A Brief Introduction to Truth-Table Logic. Kent Slinker Pima Community College

Transcription

1 ` A Brief Introduction to ruth-able Logic Kent Slinker Pima Community College

2 Earlier in this class, we learned that all arguments are either valid or invalid. Additionally, we learned that certain valid arguments have a characteristic form. It is now time to rovide students with the tools which will enable them to determine whether any argument form is valid or invalid. We will first start by reviewing the valid forms only. hey are: Modus Ponens If, then q herefore q Modus ollens If, then q Not q herefore not Hyothetical Syllogism If, then q If q, then r herefore if, then r Conjunction q herefore both and q Disjunctive Syllogism or q not herefore q Do not worry about the names now. I have rovided their traditional names for reference and historical uroses. or now, the main question we want to answer is, How can we rove that each of the above arguments really is valid? irst we note that these are not full-fledged arguments. hey are the bare-bones of arguments. In each case, the letters used;, q, r, etc stand for roositions, and roositions are statements which can be either true or false. Hence: omorrow is either June 13th or June 14th It is not June 13 th herefore it is June 14 th.

3 Is an examle of Disjunctive Syllogism, where stands for: omorrow is June 13 th, And q stands for: omorrow is June 14 th Notice the entire argument could be re-written without loss of meaning as: omorrow is June 13 th, or tomorrow is June 14 th. omorrow is not June 13 th. herefore tomorrow is June 14 th. Sometimes it may be helful to re-write an argument so that its structure is exactly the same as one of our argument forms. his is ermissible as long as there is no change in meaning. It is now time to rovide the first tool for argument analysis. Proositional Variables (or letters) Lower-Case Variables We will use lower-case letters to stand for roositions. hose letters can be: a, b, c, d, e, g,, q, r, s, m, x, k, etc. In our class, the most common ones we will use are, q, r, s. We will not use the letters, t, f, or v since these letters will be reserved for something else, and we want to avoid confusion whenever ossible. hese letters will be called, roositional variables, or just variables for short. here is one more restriction laced on our use of variables. Each variable,, q, r, s, m, n, etc must stand for a roosition which is truth-functionally indeendent from the other variables in the list. ruth-functional indeendence is just another way of saying that the roositions which the letters stand for are not related truth functionally, meaning if, for examle, is true or false, its truth or falsity does not affect the truth or falsity q. Exercise: According to our defined usage, is the argument on the right a correct symbolization of the argument on the left?

4 omorrow is June 13 th, or tomorrow is June 14 th. omorrow is not June 13 th. herefore tomorrow is June 14 th. or q not herefore q Since and q are lowercase letters, each must stand for roositions which are truth-functionally indeendent of each other. But, if tomorrow is June 13 th turns out to be true, then the roositions that tomorrow is June 14 th must be false. Hence, these two roositions are not truth-functionally indeendent, and cannot be symbolized using lower case roositional variables. he above exercise leads us to another extension of our first tool. Uer-Case Variables We will use uer-case letters: K, N, M, L, X, A, etc. to reresent roositions which are not truth functionally indeendent, or their truth functional indeendence is unknown, or to stand for Logical sentences made u of lowercase variables and logical connectives. Again, the same restrictions will aly so that we will not use,, or V. his allows us to write the above argument as: K or M Not K herefore M his distinction requires some thought on the art of the student (or Logician) as to which variable is to be used to symbolize an argument. Note that a single argument can use both uer-case and lower-case variables. I will discuss this more in class. However, both tyes of variables have something very imortant in common. Which leads us to another imortant tool. ruth-unctional Interretation of Variables Both tyes of variables, both uer case and lower case are used to reresent roositions, and roositions are sentences that are either true or false. By analogy to algebra, a variable like x or y can stand for any number. Likewise in Logic our variables can be either true or false, which we denote with the letters:, (or the lowercase: t, f). hat is the reason we do not use, (or t, f) as variable letters themselves (by analogy, in algebra, the unknowns are reresented as x, y, z, etc, and not 2 25, or 13 ). A New Symbol and a Primitive Logical Oeration Since we know that the only values our variables can take are either true or false (usually written simly as t or f), we introduce the most simle logical oerator

5 which does the most simle of all logical oerations. We will call it negation, and give it a simle function an symbol. Name Symbol Negation ~ Note: Other textbooks use rather than ~ he ~ oeration flis the truth value of the variable or Logical sentence to its immediate right to its oosite value. Hence if its true, ~ is false, or if K is false, ~K is true. We can distill this information in a table: ~ ~ Negation is quite simle. he most common mistake made by students is when they make a false analogy to algebra and conclude negation works like multilication by -1. his is not the case! Negation only changes one value. Again, negation only changes one value! Let us now turn to Logical oerators which require more than one variable. Conjunction our first Logical oerator Conjunction is just the logical oeration otherwise known as, and. Lets take a concrete examle by letting and q stand for roositions. We will say: = Mary checked out a book on Logic at the Library. q = Mary checked out a book on Italian cooking at the Library. Both of these sentences are truth-functionally indeendent. In other words, the truth or falsity of does not affect the truth or falsity of q (and visa versa). Let us now ut both of these sentences together with the word, and : Mary checked out a book on Logic at the Library and Mary checked out a book on Italian cooking at the Library. Now we want a way to symbolize the conjunction of these two sentences. Just like we did for negation, we will create a symbol for conjunction, and make the symbol do something, what it does will be called a Logical oeration. irst the name and symbol:

6 Name Conjunction Symbol Hence, the conjunction of and q will be written as, q Note: Other textbooks use & or, In set-theory the symbol is used. his is an examle of a simle logical sentence with one logical oerator. echnically, the single logical variable ( all by itself, as an examle) counts as a logical sentence as well (other textbooks use the name well-formed formula, abbreviated as wffs ). Since this is an introductory course, I will say no more about Logical sentences other than they lay an imortant role in a comlete formalized system of Logic. In this class, a Logical sentence will mainly refer to more than one variable connected with Logical oerators like conjunction. Now let us define conjunction in the same way we defined negation. he definition of logical oerators turns out to be equivalent to an exlanation of what they do to two truth values laced on either side of the logical oerator. In the case of Logical oerators other than negation, there will always be four truth ossibilities, as shown below. In the case with Mary and the books she checked out (or did not check out) at the library we have: = Mary checked out a book on Logic at the Library. q = Mary checked out a book on Italian cooking at the Library. We can now consider when the entire sentence, Mary checked out a book on Logic at the Library and a book on Italian cooking... is true. A little reflection tells us that the entire sentence is true only when both conjuncts are true. In other words, when Mary checked out a Logic book and a book on Italian cooking. Only when both are true is the entire sentence true. In any other case, the entire sentence turns out to be false. his leads us to the definition of conjunction. It can be exressed like this:

7 his lists all ossible combinations of and with the oerator in-between, and for each combination a result of the oeration is given. Really that is all there is to truth functional evaluation of Logical sentences. A Logical oerator, like, is defined by listing the result of alying that Logical oerator on the combinations of and as listed above. Such a definition is called the truth functional definition of the Logical oerator in question. In an analogy to Math, logical oerators are functions on two variables which return a unique value for four combinations of and. In other words, to define a Logical Oerator, we simly fill in the following table, by deciding on a symbol and an outcome (always or ) for each of the four combinations: (logical oerator symbol) (logical oerator symbol) (logical oerator symbol) (logical oerator symbol) or, the above table becomes: Disjunction or Inclusive or Careful review of the valid forms of argument listed at the beginning of this introduction reveals two more Logical Oerators besides negation and conjunction. One is the or found in disjunctive syllogism, and the other is the if... then found in Modus Ponens and Modus ollens and Hyothetical Syllogism. Let us turn now to the or. However, before we define or we need to note that or comes in two different tyes, or has two different meanings. One meaning is equivalent to and/or and is called inclusive or. Inclusive or simly means that both roositions on either side of the or can be true. or examle, in the sentence, Place a check in the box if you are self-emloyed or received over $1000 in untaxed income during the last year. his use of or clearly indicates that both ossibilities can be true. By the way, in sentences with or like X or Y, the two roositions on

8 each side of the word or are called disjuncts. With inclusive or, both disjuncts can be true. he second meaning of or called the exclusive or is usually written as: X or Y but not both. his is a common meaning of or and many times is understood to be the case, as in a menu item at a restaurant when you are given the otion of a house salad or fries. he hrase but not both is rarely written. In Logic, both tyes of or have a truth functional definition. In our class, we will use the inclusive or most of the time. But it will be imortant to know both tyes if our goal is to analyze arguments. irst the name and symbol: Name Disjunction (inclusive or ) Symbol In set-theory the symbol is used. Notice that our symbol for or looks very much like the lower case letter v. hat is the reason that symbol is excluded from the list of ossible variables. Inclusive or can be defined truth-functionally, just like and. Its definition is: his definition certainly seems to reflect our normal usage of the inclusive or. Going back to Mary and her activities at the Library, let us change the sentence by connecting the two clauses with the word or : = Mary checked out a book on Logic at the Library. q = Mary checked out a book on Italian cooking at the Library. q Mary checked out a book on Logic at the Library or she checked out a book on Italian cooking. Notice how the entire sentence is true if just one of the disjuncts is true, and it is certainly true if both of the disjuncts are true. Perhas the easiest way to memorize the truth table definition of inclusive or is to remember that it is true in all cases but one, when both disjuncts are false.

9 Before turning to the if then Logical Oerator, I will also give the truth functional definition of the exclusive or (X or Y, but not both). he last hrase, but not both gives a clue to how we will define it. Name Xor (Exclusive or ) Symbol Many textbooks do not give this a symbol. hose that do vary. Some list, others or the most art, we will only use the inclusive or in class. Indeed, it is the inclusive or rather than the exclusive or that makes u the or in disjunctive syllogism. We now need only to define the truth table for if... then. he Conditional if... then Just like all of our logical oerators, if... then will be given a symbol and a truth table definition. Name Conditional (If... then) When considering the logical sentence q, is called the antecedent and q is called the consequent Symbol Other common symbols include:

10 he truth table definition for the Conditional is: I will justify the above truth-table definition for the conditional in class. here is, however, one imortant asect about the conditional not shared by conjunction or disjunction. With the conditional, order matters. In the language of math, the conditional does not commute. What do I mean by this? irst let us examine the roerty of commutation as exressed in addition and multilication: 3+ 4 = = 13 7 In other words, it does not matter which comes first. he same is true with conjunction and disjunction. q = q q = q However, when we get to the conditional: q q or that reason, there are two additional distinctions we make to any conditional statement. What comes after the if but before the then is called the antecedent. What follows the word, then is called the consequent. Later on in our class, we will be able to rove the non-commutative roerty of the conditional. or now, just familiarize yourself with the truth-table definitions of each, as these are the foundation of truth functional analysis. Next week we will use these tools to determine whether any argument is valid or invalid! One last symbol. We will use the symbol to stand for therefore which marks the conclusion to an argument. (Other textbooks use to stand for therefore ) We can now write the valid forms listed at the start of this introduction in urely symbolic form:

11 Modus Ponens: q q Modus ollens: q ~ q ~ Hyothetical Syllogism: q q r r Conjunction: q q Disjunctive Syllogism: q ~ q