Logic does not let us know whether specific statements or claims are true or false.
Logic allows us to test whether propositions or arguments are valid or invalid and on a higher level if arguments are valid or fallacious. A fallacy is a provable error in a chain of reasoning due to the form of the argument itself. The conclusion does not follow or is not proven by the preceding argument or set of presented facts, assumptions, and statements.
Single Statements 
A proposition is a statment that can be either true or false. It is a claim.
Consider these statements:
I am human. 1 + 1 = 2 Roses are red and violets are blue.
These statements can be either true or false. Negations of these statements are:
I am not human. 1 + 1 ≠ 2 It is not the case that roses are red and violets are blue.
which are also statements.
As for the operators, the basic operators are as follows:
If p and q are both propositions,
- Negation: Not p (meaning p is not true)
- Conjunction: p and q are both true
- Inclusive Disjunction: p or q or both
- Exclusive Disjunction: p or q but not both
- Conditional: If p is true then q is true (but not necessarily the other way around)
- Biconditional: p is true if and only if q is true (meaning they are either both true or both false)
The NAND, NOR and XOR are all combinations of these (not-and, not-or and not-biconditional respectively)
Multiple Statements 
Two or more statements (which we will call "X" and "Y") can be joined together with logical operators to form larger statements (which we will call "Z"). These include but are not limited to:
- AND - Z is true if both X and Y are true. If either or both are false, Z is false.
- OR - Z is true if either of X or Y is true. Z is only false if both X and Y are false.
- NAND - Z is true if at least one of X or Y is false. Z is false when both X and Y are true.
- NOR - Z is true if both X and Y are false. Z is false when X or Y or both are true.
- XOR - Z is true if either X or Y is true. Z is false when both X and Y are true, or when both X and Y are false.