What are TruthTables and What Are They For?


 Eustace Norton
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1 PY114: Work Obscenely Hard Week 9 (Meeting 7) 30 November, 2010 What are TruthTables and What Are They For? 0. Business Matters: The last marked homework of term will be due on Monday, 6 December, at 4pm. Please see separate sheet with the problem set. Please read Tomassi, 2.1 & Office Hours this week changed from Thursday 12 to Friday 34. Regular OH Monday Review  Upper case letters (P, Q and R) are sentential constants, or sentence letters.  When translating a sentence from English to Propositional Logic, we require a key.  There are five logical connectives in Propositional Logic: Negation (~), Conjunction (&), Disjunction (v), Conditional ( ), and Biconditional ( ).  Negation is a unary (oneplace) connective, the others are all binary (twoplace).  Logical connectives connect either atomic or compound formulas.  Brackets (parentheses) are used to specify the scope of a connective. 2. Truth and Falsity  Arguments can be valid or invalid, sound or unsound.  Sentences, on the other hand, are either true or false. As logicians say, there are two truthvalues: true and false.  This is known as the Principle of Bivalence, which states: every sentence is either true or false but not both and not neither. 3. Scope Every connective has a specific scope. The scope of the tilda (~) is the formula that is being negated. In the simplest case this will be a simple atomic proposition, but in other cases it will be some more complex wff (wellformed formula). To determine the scope of a tilda, then, we simply ask: exactly what is being negated here? Consider these examples: ~P scope of negation (what is being negated): P ~ (P & Q) scope of negatation: (P & Q) ~(P & Q) R scope of negation: (P & Q) ~((P & Q) R) scope of negation: ((P & Q) R) For a two place connective, the scope of the connective are the TWO wffs that are being connected. In the following examples I have used underlining to indicate the scope of the ampersands. P & Q P & ~Q (P v Q) & (~P Q) 4. Exercise: Using underlining, show the scope of the indicated connective. Indicate the scope of the tilda ~(P v ~P) Indicate the scope of the arrow P (P & Q) Indicate the scope of the second arrow ~(((P Q) (Q R)) & ~(P R))
2 5. SKILL: Find the Main Connective The main connective in a formula of propositional calculus is the connective whose scope encompasses the whole formula. The main connective tells us what kind of compound formula we are dealing with: whether a negation, a conjunction, a disjunction, a conditional, or a biconditional. The simplest compound formulae have only a single connective, which accordingly serves as the main connective. Hence the ampersand is the main connective in (P&Q); the whole formula is accordingly a conjunction. A crucial skill in interpreting the propositional calculus is the skill of finding the main connective in moderately complex formulae. This is a bit like finding the main verb in an English sentence that has various subordinate and relative clauses. Read the following formulae (aloud if you can) and find the main connective. Specify what kind of formula is being expressed: a) P (P & Q) b) P & ~P c) ~(P v ~P) d) P (Q P) e) (P & Q) (Q & P) f) ((P Q) (Q R)) (P R) g) ~(((P Q) (Q R)) & ~(P R)) h) (P Q) v (Q P) 6. Truth Functions and Truth Tables A truth table is a logical calculating device. In order to understand its distinctive utility, we need to start by recalling some crucial background. 1.. In the formal symbolic language of the propositional calculus, a sentence is called a formula (or a wellformed formula a WFF). 2. As we have seen, there are two classes of formulae in the propositional calculus: atomic and compound. An atomic formula is represented by a single sentential constant (e.g., P ) and represents a simple declarative sentence such as Socrates is wise. A compound formula is formed by combining atomic formulae using the logical connectives (~, &, v,, ). Hence the sentence If Socrates is not wise then Plato was a fool can be expressed as a compound formula of the propositional calculus: (~P Q), where P represents the proposition that Socrates is wise and Q represents the proposition that Plato is a fool. 3. Now here is a key point: the truth or falsity of a compound formula is a function of the truth value of the atomic formulae that figure in it. Don t be put off by the abstract talk of functions here. The point is really as simple as this: Suppose I want to know whether the following claim is true or false: Today is Monday and the weather is fine. In order to figure out whether this is true I need to know whether each of its constituent claims is true. That is, Is today Monday? And is the weather fine? Once I know the answers to these questions, and the meaning of the logical term and I can determine the truth value of the compound sentence. That is: the truth value of the compound formula is a function of the truth value is its atomic sentences. 2
3 4. Now truth tables are nothing more than a device for spelling this out exactly and in full detail. A truth table is divided into two sides, left and right. On the left hand side is a specification of all the possible combinations of truth values for the atomic propositions that figure in a compound formulae. The right side specifies the truth value of the compound formula as a function of the truth values of the atomic formulae. Hence for instance the truth table for conjunction shows under what circumstances the compound formula is true and under what circumstances false. P Q P&Q Starting from the left side and reading one row at a time, we can see the following: If both atomic propositions are true then the conjunction is true. (That is the first row of under the horizontal line.) But for any other combination of truth values, the conjunction is false. (That is represented in the last three rows of the table.) USING TRUTH TABLES