Logical Foundations

Even though I want to dive right in to posting Deontic proofs and proposing dialogues on the differences between s4, s5, and M logical systems, I figured I should lay some groundwork. I didn't want to direct you to just a Wikipedia article to lay the groundwork as the articles tend to go off on tangents at a rapid pace, although I will occasionally reference some articles for odd bits.

Below is a set of operators:

v - Disjunction (or)
& - Conjunction (and)
¬ or ~ - Negation (not)
→ - Conditional (implication)
↔ - Bi-conditional (Equivalence)

Some less common operators:

↓ - Pierce Arrow - A↓B = ~(AvB)
↑ or | - Sheffer Stroke - A↑B = ~(A&B)

Parenthesis are used where order of precedence in applying operators may need to be modified for the current statement, or in cases where the author of the statement may just want to make their case more clear. Precedence will be discussed later.

A common way to discuss the validity of certain statements is with a truth table. Below is an example of such a table.

While I believe that truth tables are useful in demonstrating the differences between tautologies and contingent truths to intro students, I feel it is best to breeze straight past them and on to Semantic Tableaux in the next issue.