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Truth-Functional Logic A. The CHAPTER 4: TRUTH-FUNCTIONAL LOGIC A. THE PROPOSITIONAL CALCULUS Consider the following two-premise deductive arguments: Argument Argument Form (a) Today is Wednesday or today is Thursday. A or B Today is not Thursday not-B Therefore, today is Wednesday. A (b) If the gas tank is empty, then the car will not start. If C then not-D The gas tank is empty. C Therefore, the car will not start. not-D Now, these two examples embody two common argument forms. It will be the purpose of this chapter to develop a system of logic by means of which we can evaluate such arguments. This system of logic is called Truth-Functional or modern Logic. In truth-functional logic we begin with atomic or simple propositions as our building blocks, from which we construct more complex propositions and arguments. The propositional calculus provides us with techniques for calculating the truth or falsity of a complex proposition based on the truth or falsity of the simple propositions from which it is constructed. These techniques also provide us with a means of calculating whether an argument is valid or invalid. 185 I. Atomic (simple) Propositions In the propositional calculus atomic or simple propositions are used to build compound propositions of increasingly high levels of complexity. The truth value of a compound can then be calculated from the truth values of its simple propositions. The predicate calculus provides us with a way of reconstructing the categorical propositions of traditional Aristotelian logic so that they can also be represented in modern truth-functional terms. Individuals and Properties: I. Constants In truth-functional logic, we specify a universe of discourse by specifying the individuals that exist in that universe, and the properties those individuals may have. The names of individuals in our universe of discourse are abbreviated using lower case alphabets a, b, c, .. .w. The names for the properties possessed by these individuals are abbreviated using upper case alphabets A,B, C, … Z. For example, let j designate the individual John, and W designate the property of being wet. Then Wx = x is wet and Wj = John is wet. And if we let d designate Debbie and A designate the property of being asleep, then Ax = x is asleep and Ad = Debbie is asleep. If the names “John” and “Debbie” refer to actual individuals, then the propositions symbolized as Wj, Wd, Ad, and Aj are each actually either true or false. The individuals designated by j and d remain constant, no matter what property either is alleged to have. Likewise, the properties designated by W and A remain constant, no matter what individual is alleged to possess those 186 properties. Once a name has been assigned to a particular individual or property within a given discourse, it does not change. II. Variables The propositional form Wx attributes the property of being wet to an individual, x, where x could refer to any of the individuals in our universe of discourse. x is an individual variable, and Wx is a propositional form that becomes an actual proposition when x is replaced by the (abbreviated) name of an actual individual. Individual variables are place-holders for the names of individuals. The lower case alphabets x, y, and z are typically used as individual variables. A propositional form is neither true nor false, but it becomes true or false when its individual variables are replaced by individual constants. In a similar fashion, every proposition that attributes a property to a particular individual, j, has the propositional form Φj (read “phi of j” or “j has the property Φ; Φ is the Greek alphabet pronounced “phi”). Greek alphabets are used as place holders for properties that individuals may have, and are called predicate variables. Thus, Φj is a propositional form that becomes an actual proposition when Φ is replaced by the name of an actual property (e.g. W or A), which is then either truly or falsely attributed to John. In the examples used so far, some individual, x, is alleged to have some property, Φ. When a property can be attributed to a single individual, it is called a monadic or one-place predicate. Thus, Wd can be true independently of whether any other individual in our universe of discourse is wet. The following examples of one-place predicates are designated by sub- scripted alphabets. In this way, an infinite number of predicates can be generated and combined with the names of individuals to produce statements that are either true or false. 