One of the most fascinating discoveries I made in college was in a formal logic class, when we learned the foundations of logic.
Why does logic work?
Logic is the basis of mathematics.
Logic is the basis of grammar.
Logic is the relational structure upon which mathematical rules build upon.
But what is logic built upon?
Introduce: the laws of thought.
What are the laws of thought?
There are three laws that govern our thoughts, and the reason and logic they produce, and three signs which represent the contents and operations of the mind.
The abstract contents of our thoughts can be represented by three signs:
1. Symbols, which represent things that are subject to conception [A,B,p,q,x,y, etc]
2. Operations, which represent how the mind combines or resolves things into new forms [and, or, not, if then, etc]
3. Identity, which represents equality or equivalence [=]
There are three laws:
•The law of identity: (A=A)
•The law of non contradiction: ~(A&~A)
•The law of excluded middle. (A or ~A)
There are also three operators, or logical connectives:
“is” represents equality [=]
“or” represents disjunction, or this or that or both [+,∨]
“not” represents negation or [~,-]
From these operators we can build other operators:
“and” represents conjunction or grouping [&,∧, •, ×] ~(~p or ~q)
“conditional” implication, or if this, then that (essentially cause and effect) [→, ⊃,⇒] (~p∨q)
“Biconditional” representing equivalence [=,↔︎,≡]
The law of identity states that a thing is equal to itself. A is A. It cannot be the case that something is equal to something that is not itself. This is tautological. Ex: 3=2+1=1+1+1. Three is a composite of integrals, of parts. The whole must be equal to the sum of parts. 1lb = 16oz. You can imagine the consequences if things were equal to things other than what they were. Without the law of identity, you could equivocate anything, and all would fall apart and be meaningless. There would be no clear distinctions of what things are. The sun is not the moon. The sun is a star. The moon is a planet.
The law of non contradiction states that something cannot be and not be at the same time. You cannot exist and not exist simultaneously. You cannot have a dollar and not have a dollar simultaneously. It cannot be Noon and Midnight.
The law of excluded middle states that a thing is one thing, or it is something else. It cannot be both. Something is either alive or dead. Not half alive or half dead. You exist or you do not exist, not partly exist. There is no “middle”.
All of these laws assume a permanent or static context, without change, and therefore a time constant. Because we know that something can be the case in one moment, and not the case in the next moment. So logic requires we define a context, or “domain of discourse” which we assign a range of values to variables.