The System of Mathematics Syllogism, Validity, and Soundness Geometry – Mr. Ferguson

Wait….Why in the world did we just learn all of those definitions?!?!?

More importantly, how does that get us any closer to understanding how Geometry works?

If I win the lottery, my life is ruined.

For your consideration… If I win the lottery, then I become rich.  If I become rich, then I spend a lot of money.  If I spend a lot of money, then I don’t pay attention to finances.  If I don’t pay attention to finances, then spend too much money. If I spend too much money, then I run out of money.  If I run out of money, then I lose my house.  If I lose my house, then my life is ruined.   Therefore, if I win the lottery, then my life is ruined.    

Syllogism = chain of logic The first conditionals in the syllogism are called the premises. The final “therefore” statement is called the conclusion of the syllogism. Together, they are called an argument. If sirens shriek, then dogs howl.  If dogs howl, then cats freak.  If cats freak, then mice frolic.  Therefore, if sirens shriek, then mice frolic.

Why are syllogisms important? A proof is just an extended syllogism! In a proof, we are provided with a set of statements that is accepted or has been established as true, called the given. We use the given information, along with the rules of logic and geometric reasoning to move from one true statement to the next, until we reach the conclusion, called the prove. We show that the prove must follow, logically, from our line of reasoning.

Plan of Attack for Syllogisms: Identify the hypothesis of the syllogism – this is the hypothesis of the first conditional of the premises.  Start the syllogism with the premise that uses the first “if” statement.  Start the next part of the chain by linking the conclusion of the first premise with the hypothesis of the next premise.  Rinse and repeat until you reach the conclusion of the conclusion of the syllogism statement.  What if the conditionals of the syllogism are out of order?  Then put them in order!!  

Validity vs. Soundness An argument is valid iff when the premises are all true, the conclusion is true.  An argument is sound iff the argument is valid AND all the premises are true.  While the differences are subtle, they are important. In Geometry, we want all our proofs to be sound.

What’s the difference? Valid arguments have an existing logical chain between premises but aren’t necessarily true! Valid arguments can produce true conclusions, even when their premises are not true 1. All multiples of 5 are even. 2. 8 is a multiple of 5. Therefore, 8 is even. Sound arguments are both valid (have a logical chain between statements) and have true premises; therefore, they will always produce true conclusions.

(Is this argument valid? Is this argument sound ?) Valid, Sound, or Neither? If I fail a test, then I fail the class.  If I fail the class, then I drop out of college.  If I drop out of college, then I invent Facebook.  If I invent Facebook, then I become a billionaire.    Therefore, if I fail a test, then I become a billionaire.   (Is this argument valid? Is this argument sound ?)  

(Is this argument valid? Is this argument sound ?) Valid, Sound, or Neither? If Jesuit wins district, then they go to regionals. If they win regionals, then they go to state. If they win state, then they are state champions. Therefore, if Jesuit wins district, then they are state champions. (Is this argument valid? Is this argument sound ?)

(Is this argument valid? Is this argument sound ?) Valid, Sound, or Neither? If 3(x+1) – 4x + 2 = 8, then 3x + 3 – 4x + 2 = 8 If 3x + 3 – 4x + 2 = 8, then –x + 5 = 8 If –x + 5 = 8, then –x = 3 If –x = 3, then x = -3 Therefore, if 3(x+1) – 4x + 2 = 8, then x = -3 (Is this argument valid? Is this argument sound ?)

The Law of Detachment 2. P Therefore, Q. The Law of Detachment provides a basic structure for concluding that something is true from a statement: 1. If P, then Q 2. P Therefore, Q.

The Law of Detachment at work… Example 1. If I am a freshman, then I wear khakis. 2. I am a freshman. Therefore, I wear khakis.

How does this relate to Geometry? If the measure of an angle is greater than 0 degrees and less than 90 degrees, then the angle is acute.  The measure of angle ABC is 40 degrees.  Therefore,  angle ABC is acute.

Another one! If the measures of two angles sum to 180 degrees, then the angles are supplementary.  The measure of angle DEF is 80 degrees.  The measure of angle GHI is 100 degrees.  Therefore, angles DEF and GHI are supplementary.

What does this look like in practice? (Our first proof!) All radii of a given circle are congruent.