187 A1x = x is a Republican B1x = x is a human A2x = x is a Democrat B2x = x is a college professor A3x = x is an American. B3x = x is a woman A4x = x is the USA president B4x = x is a man A5x = x is a Christian B5x = x is a musician A6x = x is hardworking. B6x = x is a Muslim A7x = x is asleep B7x = x is a Buddhist A8x = x is a Hindu B8x = x is wealthy A9x = x is an engineer B9x = x is a soccer player A10x = x is Jewish B10x = x is a basketball player 4.I.1. Exercise: Using lower case letters to designate the names of actual individuals, construct one true and one false proposition for each of the above predicates. III. Relational Properties But not all properties can meaningfully be attributed to just a single individual. Many properties point to a relationship between two or more individuals rather than to an attribute of a single individual. For example, if M designates the property of being married, then Md (Debbie is married) is either true or false. But being married does not refer to the property of a single individual, but designates a relationship between (at least two) individuals. Thus, construing the property of being married as if it were like the property of being wet is misleading. 188 While Wd can be true independently of whether any other individual is wet, Md cannot be true independently of whether any other individual is married. A statement of the form Mx is really short for Mxy, which reads x is married to y. Thus, if Md is true, then there must be some other individual, e, such that Mde. Being married is not a one-place predicate, but (in a monogamous universe) a two place predicate. A two place predicate designates a relationship between two individuals, rather than a property of one individual. Following are some examples of statement forms with two-place predicates: Nxy = x is next to y Wxy = x weighs more than y Txy = x is playing tennis with y Exy = x is an employee of y Mxy = x is the mother of y Fxy = x is the father of y Note that while it is true that (if Nxy then Nyx), it is false that (if Wxy then Wyx). Some two- place predicates are commutative, but many are not. Following are some examples of two-place predicates that are often expressed as if they were one-place predicates: One Place What is meant Two Place x is in love Lx x is in love with y. Lxy x is angry Ax x is angry at y Axy x is a parent. Px x is a parent of y. Pxy x is strong. Sx x is stronger than y. Sxy x is a veteran Vx x is a veteran of war y Vxy x is reading Rxy x is reading y Rxy x is eating Ex x is eating y Exy Some properties refer to relationships between three individuals. Thus, ‘x is between y and z’ = Bxyz. Likewise, ‘x is a child’ is shorthand for ‘x is a child of y and z’ or Cxyz. 189 ‘Between’ and ‘is a child’ are three-place predicates. Some predicates actually refer to a relationship between four individuals, yet appear to be about one individual. Thus, ‘x is playing tennis doubles’ (or TDx) may seem to be saying something only about x. But it is shorthand for ‘x is playing tennis doubles with y, z and w’ ( or TDxyzw). We symbolize 1-place, 2-place, 3- place, and n-place predicates as Ax, Bxy, Cxyz, Dwxyz, … These statement forms become actual statements that are true or false when each variable is replaced by a constant. 4.I.2. Exercises: Cxyz = “x is between y and z.” Let a, b, and c be distinct points on a straight line. Answer the following: 1.) C is a ____-place predicate. 2.) If Cabc is true, then Cbca must be ____(True/False) 3.) If Cabc is true, then Cacb must be _____(True/False) 4.I.3. Exercises: Mxy = “x is married to y.” Pxyz = “x and y are the parents of z.” Bonnie = b, Joe = j, and Ted = t. Answer the following: 1.) M is a ___-place predicate. 2.) P is a ____-place predicate. Symbolize each of the following statements: 3.) Bonnie is married to Joe. _______ 4.) Ted and Joe are the parents of Bonnie. _______ 5.) Ted is married to Joe. 6.) Ted is married to Bonnie. _______ 7.) Joe and Bonnie are the parents of Ted. _______ 190 II. Compound Propositions A compound proposition is one that contains other propositions as component parts. For example, the proposition, “Today is Wednesday or today is Thursday” contains two component propositions: “Today is Wednesday” and "Today is Thursday.” Similarly, the proposition, “If the gas tank is empty, then the car will not start” is a compound proposition that contains the two component propositions: “The gas tank is empty,” and “The car will not start.” Propositions that do not contain other propositions are called simple propositions.
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