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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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symbolic notation for statements
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: ____  ____
slide3

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
symbolic notation for statements1
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: ~ ___  ~ ____
notes
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.
slide6
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
notes1
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!
slide8
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.
lewis carroll deductive reasoning activity
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?
try one on your own
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.