L07 Invalidity and More

MODR 1760 A & E 2024-25 L07 Propositional Logic: Invalidity & Shortened T-T Method, More Translations This lecture has the following parts: I. Invalidity Defined II. Shortened Truth-Table Method to prove Invalidity - Method w Invalidity example - Summary of shortened T-T instructions - Method w validity example III. More Translations - More about “and” - Exclusive and Inclusive “or” - More about “if, then” -----------------------I. Invalidity An argument is either valid, or it is not valid. An argument that is not valid is invalid. The definition of invalidity is: An argument is invalid means that there exists a truth possibility that makes the premises true and the conclusion false. To prove a propositional logic argument is invalid you must construct one truth possibility that shows the premises are true and the conclusion is false. You are constructing a counter-example to the argument on the basis of the logical connectives in the argument. Of course, you may begin to construct a full truth table, and then find that one row makes the premises true and the conclusion false. When you have such a row, you have the proof that the argument is invalid. But you might be wondering, isn’t there an easier way to find or demonstrate validity or invalidity other than constructing an entire truth table? Yes, there is. The central idea of the shortened truth-table method is this: If an argument is invalid, then there is at least one truth possibility that makes the premises true and the conclusion false. We attempt to construct this truth possibility by making the conclusion false, and see if we can make all the premises true in this same truth-possibility. If the argument is invalid, we can succeed in doing this. If the argument is valid, then we will not be able to successfully construct such a row. 1 II. Shortened Truth-Table Method to Prove Invalidity Here is an invalid argument that we saw in our last lecture. P -> Q ~P ~Q Here are the general instructions for constructing a shortened truth-table along with an illustration of how it works on this example. 1. Draw a truth- table pattern listing atomic sentences, the premises, and the conclusion. Notice I am separating each sentence by 2 lines here. It helps me to remember that I am dealing with three separate sentences. P Q P ->Q ~P ~Q or if you like it better you could create a table like this: P Q P1 P - >Q P2 ~P Conc ~Q 2. Make the conclusion of the argument false. P Q P - >Q ~P ~Q F 3. Fill in the atomic sentence letters forced to be T or F by step 2. P Q T P - >Q ~P ~Q F Whenever ~Q is false, Q must be true. That is the definition of ~. 4. Make one of the premises true, and fill in any atomic sentences determined T or F by doing that. Repeat this until all the premises are true, or until all the atomic sentences are filled in. P F Q T P - >Q ~P T ~Q F When we make ~P true, we make P false. Now all the atomic sentences are filled in. 2 5. If all the atomic sentences are filled in but the premises are not all true, we must check that any other premise(s) is true given these atomic sentence values. What is the value of P -> Q when P is false and Q is true? The definition of “->” says it’s true. So you fill that in. P F Q T P -> Q T ~P T ~Q F We have found a truth possibility that makes all the premises true and the conclusion false, so we have proved the argument is invalid. Summary of the Shortened Truth-Table Method, for easy reference: 1. Draw a truth- table pattern listing atomic sentences, the premises, and the conclusion. You can add 5 or so rows to the table to show your work for each step that you make, or you can keep everything in the one row. 2. Make the conclusion of the argument false. 3. Fill in the atomic sentence letters forced to be T or F by step 2. 4. Make one of the premises true, and fill in any atomic sentences determined T or F by doing that. Repeat this step until all the premises are true, or until all the atomic sentences are filled in. 5. If all the atomic sentences are filled in but the premises are not all true, we must check that any other premise(s) is true given these atomic sentence values. At this point you should have your answer. If you can do this without having to assign any sentence both true and false, then you have found a truth-possibility that makes the premises true and the conclusion false. This means the argument is invalid. If you encounter a sentence that has to be made true and false in order for this to work, you have found a contradiction. And that means it is not possible to make the premises true and the conclusion false. This means the argument is valid. [Warning: Some cases will require you to build a very complicated table and this presentation of the method does not account for this. To have a fully complete method that works to illustrate every case, you need something called the truth tree method, and that is not shown in this course. For us, to show validity, you need to show a full truth table. My examples and questions will not require you to worry about this, and nor do Govier’s.] 3 Second Example: Let’s look at this example, where we try the method on a valid argument. P -> (Q V R) P QVR 1. Draw a truth- table pattern listing atomic sentences, the premises, and the conclusion. P Q R P - > (QVR) P QVR 2. Make the conclusion of the argument false. P Q R P - > (QVR) P QVR F 3. Fill in the atomic sentence letters forced to be T or F by step 2. P Q F R F P - > (QVR) P QVR F Since there is only one way to make an “V” sentence false, this is simple. Both Q and R must be false. (If the conclusion was a “&” statement, then you would have to look at all three ways the conclusion could be false. I will not give you one like that to do, but remember the warning.) 