Michael Lee's Mathematics and Physics
Deductive Logic
The truth of all the premises (reasons of belief) guarantees the truth of the conclusion. In other words, for any valid argument, if all its premises are true, then it must be the case the conclusion is true (the conclusion is sound.) However, if all the premises are true, but the conclusion is false then the argument is said to be invalid.
For example,
Premise (1) "John may either have his cake or eat it but not both." Premise (2) "John ate his cake." Therefore, Conclusion (1) "John no longer has his cake."
or
Premise (1) "John may either have his cake or eat it but not both." Premise (2) "John did not eat his cake." Therefore, Conclusion (1) "John has his cake."
The arguments here take on the following structure; it's called logical conjunction and is a valid form of reasoning: A: John has his cake B: John ate his cake.
Premise (1) A or B and not both Premise (2) B Therefore, Conclusion (1) not A
or
Premise (1) A or B and not both Premise (2) not B Therefore Conclusion (1) A
Another form of reasoning concerns conditional statements. They have a structure as follows...
if A then B A is referred to as the antecedent and B is the consequent.
For example,
By Modus Pollens... Premise (1) "If it is raining outside, then the concrete walk is wet." Premise (2) "It is raining outside." Therefore, Conclusion (1) The concrete walk is wet. This is valid reasoning called Modus Pollens or the 'mode of affirming.'
By Modus Tollens... Premise (1) "If it is raining outside, then the concrete walk is wet." Premise (2) "The concrete walk is not wet." Therefore, Conclusion (1) "It is not raining outside." This is another form of valid reasoning called Modus Tollens or the 'mode of denying."
Structurally these lines of reasoning looks like this,
Modus Pollens Premise (1) If A then B Premise (2) A Therefore, Conclusion (1) B
Modus Tollens Premise (1) If A then B Premise (2) not B Therefore, Conclusion (1) not A
Invalid Deductive Logic It's important to remember with conditional statements you do not make the mistake of affirming the consequent or denying the antecedent.
Affirming the consequent (faulty logic) looks like this: Premise (1) If A then B Premise (2) B Therefore, Conclusion (1) A
Denying the antecedent (faulty logic) looks like this: Premise (1) If A then B Premise (2) Not A Therefore, Conclusion (1): Not B
Showing a deductive argument is invalid amounts to showing all the premises are true but the conclusion is false.
In our example,
Affirming the consequent (faulty logic) Premise (1) If it is raining outside the concrete walk is wet. Premise (2) The concrete walk is wet Therefore, Conclusion (1) It is raining outside
This is faulty logic because even if the walk is wet, it might not be raining outside. The sprinkler system could be in operation for example.
Denying the antecedent (faulty logic) Premise (1) If it is raining outside the concrete walk is wet. Premise (2) It is not raining outside Therefore, Conclusion (1) The concrete walk is not wet (i.e. dry).
This is faulty logic too. Suppose it is not raining outside, it doesn't necessarily follow the walk is dry. Again, the sprinkler system may be in operation.