OK, so what is the point of this? What is a truth table for? As we shall see, truth tables turn out to be quite a powerful (if also somewhat clumsy) tool for a number of interrelated tasks in logic. Let s consider them one at a time. A. Using Truth Tables to Define the Logical Connectives. Logical languages depend on perfect clarity and the absence of ambiguity. While the logical connectives all have natural language correlates, they cannot be defined by appeal to natural language without importing the ambiguities of natural language into the symbolic systems. We have already seen one example of this with the ambiguity of the English word, or, which can be used either inclusively or exclusively. Similar ambiguity infects the English word and. If someone says I got the money at the bank and I went to buy the car, that would typically mean that they first got the money at the bank and then went to buy the car. In short, the word and sometimes conveys temporal information. But it need not. If I say that I have a bike and a scooter, I am not saying anything about which I got first. In order to provide proper definitions of the logical connectives, therefore, we need a way of defining them more exactly than is possible by simply providing natural language correlates. Truth tables provide the tool for this purpose. In logic, the logical connectives are defined as truthfunctions. (P&Q) is defined as the formula which is true if and only if both of its constituent propositions are true. (PvQ) is defined as the formula which is false if and only if both of its constituent propositions are false. The truth tables for each connective spell out these definitions exactly, and without circularity. That is, they specify the truth value of the compound formula given any possible combination of truth value of its constituents. After all, that is exactly what truth tables do. a) Negation  The truth functional nature of each of the logical connectives is represented by a truthtable.  The truthtable for negation is written as follows: P ~P T F F T  Negation is a truthfunction that reverses truth values. b) Conjunction  The truthtable for conjunction is written as follows: P Q P&Q  A conjunction is true only when both conjuncts are true. The sense of temporal direction that exists 3
4 in the English and is not present in &. c) Disjunction  The truthtable for disjunction is as follows: P Q PvQ T F T F T T  A disjunction is true when either or both of the disjuncts are true. It is false only when both are false.  Disjunction in Propositional Logic is inclusive (it is true when both disjuncts are true), rather than the exclusive or that we often find in natural language. d) Conditional  The truthtable for conditional is written as follows: P Q P Q F T T F F T  There only way in which a conditional comes out false is if its antecedent is true and its consequent is false.  The conditional is, intuitively, the least close to its natural language equivalent if then. e) Biconditional  The truthtable for biconditional is written as follows: P Q P Q F F T  A biconditional is true when both sides have the same truth value. B. Using Truth Tables to Interpret Complex Compound Formulae. Once we have truthfunctional definitions of the five connectives, we can put truth tables to work for other purposes. As an example, consider this fairly simple compound formula: P v ~(P v R) For the logician, an interpretation of this compound formulae must tell us its truth value (that is, whether it is true or false) for every combination of the truth values of its constituent atomic propositions. But that is exactly the job for which truth tables are designed. P Q P v ~ (P v Q) T F T T F T T F T F F T T T F mc What this truth table shows is that this compound formula is false only in the case where P is false and Q is true (that is the second row from the bottom of the table). It is true in every other case. 4
5 EXERCISES Construct truth tables for the following formula P & (Q P) (P & Q) R ((P v Q) & ~P) Q C. Using Truth Tables to Sort Formulae into Tautologies, Contingencies, and Inconsistencies (Contradictions): A fourth use of Truth Tables will be particularly important for establishing a further set of tools of proof in the propositional calculus. We can use truth tables to sort formulae into three different groups. E. Exercise: Tautologies are formulae that are always true, no matter what the truth value of their constituent atomic propositions. If today is Thursday then today is Thursday is a simple tautology. But other tautologies are more interesting and will provide us with a set of logical transformation rules. An example is ((P v Q) & ~Q P). A compound formula is a tautology if and only if the compound formula comes out as true in every row of its truth table. Contradictions (or Inconsistencies) are compound formulae that are never true, no matter what the assignment of truth values to the atomic propositions that comprise them. An example of a primitive contradiction is (P & ~P). A compound formula is a contradiction if and only if the compound formula comes out as false in every row of its truth table. Contingent formulae are all the rest. In order to know whether a contingent proposition is true or false you need to know more than its logical form; you need to know the actual truth value of its constituent atomic propositions. An example of a contingent proposition is Tomorrow is Friday and I am going down to the pub. A compound formula is contingent if and only if the compound formula comes out as false in some rows of its truth table and true in others. Consider the following propositions. Without constructing a truth table, try to determine whether each one is a tautology, a contradiction or a contingent proposition. After placing your bets we can split up the list and use truth tables for each one. a) (P Q) & ~(P Q) b) (P v Q) & Q c) (P Q) & (~Q ~P) d) ~P (P Q) 5
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