It’s always worth re-visiting first principles i.e. the theory from time to time…today is a very basic recap of a fundamental logic using facts, rules and definitions as the basis for reason; deductive reasoning.
Lets look at a very basic statement as an example to illustrate what I’m on about:
If you make the effort then you will get the result.
The part after the “if”: you make the effort – is called a hypotheses and the part after the “then” – you will get the result – is called a conclusion.
Hypotheses followed by a conclusion is called an if-then statement or a conditional statement.
In mathematics, this is noted as
p → q
and reads – if p then q.
A conditional statement is false if the hypothesis is true and the conclusion is false. Make sense?
What we’re saying is pretty straight forward and that is our example above would be false if it said “if you make the effort then you will not get a result”.
If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.
For example a conditional statement is:
If we spend 50% of our money on things we don’t need (p) then we will have 50% of our money left to spend on things we do need (q).
p → q
If we exchange the position of the hypothesis and the conclusion we get a converse statement:
If we spend 50% of our money on things we need then we will have 50% of our money left to spend on things we don’t need.
q → p
If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing though. In fact they have very different meanings. Let me explain:
If we negate both the hypothesis and the conclusion we get a inverse statement:
If we don’t spend 50% of our money on things we don’t need then we will not have 50% of our money left to spend on things we do need.
~ p → ~ q
The inverse is not true just because the conditional is true. The inverse always has the same truth value as the converse.
We could also negate a converse statement, this is called a contrapositive statement:
If we don’t spend 50% of our money on things we do need then we will not have 50% of our money left to spend on things we don’t need.
~ q → ~ p
The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.
A pattern of reasoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism. Slow down I hear you saying, my head hurts! Maybe another example might help:
If we turn off the computer on our desk, then the monitor will stop showing.
If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:
[(p → q) ^ p] → q
The law of syllogism tells us that if p → q and q → r then p → r is also true.
Again in mathematical speak which leaves nothing to question:
[(p → q) ^ (q → r)] → (p → r)
A small jump here but if you’re still with me then we’re almost done! So if the following statements are true:
If we turn off the computer on our desk (p), then the monitor will stop showing (q). If the monitor stops showing (q) then we don’t see anything on the screen any more (r).
Then the law of syllogism tells us that if we turn off the computer (p) then we don’t see anything on the screen any more (r) must be true.
This basic sequence of events [despite my hack attempt at trying to explain it] is so fundamental to our existence. It’s one the three basic skills we all need to decipher the fake from the real. The truth from the lies.