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Phi 1103 - Critical Thinking

by: Emilia Notetaker

Phi 1103 - Critical Thinking PHI 1103

Emilia Notetaker

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About this Document

How to create truth tables and translate logical statements to english
Critical Thinking
Bryan Mitchel
Study Guide
philosophy, logic, truth, truth tables, tables
50 ?




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This 8 page Study Guide was uploaded by Emilia Notetaker on Friday April 29, 2016. The Study Guide belongs to PHI 1103 at University of South Florida taught by Bryan Mitchel in Winter 2016. Since its upload, it has received 13 views. For similar materials see Critical Thinking in PHIL-Philosophy at University of South Florida.


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Date Created: 04/29/16
Truth Tables Truth tables are tools used in logic to construct valid arguments by testing if the premises are true of false. Premises The premises (A, B) are variables that represent propositions and they can be either true or false Therefore you can say: A: I have a test In this statement it can either be true or it can be false T: I have a test F: I don’t have a test Truth tables should include any combination of true and false so if you are just considering A it would look like this: A T F Next you can add another variable: B B: I will Study Again this statement can be true or false T: I will study F: I will not study B T F Combining Premises You can combine both premises in a table. To do it, you need to include every possible combination of true and false. When you only have two premises it is simple. They can either both be true, both false, or one true and one false and vice versa. The point of truth tables is to help you keep tract of true and false statements A B Premises T T Both are true T F A is true, B is false F T A is false, B is true F F Both are false This is should always be the beginning of your table and it is a convention of organization Statements (Combining with logical operators) You can combine these two premises with different operators A → B This is a conditional statement, which in this case means that IF A happens THEN B will happen too It reads: If A then B In the previous example, you would translate it to IF I have a test THEN I will study The truth table would look like this A B A → B Explanation T T T This is TRUE because IF A is true (which it is) THEN B is true (which it also is) T F F This is FALSE because IF A is true (Which it is) THEN B is true WHICH IT IS NOT F T Ⓣ This is problematic (explanation below) F F Ⓣ This is problematic (explanation below) Ⓣ: this symbol means that there is a problem because the statement is not actually being tested Think about it this way: If you do not have a test (A is False) then how can you know if you will study? It will become clearer later, but generally if the antecedent is false then the statement is problematic. For a conditional statement to be true, both the antecedent and the consequent need to be true. A & B This is a conjunction and it is read A and B I have a test AND I will study In this case you want BOTH premises to be true Think about it this way: if you want to pass a class, you need to do the homework AND take the exams. You cannot just do the homework, or just take the exams and expect to pass. You HAVE to do BOTH A B A & B Explanation T T T This is TRUE because BOTH A AND B are true T F F This is FALSE because even though A is true B is false F T F This is false because A is false and B is true F F F This false because they are both false A v B This is a disjunction and it is read A or B Note that this operator means you want “either, or” in other words, you would be fine if you had only A, as if you had only B, or if you had both So if you are eating pizza, you can have pepperoni, or you can have Hawaiian or you can have both! You read it: I have a test OR I will study A B A v B Explanation T T T This is TRUE because BOTH A AND B are true T F T This is TRUE because at least one of them is true F T T This is TRUE because at least one of them is true F F F This false because they are both false NEGATION The negation ~ means NOT or IT IS NOT THE CASE THAT So if you see ~ A it means NOT A If you see ~ (A v B) means: IT IS NOT THE CASE THAT A OR B (or whatever is in the parenthesis) NOTE that the negation IS NOT distributive so do not think of it as math. The sign negates the whole parenthesis so: ~ (A v B) IS NOT THE SAME AS ~ A v ~ B The easiest way to solve these in the truth tables is solving the parenthesis first and then just fliping