Truth-Functional Propositional Logic by SidneyFelder Truth-functional propositional logic is the simplest and expressively weakest member of the class of deductive systems designed to capture the various valid arguments and patterns of reasoning that are specifiable in formal terms. The term ‘formal’ in the context of logic has a number of aspects. (Some of these aspects—those connected with the distinction between form and content—will be discussed a bit later). The aspect concerning us most right nowisthat connected with the capacity to specify meanings and drawinferences by the more or less automatic application of predetermined rules (rules previously either learned, deduced, or arbitrarily stipulated). This involves far more than the substitution of simple symbols for words. The examples to have inmind are the rules and oper- ations employed in arithmetic and High School algebra. Once we learn howtoadd, subtract, multi- ply,and divide the whole numbers {0,1,2,3,...} in elementary school, we can apply these rules, say, to calculate the sum of anytwo numbers, automatically,without thinking, whether we understand whythe rules work or not. The uniformity,simplicity,and regularity of these arithmetical rules, and their applicability with minimal understanding, is shown by the existence of extremely simple artifi- cial devices for effective arithmetical calculation such as the ancient abacus.Before anysystem can be “automated” in this way (that is, before it can be converted into a system for calculation (i.e., a calculus), its various elements must be represented in an appropriate form.For example, if each natural number was represented by a symbol possessing no relationship to the symbol representing anyother number,nomethod could exist for adding twonumbers by operations employing paper and pencil. It is necessary to represent the system of numbers in a systematic form, to formalize their representation, if such mechanical operations are to be possible. This can be done with various fragments of logical reasoning also, and hence we speak, for example, of the propositional and first- order predicate calculus as well as of propositional and first-order predicate logic. Truth-functional propositional logic, also known as sentential logic,the sentential calculus,the state- ment calculus,etc. studies the expressive and deductive relationships among certain combinations of propositions. The only objects of propositional logic that possess autonomous expressive significance are complete sentences, statements, or propositions that makeassertions possessing definite truth values.(There is asubtlety here that we will illuminate shortly). All the deductive logics we are examining in this course are bivalent,meaning that (for the purposes of this course) each sentence is to be considered either true or false.(There are logics in which more than twotruth values are admitted, and logics in which there are “truth-value gaps”, but unfortunately we will not have time to consider them here). Toreinforce the convention that we are admitting no alternative toasentence’struth or fal- sity (meaning that the values ‘true’ and ‘false’ are exhaustive), it is best to interpret ‘false’ in this context as equivalent to ‘not true’, meaning that the denial of a sentence’struth is considered equiv- alent to the assertion of its falsity. The symbols A1,A2,A3,... represent the atomic (elementary) sentence forms from which all others are formed. Although it is probably best to imagine these symbolic complexesaspotentially stand- ing for comparatively simple statements such as “It is snowing”, “It is sunny”, “Albert Einstein pub- lished the Special Theory of Relativity in 1905”, and “The moon orbits the earth”, a letter can rep- resent an arbitrarily complexstatement: There is nothing in principle to prevent us from letting A1 represent the conjunction of all the assertions made either in the text of Edward Gibbon’s The Decline And Fall Of The Roman Empire or in the text of Whitehead and Russell’s Principia Course Notes Page 1 Truth-Functional Propositional Logic Mathematica.Because in logic we are concerned with the manner in which the truth values of cer- tain compound expressions are related to the truth values of both atomic sentences and other com- pound expressions, the atomic sentences are best conceivedasplace-holders for truth values rather than as ordinary sentences whose truth or falsity is susceptible to evaluation. The decision to repre- sent a sentence by a letter with a subscript commits us to treating it as indivisible or atomic in the context in which that representation holds. Operationally,this has a two-fold significance. Syntacti- cally,atomicity signifies that an atomic sentence possesses no inner structure, and in particular no connectives. Semantically,atomicity signifies that truth-values are