Deductive Reasoning

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# Deductive Reasoning - PowerPoint PPT Presentation

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

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
• 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: ____  ____

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 statements
• Negation: uses this symbol: ~
• ~p is read not p
• Statement: p  q
• Inverse: ~p  ~q
• Contrapositive: ~q  ~p
• 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
• 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.

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
Notes
• Law of Syllogism: says
• If p q is true and q r is true, then p r is true also
• 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!
• 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
• 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?
• 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.