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Logic Chapter 7 Logic: A Brief Introduction Ronald L. Hall, Stetson University PART III - Symbolic Logic Chapter 7 - Sentential Propositions 7.1 Introduction What has been made abundantly clear in the previous discussion of categorical logic is that there are often trying challenges and sometimes even insurmountable difficulties in translating arguments in ordinary language into standard-form categorical syllogisms. Yet in order to apply the techniques of assessment available in categorical logic, such translations are absolutely necessary. These challenges and difficulties of translation must certainly be reckoned as limitations of this system of logical analysis. At the same time, the development of categorical logic made great advances in the study of argumentation insofar as it brought with it a realization that good (valid) reasoning is a matter of form and not content. This reduction of good (valid) reasoning to formal considerations made the assessment of arguments much more manageable than it was when each and every particular argument had to be assessed. Indeed, in categorical logic this reduction to form yielded, from all of the thousands of possible syllogisms, just fifteen valid syllogistic forms. Taking its cue from this advancement, modern symbolic logic tried to accomplish such a reduction without being hampered with the strict requirements of categorical form. To do this, modern logicians developed what is called sentential logic. As with categorical logic, modern sentential logic reduces the kinds of propositions to a small and thus manageable number. In fact, on the sentential reduction, there are only two basic propositional forms: simple propositions and compound propositions. But it would be too easy to think that this is all there is to it. What complicates the sentential formal reduction is that there are a number of different kinds of compound propositions. These different kinds of compound propositions are called propositional forms and are determined by the several different ways that simple propositions can be combined to form compound propositions. Fortunately, there is a small and manageable number of these connections, hence a relatively small number of different compound propositional forms. In this Chapter we will investigate five compound propositional forms. These forms are as follows: 119 1. Conjunctions 2. Negations 3. Disjunctions 4. Conditionals 5. Biconditionals. This reduction of every compound proposition to one of these forms is almost as neat as the categorical reduction to the four standard-form categorical propositions, A, E, I, and O. However, the flexibility and power of the sentential system of logic, as we will see, is much greater than we had in categorical logic. For one thing, in sentential logic there is no necessity for reformulating arguments as standard-form syllogisms. There is no better way to see the flexibility and power of the sentential system of symbolic logic than to jump right into an explanation of these five kinds of compound propositional forms. Before we do this, however, we need to say something more about the basic distinction between simple and compound sentential propositions. A good place to start is with some definitions. So we will define the two as follows: Simple Proposition: An assertion (either true or false) that cannot be broken down into a simpler assertion. Example: “The cat is on the mat.” Compound Proposition: An assertion of a combination of simple propositions. Example: “The cat is on the mat and the dog is in the yard.” Compound propositions are like simple propositions insofar as they are either true or false. However, the truth or falsity of a compound proposition is a function of the truth or falsity of its component simple propositions. Because this is so, sentential logic is sometimes called truth-functional logic. All this means is that the truth or falsity of a compound proposition is determined by the truth-values of its component simple propositions. We will say more as to how this determination is made as we proceed. Sentential logic has adopted the convention of using upper case letters to stand for particular simple propositions. So in this system, C can be used to stand for the simple proposition, “The cat is on the mat.” It is common practice to use a letter that reminds us of the content of the particular proposition that we are symbolizing, but we are free to use any letter we choose. We use “C” because it reminds us of “cat.” To symbolize a compound proposition, we simply join more than one simple proposition together, symbolizing each simple proposition with an upper case letter. So, to symbolize the compound 120 proposition, “The cat is on the mat and the dog is in the yard” we simply substitute the following symbolic expression: “C and D.” The “and” here is called a truth-functional connective. It is called a connective since it connects two or more simple propositions to form the compound proposition. It is said to be a truth-functional connective because the truth-value of the compound proposition that it forms is a function of the truth- values of its component simple propositions. One way to characterize the 5 kinds of compound propositional forms that we are now going to look at is to say that they represent 5 different truth-functional connectives, that is, 5 different ways to combine simple propositions into compound ones. The name of each of these several kinds of compound propositional forms is determined by the kind of the truth-functional connective that forms it. Enough said then by way of introduction. Let’s turn to our first kind of truth-functional connective and accordingly to the first kind of propositional form, namely, the conjunction. 7.2 The Conjunction Usually conjunctions are formed by joining two or more simple propositions together with the word “and.” This word “and” has a function and that function is to join two or more simple propositions together to form a compound proposition known as a conjunction. The simple propositions that are joined together to form a conjunction are called conjuncts . The word “and” in English is not the only word that can have this function. Other English words, as well as grammatical techniques, can be surrogates for “and” and serve to function exactly as it does. Such surrogates include “but,” “nevertheless,” “moreover,” “however,” “yet,” “also,” and others, and sometimes simply a comma or semicolon. For example, it is clearly the case that the compound proposition, “The cat is on the mat and the dog is in the yard,” has exactly the same sense, reference, and truth-value as the compound proposition, “The cat is on the mat but the dog is in the yard,” or as the proposition, “The cat is on the mat; the dog is in the yard.” As well, sometimes “and,” does not serve this function of conjoining two or more propositions together to form a conjunction. For example, “The dog and the cat were fighting each other” need not be seen as a compound proposition; that is, it can plausibly be seen as a simple proposition. Even though we have the word “and” in this sentence, it is not necessarily functioning to join two or more simple propositions together. To capture and express the function of “and,” we will adopt the following symbol "•" (the dot) to stand for this truth-functional connective. Using this symbol, we can express symbolically the proposition, “The cat is on the mat and the dog is in the yard” as follows: “C•D.” This expression is a conjunction with two simple conjuncts “C” and “D.” The symbol expresses the “and” function whether or not the word “and” is used, but only when that function is intended in the expression that is being symbolized. 121 The symbol for "and" is a truth-functional connective, insofar as the truth-value of the compound proposition it forms is a function of the truth-values of its component simple conjuncts. I will specify how we determine the truth-value of a conjunction in just a moment, but first I must introduce another distinction. It is helpful at this point to be reminded of the distinction between actual propositions and propositional forms . In sentential logic we mark this distinction by adopting the convention of using upper case letters to stand for actual propositions and lower case letters, usually “p-z,” to stand for propositional forms. Accordingly, “C” is used to stand for an actual proposition, and “p” is used to stand for any proposition whatsoever. For example, we could use “p” to stand for “C” or for “C•D” or indeed for any simple or any compound proposition whatsoever. Adopting this convention, we now see that, the symbolic expression, “C•D” may be used to stand for the particular proposition, “The cat is on the mat and the dog is in the yard.” We also see, by contrast, that the expression, “p•q” does not stand for any actual proposition. Rather, this last expression is a propositional form and can be read as standing for any conjunction whatsoever. So clearly the proposition “p•q” can stand for the particular proposition “C•D.” As we say, “C•D” is a substitution instance of the conjunctive propositional form “p•q.” The latter propositional form, however, has an infinite number of substitution instances, namely, any conjunction whatsoever. So the latter expression is much broader and much more useful than the first. With this distinction, we are ready to specify how the truth-value of the conjunction is determined. As I have said, a conjunction is truth-functional, that is, the truth-value of the conjunction (as a compound proposition) is a function of the truth-values of its component conjuncts (simple propositions).
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