4. Make one of the premises true, and fill in any atomic sentences determined T or F by doing that. Repeat this until all the premises are true, or until all the atomic sentences are filled in. P T Q F R F P - > (QVR) P T QVR F In this example, that just means copying P is true into the atomic column. 5. If all the atomic sentences are true, we must check that any other premises not yet looked at are true. In this example, the values of the atomic sentences we have on the left make the conditional P - >(QVR) false. But it is a premise, so it must be true. This is an impossibility. 4 In order to make the premises, true the conditional must be true. But when P is true and Q and R are false, the conditional is false. This tells us that the argument is valid. P T Q F R F P - > (QVR) F but (T) P T QVR F Problems for demonstrating invalidity or validity using the shortened truth-table method will be in the file, “Lecture 7 Questions.” Some will be created by me, and some will be taken from Govier’s text, p. 227-8, Exercise 2B. III. More Translations More about “&” a. statements vs. compound subjects “&” for prop logic makes sense with two statements “Joey and Marianna are York students” is translated as J & M which could stand for Joey is a York student and Marianna is a York student. “Vodka and orange juice is a nice drink” is not translated as V&J (Vodka is a nice drink and orange juice is a nice drink). There is a single subject expressed as a compound in this sentence. When “&” is used in a compound subject make sure it is applying to two sentences before conjoining them as a translation. b. Each of the words below can be translated with an “&” symbol: - A “although” B - A “but” B - A “however” B - “A “and yet” B” - A “nevertheless” B, A “notwithstanding” B, A “despite the fact that” B. In these cases, the English tells us that both of A, B happened. In addition they convey the idea that there is a contrast between A and B, or that A is surprising or unexpected given B. This does not change the fact that they assert that A as well as B happened. And when we are assessing the validity or invalidity of the argument, that is what is important to us. Each can be translated A&B. Now let’s consider the phrase, “A and then B.” 5 Given this key: A: Kamala took off her socks. B: Kamala jumped into bed. Kamala took off her socks and then jumped into bed. A&B Kamala jumped into bed and then took off her socks. B&A These two English sentences do not mean exactly the same thing. The English sentences tell us which one of the events happened first. But the logical translation just tells us that each of the two events happened. She took off her socks. She jumped into bed. The phrase “and then” adds more than “and” alone, but it includes the meaning of “&.” A&B is logically equivalent to B&A. All of the English phrases introduced in this section include the meaning conveyed by “&” that each of the events did occur. So for the purposes for evaluating the validity of the argument for Propositional Logic, we can translated them with “&.” Exclusive vs. Inclusive Or Inclusive Disjunction: AB “Either Abdul bakes or Bridget drives.” The whole sentence is a disjunction, its main connective is . Each component sentence, A, B is called a disjunct. A T T F F B T F T F AB T T T F In the case where A and B are both true the disjunction is true, there are three of these. means that our defined truth-table for disjunction is inclusive. But we can easily make an exclusive meaning for disjunction if we want to by using conjunction and negation. Suppose I say, “Either Abdul bakes or Bridget drives, but not both.” I mean that one or the other is true, but it is false if both of them are true. This is translated ( AVB ) & ~( A&B ): This is exclusive disjunction given by “( AVB ) & ~( A&B ): 6 A T T F F B T F T F AB T T T F A&B T F F F ~(A&B) F T T T ( AVB ) & ~( A&B ) F T T F The truth-table confirms the meaning that we want, it is true when one of A or B is true, but not when both are true. This is the exclusive meaning of “or.” More about “If, then” The following words in English have the same or a very similar meaning to “if, then.” a. A implies that B, A-> B abbreviated way of stating it “Human free will implies the ability to choose.” this means the same thing as “That human beings have free will implies that humans have the ability to choose. “ F: Human beings have free will C: Human beings have the ability to choose. This can be represented as F -> C. b. A provided that B, A -> B Aaron can run the race provided that he stays in good shape. A: Aaron can run the race B: Aaron stays in good shape. A -> B c. A only if B Samuel receives a driver’s license only if Samuel passes the in-person driving test. A: Samuel receives a driver’s license. B: Samuel passes the driver’s test. A -> B d. B unless A The soccer match will be great unless Ronaldo does not play. G: The soccer match will be great R: Ronaldo plays in the match G unless ~R 7 ~(~R) -> G or R -> G. or G V ~R. Each of these answers is equivalent to the other. Exercises for practice will be taken from Govier p. 241-3, along with some Pelham questions. These will also be placed in the file “Lecture 7 Questions.” 8