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Deductive Reasoning

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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

- 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: ____ ____

- Example:

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

- Biconditional Statement: The weather is good if and only if the sun is out

- 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: ~ ___ ~ ____

- 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.

- Law of Detachment: says that if an if-then statement is true and its hypothesis is true, then its conclusion must also be true.

- 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

- 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!

- 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?

- 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.

- 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.