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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

• 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: ____  ____

• 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

• 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: ~ ___  ~ ____

• 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

• 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!

• 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.

• 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.