer a is odd." Then there are complex propositions which are constructed from the basic propositions by connectives: conjunction ("p and q"), disjunction ("p or q"), negation (not-p), and implication ("if p then q). These connectives are truth-functional, meaning that the truth-value of the complex sentence is a function of the truth-values of the connected basic sentences. Each connective can be considered a truth function.
Truthfunctions are formally defined. Like any other function, a truth function is associated with two sets, a domain set and a range set. A function is a rule that associates each unique object in the first set with exactly one object in the second. In the case of truth functions, the first set is the set of finite sequences of truth-values; the second is simply the set of truth-values.
Usually, systems of logic only permit two possible truth-values: true (T) and false (F). Such systems are known as binary logics. In some philosophical circles, other truth-values are permitted; for instance "unknowable" is sometimes considered a truth-value in its own right.
A generic binary truth function would take a sequence of T's and F's (e.g., TTFTFFTTT) and assign to it a value of either T or F. For example, let v be a truth function, where: v(TTFTFFTTT) = T.
This definition of the truth function may seem obscure, but it is actually just an abstract formalization of a pattern of reasoning that we feel quite comfortable with. Consider negation. Suppose we have a set of propositions p, q, etc., that can be made about the world. For example, sentences of the sort "The sky is blue," "Jack loves Jill," or "Elvis is alive." Each of these can be assigned a truth-value, T or F, depending of course on whether or not these sentences are true or false. Suppose p is true (T). Then we quite naturally reason that its negation (not-p) must be false (F). If "the sky is blue" is a true statement, then "the sky is not blue" is a false statement. Conversely, if p is false, then not-p is true.
This reasoning is mathematically modeled by the truth function. Recall our definition. Consider the following truth function "n." It tells us what to do with a one-place sequence of truth-values (i.e., T or F). Whenever you feed either a T or an F into the n function, it spits out the opposite of what you put in. So
n(T) = F, and
n(F) = T.
In other words, if we put the truth-value of a sentence into the function n, we get the truth-value of the negation of the sentence! This is what we do when we reason that the negation of a true sentence must be false. Essentially, we apply the n function - even though we do not think about it in formal mathematical terms.
Like negation, the other logical connectives (conjunction, disjunction, implication, etc.) can also be understood as truth functions. The conjunction of two propositions (e.g., pandq) is true whenever both p and q are true; otherwise it is false. Think about the sentence: "The sky is blue and Elvis is alive." Clearly this sentence is true only if the sky really is blue and Elvis really is alive. If the sky is not blue, or if Elvis is dead, then the sentence, as a whole, is false. The corresponding conjunction truth function (call it c) takes two-place sequences of T's and F's (i.e., TT, TF, FT, FF) and converts them into either true or false according to the following plan:
c(TT) = T
c(TF) = F
c(FT) = F
c(FF) = F
When we figure out the truth-value of a conjunction from the truth-values of its constituent parts (conjuncts), we are effectively applying the c function. Analogously, implication disjunction are also representable by truth functions. In fact, in formal logic, it makes sense to simply define the meaning of the connectives in terms of their truth functions.
Complex compound propositions can also consist of basic propositions joined together by multiple connectives ("The sky is blue and Elvis is alive, or the sky is beige."). We can calculate the truth-value of a compound proposition from the truth-values of the basic propositions using truth functions. These complex functions would be compositions of many "simple" truth functions. The domain set of these functions must consist of longer truth-value sequences. For every basic proposition there is one truth-value, and so one letter in the sequence.
The truth function is closely related to the truth table. Truth tables are schematic representations of truth functions.
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