freely assignable to the class of atomic sentences, meaning that the truth-values assigned to anyset of atomic sentences places no constraints on the truth-values assignable to anyother set of atomic sentences. This implies, for example, that the truth values freely assigned to an arbitrarily chosen set of atomic sentences S can never“collide” with the truth values that we choose to assign to anyset of elementary sentences outside S.Onthe other hand, the fact that the truth values possessed by non-atomic sentences can in general collide with each other underlies the possibility of deducing one set of sentences from another. Truth-Functional Connectives At the expressive and deductive heart of propositional logic are the truth-functional logical connec- tives.The application of these connectivespermit us in the first instance to construct sentences, for- mulae, or expressions of arbitrary degrees of formal complexity.Propositional logic is ‘truth-func- tional’ because the rules of formation for the construction of compound expressions out of simpler expressions are such, and the semantic properties of the connectivesare such, that together they imply that the truth value of anyexpression (compound of sentences) composed in a legalway is uniquely determined by the truth values of the atomic sentences appearing in the expression. (The proof of this fact is not trivial, but we here takethis property,called unique readibility,asgiv en). Thus a function (an assignment or valuation)that associates a definite truth value to each element of aset of atomic sentences S uniquely determines the truth value of all compound expressions com- posed solely of the atomic sentences of S.Toelaborate: Consider anyset of atomic sentences S and the set T that includes together with all the elements of S all the non-elementary expressions that can be formed from the elements of S by applying the truth-functional connectivesanarbitrary finite number of times. (Thus T contains the same atomic sentences as S as well as all compound expres- sions consisting solely of the atomic sentences of S). It can be proventhat anytwo assignments v and w that agree on what truth values theyassociate with the atomic sentences of T (i.e., agree on S)agree on all sentences of T.1 We will nowdiscuss the intuitively most accessible connectivesemployed in the construction of complexformulae from combinations of simpler formulae. And,also designated conjunction,ismost commonly represented by an ∧ or an ampersand (&). The formula A ∧Bproduced by combining arbitrary sentences A and B, either atomic or composite, by this connective istrue if and only if A is true and Bistrue; in other words, the sentence A ∧Bis true under the circumstances, and only under the circumstances, that both sentence A and sentence 1 Givenafunction f defined on a domain (set) S,any function g defined on a superset T of S is called an extension of f if f and g both agree on their common domain of definition S.Ingeneral, functions admit of an infinite number of distinct extensions, but there is only one extension from truth value assignments defined on anyset of atomic sentences S to truth value assignments defined on expressions consisting solely of occurrences of S. Course Notes Page 2 Truth-Functional Propositional Logic Bare true. Since A is true under the condition that the state of affairs corresponding to A in fact holds, and since B is true under the condition that the state of affairs corresponding to B in fact holds, the sentence A ∧Bistrue under the (in general more stringent) condition that both the state of affairs corresponding to A holds and the state of affairs corresponding to B holds. Thus let A be “Jack is ill” and B be “Joan won the game”. If Isay “Jack is ill and Joan won the game”, I will be making a true statement if and only if both “Jack is ill” is true (that is, Jack is ill) and “Joan wonthe game” is true (that is, Joan did win the game). The case in which the conjunction “Jack is ill and Joan won the game” is true is the case in which True is assigned to “Jack is ill” and True is assigned to “Joan won the game”, TT for short. On the other hand, “Jack is ill and Joan won the game” is false under the conditions in which 1) "Jack is ill” is true and “Joan won the game” is false (TF); 2) "Jack is ill” is false and Joan won the game” is true (FT); and 3) "Jack is ill” is false and “Joan won the game” is false (FF). Thus, just as in ordinary language, the falsity of a conjunc- tion is entailed by the falsity of evenasingle one of its conjuncts.This means that if I makea claim such as A ∧B∧C∧D∧E, the discovery that evenasingle one of these conjuncts is false—either AorBorCorDorE—is sufficient to falsify my claim. To makethis quantitatively precise, there is one case (TTTTTTTT) in which A ∧B∧C∧D∧Eistrue and 255 cases in which A ∧B∧C∧D∧Eis false.