Deductive reasoning
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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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Deductive Reasoning

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Deductive reasoning

Deductive Reasoning


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: ____  ____


Deductive reasoning

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.


Deductive reasoning

  • 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!


Deductive reasoning

  • 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.


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