TRUTH TABLES LOGIC (CONTINUED) Philosophical Methods

Transcription

1 TRUTH TABLES LOGIC (CONTINUED) Philosophical Methods

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3 Here s some Vocabulary we will be talking about in this PowerPoint. Atomic Sentences: Statements which express one proposition Connectives: These are used to make more complex statements. Connectives connect statements together, i.e..: and Conjunction: A conjunction connects two sentences by saying that they are both true. Disjunction: (inclusive) A disjunction connects two sentences that at least one of the statements are true, if not both. Disjunction: (exclusive) A disjunction where one of the two statements are not both. Negation: A negation in English denotes in by using the word not or more formally it s not the case that. It is a strange connective that operates in only one atomic sentence. Material Conditional: This is a stylistic variant that contains and antecedent and a consequent, i.e.: If then. Material Biconditional: This is a variant that contains same truth value for both conditions, i.e.: if and only if.

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5 Atomic Sentences Atomic Sentences Today is Thursday. The atomic number of hydrogen is 1. It is raining outside right now. Not Atomic Sentences The atomic number for hydrogen is 1 and the atomic number for helium is 2. Are you hungry? Tomorrow is not a weekday. We can symbolize atomic sentences by using a capital letter. Traditionally, logicians like to start with the letter P and go on alphabetically from there. Some use a symbol representing the sentence, such as: H for The atomic number of hydrogen is 1.

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7 : Conjunction: P: The atomic number of hydrogen is one. and Q: The atomic number of helium is two. Conjunction Symbol: Example: P Q Stylistic variants: and, but, although, in addition to The truth table for conjunction is the following: P Q P Q T T T T F F F T F F F F

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9 Disjunction: (Inclusive) P: I will roll a 7 on this roll. or Q: I will roll an 11 on this roll. Disjunction Symbol: Stylistic variants: or, either or, unless Example : P Q The truth table for disjunction is the following: P Q P Q T T T T F F F T F F F F

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11 Negation: P: The sky is blue. P: It is not the case the sky is blue Negation Symbol: Stylistic variants: it is not the case that, not, un-, non- Example: P The truth table for negation is the following: P P T F F T Going back to disjunction, we can now write an exclusive disjunction as (P Q) (P Q). Translating back to English, that says P or Q, but not both P and Q.

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13 Material Conditional: P: x > 4 Q: x > 2 Material Conditional Symbol: Stylistic variants: if then, given that, only if Example: P Q The truth table for material conditional is the following: P Q P Q T T T T F F F T T F F T

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15 Material Conditional - Continued Reflection If x > 4 then x > 2 -orsymbolically, (x > 4) (x > 2) Is this true? (Yes) Is it always true or just sometimes true? (Hmm??? Let s see ) P Q P Q T T T T F F F T T F F T 1. X = 5 Then both the antecedent and the consequent are true. 2. X = 3 Then the antecedent is false but the consequent is true. 3. X = 1 Then both the antecedent and the consequent are true. Notice that we can t find a value for x that makes the antecedent true but the consequent false. That s because this material conditional is always true. In other words, we can t create a situation like row two of the truth table for a material conditional. But we said that this material conditional is always true. And that includes the cases when x = 3 and x = 1 as well as when x =5. In other words, rows three and four of the truth table should be true!

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17 Material Biconditional Symbol: Material Biconditional P: The lightbulb will go on. iif Q: The light switch is turned on Stylistic variants: if and only if, just in the case that Example: P Q The truth table for the material biconditional is the following: P Q P Q T T T Intuitively, P if and only if Q means that P and Q should always possess the same truth value, which is reflected in the truth table. T F F F T F F F T

18 YAY!!!!!! We made it through! Time for a little Laugh Therapy!