Logic 101
There are two kinds of logic: Formal Logic and Informal Logic. Formal Logic is intended to prove a proposition (theorem). Informal Logic is designed to persuade.
FORMAL LOGIC
Formal logic follows a set of strict rules. The basic argument of formal logic is the syllogism which takes the form
- Premise 1
- Premise 2
- Conclusion
For example
- P1 – All men are mortal
- P2 – Socrates is a man
- C - Socrates is mortal
The argument must be both
valid and
sound. To be valid the conclusion need only follow from the premises. For example
- All birds can fly
- Penguins are birds
- Therefore penguins can fly
This argument is valid but it is not sound. To be sound the premises must be true and, in this example, they are not. Penguins, emus, ostriches, domestic turkeys are all birds but they cannot fly.
Arguments in Formal Logic can be written out in a notation called Boolean Algebra, named after George Boole who invented it. In Boolean Algebra a syllogism is written as
where the “+” sign means “and”. The ==> means “implies”. (It’s supposed to look like an arrow.) P1, P2 and C are statements, i.e., anything which can be assigned a truth value (TRUE or FALSE) and only a truth value. (Note that pairs such as ON or OFF, YES or NO, 0 volts or 5 volts, can also be considered truth values but I do not intend to get into that.)
The beauty of Boolean Algebra is that validity is almost automatic. Good mathematicians can avoid errors in the math and, if they do creep in, other mathematicians can find and correct them. Thus validity is not controversial. That leaves soundness, i.e., proving the truth of the premises, and that is beyond the scope of mathematics.
Before moving on to Informal Logic there is one subject I wish to briefly cover: that of inductive reasoning, a form of extrapolation. Inductive reasoning should raise a red flag in most cases but there is an exception, viz., the Axiom of Induction put forth by mathematician Guiseppe Peano.
- P1 - you have a set of statement which can be numbered, e.g., statement 1, statement 2, statement 3, etc.
- P2 - statement N ==> statement N + 1, i.e, if you assume statement N is true then statement N + 1 must also be true
- P3 - statement 1 is true
- C - all statements are true
Next I’ll cover INFORMAL LOGIC.
Please hold all questions until I have finished the series.
