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  • Use deductive reasoning to draw a valid conclusion based on a set of given facts

Assignment

  • Three vocabulary definitions
  • p49 9, 14–22, 25, 27–28 (PDF link)

With inductive reasoning, we used patterns to come up with a conclusion (conjecture). In other words, we looked at specific instances to come up what we thought was happening. Deductive reasoning works from the other direction. It takes general facts and works towards a conclusion.

Inductive vs Deductive Reasoning

Inductive Conclusions from patterns.

You have trouble sleeping whenever you drink a can of soda right before bed. Therefore, soda prevents you from sleeping well.

Deductive Conclusions from facts.

Soda contains caffeine. Caffeine is a stimulant. You drink soda before. Therefore, you won’t sleep well if you drink soda before bed.

With deductive reasoning, there are two laws that we’ll look at.

The Two Laws

The first reinforces the conditional work we did last section.

The Law of Detachment

If a conditional statement and it’s hypothesis are both true, then it’s conclusion must be true.

If you finish within 30 minutes, then you win a prize. You finished in 26 minutes. Therefore, you win a prize.

Keep in mind that this doesn’t work with the conclusion. If you win a prize, it’s not necessarily because you finished within 30 minutes. There might be another condition that triggered earning a prize.

The other law is similar to the transitive property, where if $a=b$ and $b=c$, then $a=c$.

The Law of Syllogism

Given two conditionals, where the conclusion of one is also the hypothesis of the other, then there is a third conditional linking the other hypothesis and conclusion.

If a number is divisible by 6, then it is also divisible by 2. If a number is divisible by 2, then it is an even number. Therefore, if a number is divisible by 6, then it is an even number.

Be careful with what you are connecting. Just because conditionals share the same part it doesn’t mean they can be connected using the Law of Syllogism. If a number is divisible by 2, then it’s even. If a number is divisible by 4, then it’s even. That does not mean that if a number is divisible by 2, then it is divisible by 4.