Logic - What is the explanation behind the results of logic operations? How come the compound statement "p ∧ q" is true only if both p and q are true, and is false otherwise? Likewise, how come "p ∨ q" is true as long as either p or q (or both) is true? What's the connection between the truth values of the statement's components and the truth value of the statement (conjunction, disjunction, conditional, biconditional) itself? How is the result arrived at by considering the truth values of the components?
My brain seems to need to know the "algorithm" behind these rules. I cannot seem to proceed without knowing the why and how of the results of the logic operations. I've been trying to look up for an answer, but it appears that "they just are".
Thanks in advance.
 A: There are at least two approach your question.  One is to see it as an English (or other natural language) question.  Outside math, if I say "This bird is a crow and it is black" you would only call my statement true if both parts are true.  One is to see it as a definition of the connective $\wedge$.  We define $p \wedge q$ to be true when both $p$ and $q$ are true.  The test of a definition is to be unambiguous and to be useful.  It becomes just a function $And(p,q)$ with a value of true or false depending on the truth values of $p$ and $q$.  The algorithm is just the evaluation from the definition.  
"Or" suffers from ambiguity in English.  It is not clear whether it should be true when both parts are true.  Sometimes in English we use the exclusive or, so exactly one of the alternatives must be true.  Sometimes in English we use the inclusive or, which corresponds to the mathematical $\vee$.  Math has decided which one to use.  
"If...then" is even worse in English.  It is not at all clear how to evaluate the truth value of "If A then B" when A is false.  Math has decided that $A \implies B$ will be true whenever $A$ is false or $B$ is true.  It meets the test of unambiguous and useful, but can be a trap when translating English into logical symbols.  Again, the algorithm is just the definition.  You evaluate the truth values of $A$ and $B$, then plug them into the definition to get the truth value of the compound statement.
