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Chapter 8 Lecture Notes. Deductive Arguments: Propositional Logic. Chapter 8.
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Chapter 8 Lecture Notes Deductive Arguments: Propositional Logic
Chapter 8 Categorical logic is the oldest logic that we have, but it is no longer regarded as the most basic part of logic (216) This distinction goes to propositional logic. This is the logic of propositions (think sentences) and their compound. There are four basic logical terms in propositional logic: not and or and if then The are sometimes called logical connectives.
Chapter 8 If we want to test the validity of an argument, then we can use letters to represent complete sentences and logical symbols to represent not, and, or, and if then. It is important to see that once a letter is used to represent a statement in an argument, that letter cannot be used to represent another statement in the argument. (216) For example, if we had the following argument: • I like ice-cream and I like pie. Therefore 2. I like ice-cream.
Chapter 8 This argument would look like this using letters to represent sentences and using symbols to represent the logical connectives. • I P Therefore 2. I ‘I’ represents “I like ice-cream” and ‘P’ represents “I like pie” Now we see the logical form of the argument. What we need to do now is define the symbols we will be using to represent: not, and, or, and if then.
Chapter 8 The logical connectives not, and, or, and if then will be represented by the following: –,,, and. We are going to define these symbols using truth-tables and thus they will have truth-table definitions. The point of a truth-table definition is to provide all the possible cases of truth or falsity for the values of the the sentence letters. Each connective will have its own definition.
Chapter 8 Not Not functions to change the truth of a statement to its opposite. The truth-table definition for not is quite simple. If P is true, the not P is false. If P is false, then not P is true. This is our truth-table definition of not.
Chapter 8 And And is a conjunction and connects two conjucts. The conjunction is true when both conjuncts are true and false ever where else. Here is our truth-table definition of and. The only time P Q is true is when both P and Q are true.
Chapter 8 Or Or is disjunction and it is true when either one or both of the disjuncts are true and false when both disjuncts are false. Here is the truth-table definition of or. Because we have defined or a being true when both disjuncts are true, this is called inclusiveor
Chapter 8 If then If then is a logical connective that has parts with names and is represented by the horseshoe: ‘’. The parts of a conditional are the antecedent and the consequent. So in a conditional like: If it is raining, then my car is wet. It is raining is the antecedent and my car is wet is the consequent. Since the truth of the whole conditional is based on the truth of the parts, it is important to know the parts.
Chapter 8 We are going to define the horseshoe as true whenever the consequent is true or whenever the antecedent is false. The only time a conditional statement is false is when the antecedent is true, and the consequent is false. So, if it is raining and my car isn’t wet, because it is in the garage, then the prior conditional is false. Here is the truth-table definition of the conditional.
Chapter 8 Formalization of arguments makes testing the argument for validity much easier. When we turn English sentences into formalized statements with sentence letters and logical connectives we have to be careful. There are stylistic variants of the conditional as well as uses of and that cannot be expressed with the truth-table definition. One should take special care to make sure that formalization is appropriate for a given proposition.
Chapter 8 Creating truth-tables to test argument for validity requires knowing how many sentence letters are in a given argument. We can use this formula: 2n Where ‘n’ represents the number of distinct statement letters. So, if there is one statement, there will be two rows on the truth-table; two statement letters will get four rows; three letters will need eight rows, and four letters will require 16 rows.
Chapter 8 In order to account for every possible combination of values for the statement letters, we need a standard way of creating truth-tables. Here is how you should do it. After determining how many rows will occur in the truth-table, begin at the left most statement letter and fill the table’s top half with Ts and the bottom half with Fs. Then move to the right one letter and alternate T and F making sure that half the rows of T and F from the column to the left are filled by half with Ts and Fs. The final column to the far right should ultimately end up alternating Ts and Fs. This will account for all the possible cases.
Chapter 8 Imagine the following argument: • D S • –D Therefore, 3. –S Figure 8.5 shows the truth-table for this argument. We can see that it is possible for the premises to all be true and the conclusion false, and thus, this inference is invalid. This happens to be the fallacy of denying the antecedent.
Chapter 8 Methods of shortening truth-tables can be useful. Since we are looking for cases where all the premises are truth and the conclusion is false, you only have to look at rows on a truth-table that have a false conclusion. Of course if there are no instances where all the premises are true and the conclusion is false, the argument in question is a deductively valid, propositional argument. Let’s briefly look at some issues with formalization and translation.
Chapter 8 Translating English into propositional logic can be a bit tricky. We need to watch for special cases and instances of stylistic variants. Not requires that we have a instance where the sentence in question has the opposite truth value, not merely as a contrary. See page 229 for more examples.
Chapter 8 There are instances of and that are not propositional, and there are other words that function as and. Jim and Mary are married to each other is not an case where we can use to join two proposition, because there is only one proposition. Words like ‘but’ and ‘although’ also have a core meaning of like and. The real issue is that one has to be careful when translating from English to propositional logic. See page 229-230 for more examples.
Chapter 8 Or can have two meanings: inclusive and exclusive. The inclusive meaning of or is just that statements constructed with or are true when one or both disjucts are true. Exclusive or on the other had is false when both disjuncts are true. This can be expressed in English as: You can have soup or salad, but not both. We shall default to the inclusive usage of or unless there is special reason to think otherwise. See page 230-1
Chapter 8 Conditional statements can get complicated. One such conditional is the counterfacutal conditional. These conditionals have false antecedents as a matter of course. For example, If Hitler had never been born, WWII would still have occurred. Is true under are current understanding of the conditional. There are problems with this interpretation for counterfactual (or subjunctive mood) conditional, and thus we won’t let the horseshoe stand for them. For more information about conditional see pages 231-4.
Chapter 8 Translation tips: Both…and… is P Q. Neither…nor… is –(P Q) Implies that… is P Q Provided that… is Q P Only if… P Q Unless is best understood as or -- . See pages 236-41 for detailed examples.
Chapter 8 Necessary, Sufficient, and Necessary and Sufficient conditions can be represented by conditionals. (1) Being mercury is sufficient for being a metal. (2) Passing the final exam is a necessary condition for passing the class. Both of these statements can be expressed as conditionals. • M M1(M = being mercury and M1 = being a metal) • C E (C = Passing the class and E = passing the final exam) For full definitions of necessary and sufficient conditions see pages 240-1 and the glossary of chapter 8.
Chapter 8 Sometimes argument can get too big to draw a truth-table for them. When this happens, simple proofs and logical equivalencies are useful. There are about 20 different valid and invalid logical moves that everyone should try to memorize. Some are: Modus ponens, modus tollens, transpostion, De Morgan’s Rules, Double negation and many others. (A more complete list can be found on page 224) When you cannot create a direct proof, you can use the indirect proof method of a conditional proof (227).
Chapter 8 Terms to review: Affirming the consequent Biconditional Conditional proof conditional statement Conjunction consequent Contradictory statements contrary statements Counterfactual denial Denying the antecedent dilemma argument Disjunction exclusive disjunction Horseshoe inclusive disjunction
Chapter 8 Indirect proof modus ponens Modus tollensnecessary condition Propositional logic reductio ad absurdum Sufficient condition truth table
