Kirk Rader
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The logic of declarative sentences.
The sentential calculus is a formal language for expressing simple logical arguments. Consider the following:
The first two sentences are premises that are simply stipulated to be true for the purposes of the given argument. The third sentence is the conclusion that is derived from the premises by the application of Inference Rules (amodusponens in this case). The whole argument is a theorem, i.e. it is a valid argument. An argument is valid if and only if it is possible to show that the conclusion must be true when making no assumptions other than that the explicitly cited premises are true and then applying a sequence of one or more transformations that are supported by the axioms of symbolic logic.
The preceding naturallanguage argument could be expressed as a sequence of wellformed formulas* (WFF's) of the sentential calculus, as follows:
where is, "It is raining," and is, "The streets are wet."
The primitive terms in a formula of the sentential calculus are represented by upper case letters of the Latin alphabet, and in this example. They evaluate to one of exactly two possible values, true (T) or false (F). When interpreting an expression in natural language, such primitive terms correspond to complete declarative sentences – hence the "sentential" in "sentential calculus" – such as "It is raining," "Two plus two equals four" and so on.
The symbol is one of a number of connectives that can be used to combine terms to express a logical relationship – "if ... then ..." in this case. The statements above the horizontal line are the premises of an argument and the statement below the line its conclusion. Such an argument is valid if the conclusion is necessarily true assuming that its are true.
There are a number of "connectives" or "operators." Each is defined by its truth table* and has a commonly used natural language formulation like "and," "or," "if ... then ..." etc.
Here, for example, is the truth table for , i.e. "and":
See Connectives for a complete set of such truth tables for all of the connectives in traditional logic and a more detailed explanation of their meaning.
The validity of the first argument, above, should be fairly intuitive. Assuming that the symbol means "not" (logical negation) and the symbol means "or", consider this argument:
This is less immediately intuitive, but is also valid. In fact, according to the rules of the sentential calculus it is entirely equivalent to the first argument involving the symbol, as will be discussed further below. For now, consider the following translation into the corresponding natural language argument:
That the sentence, "Either the weather is fair or the streets are wet," is logically equivalent to the sentence, "If it is raining then the streets are wet," is a theorem of traditional logic – but only assuming that "the weather is fair" means exactly the same thing as "it is not raining" and vice versa. The whole point of symbolic logic as it was originally conceived is to divorce purely logical arguments from that sort of ambiguity of natural language. It is from that point view that we can assert the equivalence:
which could be rendered into English as:
The sentence , which may be true (T) or false (F):
Another sentence, , which also may be true or false independently of the value of :
The negation of , i.e. is false if is true, and viceversa:
Two sentences joined by a connective ("inclusive or", in this case):
Grouping using parentheses:
A vacuously true statement (i.e. a tautology):
A way to visualize the grammar of the sentential calculus is by drawing formulas as tree diagrams. Here is such a tree diagram for the example of De Morgan's Law just cited:
If a node in such a tree corresponds to a primitive term, the truth value of the node is that of the primitive term. Otherwise, the truth value of a node is calculated by first evaluating the truth value(s) of the term(s) to which it is directly connected and then applying the appropriate row from the truth table for the nonterminal node. Applying such a procedure to the preceding diagram shows that the root node is always true without regard to the values of the terminal nodes.