the signs to the OPPOSITE. So what was true becomes false, and what was false becomes true A B A v B ~ (A v B) T T T F T F T F F T T F F F F T Original When more elements are involved it is easier to break it down to simpler parts and then combine A simple example would be A →~B (If A then not B) The parts of this statement are A (which we know), NOT B, and the conditional 1. Start with the original table A B T T T F F T F F 2. Add NOT B (~ B) which is the opposite of B A B ~ B T T F T F T F T F F F T 3. Determine the conditional with A and ~ B a. Remember it should be TRUE for A and True for ~ B (which is the same as false for B) b. IF A THEN NOT B A B ~ B A → ~B T T F F A is TRUE and B is TRUE, NOT WAT WE WANT SO IT IS FALSE T F T T A is TRUE and B is FALSE, so the conditional is TRUE F T F Ⓣ We do not have A so you cannot really know what happens to B F F T Ⓣ We do not have A so you cannot really know what happens to B We can take it a step further and make it more complicated. For instance saying ~ (A → ~ B) This would read as: (~)It is not the case that IF A (→) THEN NOT B Using the premises we were using earlier ~ ( A → ~ B) It is not the casethat if I have a test then I will not Study And the table would be the same, just the opposite of the last conclusion A B ~ B A → ~B ~ A → ~B) T T F F T T F T T F F T F Ⓣ Ⓣ F F T Ⓣ Ⓣ Whenever it gets longer, you just have to remember to simplify it and do it part by part Like in math, there is an order of operations, and you should work from the inside out. ~A → (~A v B) 1. The smallest part of this are the variables: A, B and ~A so solve those first a. ~A is just the opposite of A A B ~ A T T F T F F F T T F F T 2. The next smallest is the parenthesis, so add that to the table a. In this case you want ei~A or B or BOTH b. Do this using the second and third columns (B and ~A) ignore the first one (A) A B ~ A (~A v B) T T F T AT LEAST ONE is true (B) so the statement is true T F F F NEITHER is true so the statement is false F T T T BOTH ARE TRUE so the statement is true F F T T AT LEAST ONE is tru~A() so the statement is true 3. Finally, as you can see, the major operator is the conditional so now you have to solve that one with all the other parts that you have a. Remember the conditional in this case means:~A) THEN (~A v B) b. SO (~A) has to be true and (~A v B) has to be true (the whole parenthesis which you solved in the previous step) c. Now you can ignore the first two columns and only use columns three and four A B ~ A (~A v B) ~A → (~A v B) T T F T Ⓣ You don’t have ~A so you can’t know T F F F Ⓣ Problematic because you don’t have ~A F T T T T You have ~A and you have (~A v B) F F T T T You have ~A and you have (~A v B) If you think about it this makes sense You are saying “If not A then not A or B” which may seem redundant but it is true So if you are eating pizza A: pepperoni B: Hawaiian You can pick pepperoni or Hawaiian, but “If there is no pepperoni then you can either have no pepperoni, or Hawaiian” It might sound silly in regular English, but it makes sense. Let’s try a longer example ~ [(~A & B) v (A & ~B)] Translation: It is not the case [that either (not A and B) or (A and not B)] 1. Simplify it into smaller parts, you can start with the premises A B ~ A ~ B T T F F T F F T F T T F F F T T 2. Now solve for each parenthesis a. (~A & B) A B ~ A ~ B (~A & B) T T F F F T F F T F F T T F T F F T T F b. (A & ~B) A B ~ A ~ B (~A & B) (A & ~B) T T F F F F T F F T F T F T T F F F F F T T T F 3. Now combine both parenthesis with the OR (v) a. This means you should have either the first parenthesis, the second parenthesis, or both A B ~ A ~ B (~A & B) (A & ~B) (~A & B) v (A & ~B) T T F F F F F T F F T F T T F T T F T F T F F T T F F F 4. Finally apply the negation to the whole thing, which is basically just the opposite of what you just did A B ~ A ~ B (~A & B) (A & ~B) (~A & B) v (A & ~B) ~ [(~A & B) v (A & ~B)] T T F F F F F T T F F T F T T F F T T F T F T F F F T T F F F T Final Example [~A → (~A v B)] → ~ [(~A & B) v (A & ~B)] This example combines the previous two examples with a conditional The conditional becomes the main operator You can help differentiate with the grouping symbols I will use the tables be already made 1. Part One A B ~ A (~A v B) ~A → (~A v B) T T F T Ⓣ T F F F Ⓣ F T T T T F F T T T 2. Part Two A B ~ A ~ B (~A & B) (A & ~B) (~A & B) v (A & ~B) ~ [(~A & B) v (A & ~B)] T T F F F F F T T F F T F T T F F T T F T F T F F F T T F F F T 3. Combine ~A → (~A v B) ~ [(~A & B) v (A & ~B)] [~A → (~A v B)] → ~ [(~A & B) v (A & ~B)] Ⓣ T Ⓣ Ⓣ F Ⓣ T F F T T T


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