Be warned that some require quite lengthy derivations! Consider, for example, a statement of the form . The modern study of three-valued propositional logic began in the work of Jan Łukasiewicz in 1917, and other forms of non-classical propositional logic soon followed suit. A simple example would be the wff ““; that is, “ or not “. Nothing that cannot be constructed by successive steps of (1)-(3) is a well-formed formula. The language PL, as we have seen, contains operators that are roughly analogous to the English operators ‘and’, ‘or’, ‘if… then…’, ‘if and only if’, and ‘not’. A similar consideration justifies the use of indirect proof. So, for example, the following are statements: Sometimes, a statement can contain one or more other statements as parts. If the truth-value assignment makes it false instead, then is a member of , and so we have a derivation of from , since again a premise may be introduced at any time. Owing to this, all those features of a complex statement that are studied in propositional logic derive from the way in which their truth-values are derived from those of their parts. 6. However, it is also very useful for proving other metatheoretic results, as we shall see below. Definition: A statement letter of PL is defined as any uppercase letter written with or without a numerical subscript. Then, as the truth-values of those wffs that are parts of the complete wff are determined, we write their truth-values underneath the logical sign that is used to form them. One possibility, suggested by C. A. Meredith (1953), would be to define an axiom as any wff matching the following form: The resulting system is equally powerful as system PC and has exactly the same set of theorems. In a statement of the form , the two statements joined together, and , are called the disjuncts, and the whole statement is called a disjunction. Let us suppose in fact that ‘‘ is true, but might have been false. Our first topic, however, concerns the language PL’ generally. We can apply the same reasoning given in steps (3)-(5) to remove or its negation from the premise sets by the deduction theorem, arriving at the result that for every set of premises consisting of either or and so on, up until , it is possible to derive . This derivation takes the form of an ordered sequence , where the last member of the sequence, , is , and each member of the sequence is either (1) a premise, that is, it is one of , (2) an axiom of PC, (3) derived from previous members of the sequence by modus ponens. propositional logic, such as: ! Classical truth-functional propositional logic is by far the most widely studied branch of propositional logic, and for this reason, most of the remainder of this article focuses exclusively on this area of logic. Notice that both ‘‘ and “” are true, but different truth-values result when the operator ‘‘ is added. Metatheoretic result 2 (a.k.a. Obviously, whether or not a statement formed using the connective ‘‘ is true does not depend solely on the truth-value of the propositions involved. 2. Constants for contradiction and tautology may also be added. Chrysippus suggested that the following inference schemata are to be considered the most basic: Inference rules such as the above correspond very closely to the basic principles in a contemporary system of natural deduction for propositional logic. “Untersuchungen über das logische Schließen”, Herbrand, Jacques. Metatheoretic result 4 (Completeness): If is a wff of language PL’ and a tautology, then is a theorem of the Propositional Calculus. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another. This may be much clearer with an example. Either the first or the second [and not both]; but the first; therefore, not the second. Moreover, truth tables are alien to our normal reasoning patterns. Each step in the chain represents a simple, obviously valid form of reasoning. PL” differs from PL and PL’ only in that its definition of a well-formed formula can be simplified even further: Definition: A well-formed formula (or wff) of PL” is defined recursively as follows: In language PL”, strictly speaking, ‘|’ is the only operator. Even if ‘‘ and ‘‘ are not actually both true, it is possible for them to both be true, and so this form of reasoning is not truth-preserving. It is provable that an argument in the language of PL is formally valid in that sense if and only if it is possible to construct a derivation of the conclusion of that argument from the premises using the above rules of inference, rules of replacement and techniques of conditional and indirect proof. Moreover, the result is quite often not the most elegant or easy way to show that which you were trying to show. Deontic propositional logic and epistemic propositional logic are two other forms of non-truth-functional propositional logic. If is itself a statement letter, then obviously either it or its negation is a member of . The most commonly studied and most popular formal system used is (truth functional) classical propositional logic with natural deduction, which this article mostly will talk about. Applying the procedure from step (3), we get that without making use of as a premise. Unlike the other operators we have considered, negation is applied to a single statement. So, consider again the following example argument, mentioned in Section I. Each of these, as we have also seen, can be thought of as representing a certain truth-function. Metatheoretic result 3: If is a wff of language PL’, and the statement letters making it up are , then if we consider any possible truth-value assignment to these letters, and consider the set of premises, , that contains if the truth-value assignment makes true, but contains if the truth-value assignment makes false, and similarly for , if the truth-value assignment makes true, then in PC, it holds that , and if it makes false, then . Either the first or the second; but not the second; therefore the first. 3. Such deductions are only simpler in the sense that fewer distinct rules are employed. Corollary 5.1: A wff of language PL’ is a tautology if and only if is a theorem of system PC. (These notions are defined below.). There are eight possible truth-value assignments to these letters, and since is a tautology, all of them make true. See the following chart: Here we see that a wff of the form is true if either or is true but not both. Corollary 5.4: There is a derivation of the wff with as premises in the Propositional Calculus if and only if is a logical consequence of , according to their combined truth table. In sum, then, the Propositional Calculus method of demonstrating something to follow from the axioms of logic is extensionally equivalent to the truth table method of determining whether or not something is a logical truth.