The document discusses formal logic, emphasizing its structure as a formal system with atomic symbols, axioms, and rules of inference. It defines formal logic as a system where formulas can be assigned true or false values, enabling the expression of logical statements. Various types of logic, such as propositional and predicate logic, allow for the formalization of diverse statements.

1. Formal Logic
Ivan Ruchkin
Microteaching Workshop
June 26, 2012

2. Why Formal Logic?

3. Formal System
● Consists of:
– Atomic symbols. E.g.: {a, b, +}.
– Axioms. E.g.: {b+a, a}.
– Rules of inference. E.g.: {F→F+a; F→F+b}, where F
is a formula.
● Defines a set of formulas.
– E.g.: {b+a, a, a+a, b+a+b, ...}
– Other examples: arithmetic, programming language.

4. Formal Systems as Construction
Axioms
Formulas
Rules of
inference

5. Formal Logic: Definition
● Formal logic is a formal system where all formulas are
assigned one of two values: true or false.
● Allows to express logical statements (not necessarily
true):
– Ivan is teaching now.
– All husbands cheat.
– Geese usually bite, and ducks always panic.
– If something is big, it has hydrogen in it.
● There are many logics!

6. Different Logics
Propositional
Logic
Predicate Logic Temporal (Modal) Logic
Atomic symbols Concrete objects,
AND, OR, NOT, IF-
THEN.
Propositional logic +
variables + quantifiers:
for all, exists.
Predicate logic + temporal
operators: always, eventually,
until, ...
Examples of
formalizable
statements
My son is at home
and my dad is not.
It either rains or it
does not.
If the president
sleeps, then a war
cannot start.
All husbands cheat.
If all colleges are bad,
CMU is bad.
At least one chinchilla
is smarter than at least
one human.
Eventually all humans will
die.
I always tell ducks not to
panic.

7. Summary
● Formal system is a collection of atoms, axioms,
and rules of inference.
● Logic is a formal system with true or false
statements.
● There are many logics allowing to formalize
different statements.