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Conditional Propositions and Logical Equivalence

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Johnsonbaugh Chapter 1.1

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The purpose of studying logic is to determine if our reasoning is correct. It is not concerned with determining if a statement is true or false. There is a big difference between mathematical argument and rhetorical argument: the former is concerned primarily with valid, incontrovertible reasoning, while the latter is more concerned with persuasion. Persuasion depends heavily on the content or meaning of the statements used. Logic depends on the relationship between statements, and is otherwise unconcerned with content.

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For this reason, logic lends itself to a mathematical treatment, leading to a kind of algebra of symbolic manipulation. We will therefore concentrate on three steps: Translation of common English statements into symbolic notation. Why do we do this? Because English statements can often be misinterpreted or taken out of context. Consider for the example the following actually headlines [http://www.antion.com/humor/speakerhumor/headlines.htm] or [http://freespace.virgin.net/mark.fryer/headlines2.html] Add to this the line from a police blotter: “Officers sent to kick teenagers off roof of …”http://www.antion.com/humor/speakerhumor/headlines.htmhttp://freespace.virgin.net/mark.fryer/headlines2.html Symbolic manipulation of this notation. Translation of symbolic notation back into common English statements.

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First we define our "playing pieces": Definition: A proposition is a statement that is either true or false, but not both. We need to worry about this definition a little, and fine-tune our understanding of what this means. English has so many shades of meaning and is subject to so many different interpretations that it can be rather ambiguous at times.

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First of all, the statement needs to be explicit—it cannot contain any "variables" that could take on different values and change whether the statement is true or false, and must be objective. "October 26, 2001 is a Friday." is a proposition (it is true). "Charles Dickens wrote Moby Dick." is also a proposition (it is false). "Alfred Hitchcock was a brilliant director." is NOT a proposition, since it is subjective, and not everyone would agree on the truth or falsity of the claim. "Go home!" is NOT a proposition. It is neither true nor false. "JFK was shot by a lone gunman." is a proposition. It is either true or false, though we may not know which. "x = 22" is NOT a proposition. Its truth value depends on what x is. A little more subtle would be something like: "Today is Wednesday." "I danced with your grandfather."

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Sentences like these contain variables: today, I, you, your grandfather. The truth or falsity of these sentences depends on the value of these variables. However, these statements are generally made in context, and the context determines the truth of the sentence. If I said "Today is Wednesday" today, and said it again tomorrow, the sentence might be true one day, and false the next. If your grandmother said "I danced with your grandfather," the sentence would likely be true. If I said the same thing, it would certainly be false unless you have a square dancing grandfather, in which case it could be true. So this sentence depends on: who says it, to whom it was said, and which of your grandfathers was referred to! We will often accept these context-dependent sentences as propositions, as long as we are aware of their context-dependency.

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Fortunately, in mathematics, we can avoid most of the context dependency, and when we cannot, we make the context explicit. Notation: We will denote propositions with lower-case letters: p, q, r, s,... Connectives: In ordinary speech, we often combine small statements into larger, more complex statements. This happens routinely in mathematics, too. "If is invertible (has an inverse), then it is row-equivalent to I (ero’s can transform it into I), and its determinant is nonzero."

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We concern ourselves with a handful of ways to build such compound propositions from simple propositions. The main four are: AND, OR, NOT, IMPLIES. There are others (XOR, NAND, NOR,...) but they can all be constructed from the basic four. Definition: If p and q are propositions, we define the compound proposition “p and q”, denoted p q to be true whenever both p and q are true, and false whenever at least one of p and q is false.

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Notation: To compactly and precisely represent this type of compound statement, we use a “truth table”. A truth table lists all possible combinations of the truth values of the component propositions (one combination per line ), and has a column for each connective to be executed. This column gives the truth value of the compound proposition at the top of the column for each possible combination of truth values of the component propositions. As usual, this is easier to do than to explain: The truth table for AND: pq pqpq T T F F T F T F T F F F

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Example: Let p be “Alfred Hitchcock directed Rear Window” Let q be “Rear Window starred Cary Grant” Then p q is the proposition: “Alfred Hitchcock directed Rear Window and Rear Window starred Cary Grant.” This might be shortened to: “Alfred Hitchcock directed Rear Window which starred Cary Grant”, Or even: “Alfred Hitchcock directed Rear Window, starring Cary Grant”. If you know your movie trivia, you know this is a false statement. Although Hitchcock did direct Rear Window, the male lead was Jimmy Stewart, not Cary Grant. So even though “half” of the statement was true, the second half forced the combination to be false. (This statement falls into row 2 of our truth table.)

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Definition: If p and q are propositions, we define the compound proposition “p or q”, denoted p q to be true whenever at least one of p and q are true, and false whenever both p and q are false. The truth table for OR: Example: Let p be “Rear Window starred Jimmy Stewart” Let q be “Rear Window starred Cary Grant” Then p q is the proposition: “Rear Window starred Jimmy Stewart, or Rear Window starred Cary Grant”, which would probably be rewritten as: “Rear Window starred Jimmy Stewart or Cary Grant”. This is a true statement, because Rear Window did indeed star Jimmy Stewart, and the truth of this part of the compound statement was enough to make the entire statement true. (row 2 of the OR table) pq pqpq T T F F T F T F T T T F

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Example: Let p be “Rear Window starred Jimmy Stewart” Let q be “Rear Window starred Grace Kelly” Then p q is the proposition: “Rear Window starred Jimmy Stewart, or Rear Window starred Grace Kelly”, which would probably be rewritten as: “Rear Window starred Jimmy Stewart or Grace Kelly”. This is a true statement, because Rear Window did not only star Jimmy Stewart, but also starred Grace Kelly. This was a kind of overkill, because the truth of just one part of the compound statement would be enough to make the entire statement true. Note, however, that OR does not require one of its components to be false! This is called an inclusive or—the statement is true even when both components are true. If you want to use OR in the sense of “one or the other—but you can’t have both”, you need an exclusive or. This is a different connective (XOR). Don’t confuse the two!

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Definition: If p is a proposition, we define the compound proposition “not p”, denoted p, ~p, or p to be true whenever p is false and false whenever p and q is true. The truth table for NOT: Note: NOT is often referred to as “negation”. pp TFTF FTFT

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This will be a tricky one to translate, since we negate statements in a variety of ways—almost none of them have the word “not” in front of an expression....

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Example: Let p be the proposition: “Alfred Hitchcock won an academy award for directing”. Then is the proposition: “It is not the case that Alfred Hitchcock won an academy award for directing”, which would more commonly be phrased: “Alfred Hitchcock did not win an academy award for directing”.

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Exercises: 9-12, (odd), (odd). There will be a quiz on this material.

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