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Deductive Reasoning

Deductive Reasoning. Symbolic Notation for statements. Statements can be represented by symbols Example: Statement: If the sun is out, then the weather is good p: the sun is out q: the weather is good If p, then q or p  q Example Converse: If the weather is good, then the sun is out

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Deductive Reasoning

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  1. Deductive Reasoning

  2. Symbolic Notation for statements • Statements can be represented by symbols • Example: • Statement: If the sun is out, then the weather is good • p: the sun is out • q: the weather is good • If p, then q or p  q • Example • Converse: If the weather is good, then the sun is out • If q, then p or q  p • On Your Own: • Define the hypothesis and conclusion of the following statement with letters. • Write the statement and its converse in symbolic form. • If the sky is clear tomorrow morning, then I’ll go for a run. • r: ___________________ • s: ___________________ • Statement : ____  ____, • Converse: ____  ____

  3. Symbolic Notation for statements • Biconditional Statement: use this symbol ↔ • Example • Biconditional Statement: The weather is good if and only if the sun is out • p: the sun is out • q: the weather is good • P if and only if q, or q ↔ p

  4. Symbolic Notation for statements • Negation: uses this symbol: ~ • ~p is read not p • Statement: p  q • Inverse: ~p  ~q • Contrapositive: ~q  ~p • On Your Own: • For the statement below, first define the hypothesis and conclusion in symbols then write the converse, inverse and contrapositive in symbols. • Statement: If the sky is clear tomorrow morning, then I’ll go for a run. • r: ___________________________ • s: ___________________________ • Statement : ___  ___, • Converse: ___  ___ • Inverse: ~ ___ ~ ___ • Contrapositive: ~ ___  ~ ____

  5. Notes • Deductive Reasoning: uses facts, definitions, and true statements whether assumed or proved to come to conclusions. • Law of Detachment: says that if an if-then statement is true and its hypothesis is true, then its conclusion must also be true. • If p q is true and p is true then q is true • Example: • True Statement: If you over mix your biscuit dough, then it will not rise. • From the law of detachment, I can be assured that my biscuits will be flat and hard if I over mix the dough.

  6. On your own: Use the law of detachment to come up with a conclusion • If I visit Germany, then I’ll have to learn to eat sour kraut. • I’m visiting Prague this summer. • Is the hypothesis satisfied? Is it true? What can you conclude? ________________________ • What if I visit Frankfurt?____________________ • If I have to learn to eat sour kraut, does that mean I’m going to Germany?_________________________ • Confirmation of the conclusion doesn’t ensure that the hypothesis is true. • The point: the hypothesis must be true for the conclusion to be true

  7. Notes • Law of Syllogism: says • If p q is true and q r is true, then p r is true also • It’s like a road that gets you to your destination • Example: • True Statement 1: If I get into the pool, then I have to shower first. • True Statement 2: If I have to shower first, then I will be cold before I’m even in the water. • It is horrible rushing to the pool after taking that cold shower isn’t it!

  8. On your own: Use the law of syllogism to answer this question • If I want to fly to Hamburg, then I have to stop in either London or Munich • If I stop in Munich, then I must see Neuschwanstein. I have always wanted to see the most famous of Europe’s castles. • On my way to Hamburg this spring, will I get my wish to see Neuschwanstein?__________________ • Was there a link between one if-then statement and the next?__________________ • _______________________ • How could I have rephrased the second statement to make it so a conclusion could be reached?________________ • The Point: There has to be a link between the two statements, and you have to proceed from hypothesis to conclusion in your reasoning.

  9. Lewis Carroll: Deductive Reasoning Activity • Write the statements symbolically as if-then statements, along with their contrapositives, and then string together the statements that match up to arrive at a final conclusion. • 1. My saucepans are the only things I have that are made of tin. 2. I find all your presents very useful.3. None of my saucepans are of the slightest use. • p: They are my saucepans • q: they are made of tin and mine • r: They are presents from you • s: I find them very useful • r  s; s  ~p; ~p  ~q so r  ~q • If They are presents from you, then they are not made of tin • q  p; p  ~s; ~s  ~r so q  ~r • If they are made of tin, then they are not presents from you! • How are these two statements related?

  10. Try one on your own: • Write the statements symbolically as if-then statements, along with their contrapositives, and then string together the statements that match up to arrive at a final conclusion. • No potatoes of mine, that are new, have been boiled.All my potatoes in this dish are fit to eat.No unboiled potatoes of mine are fit to eat. • No ducks waltz.No officers ever decline to waltz.All my poultry are ducks. • Every one who is sane can do Logic.No lunatics are fit to serve on a jury.None of your sons can do logic.

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