This document provides an introduction to formal logic and mathematical logic. It defines key concepts like logic, propositions, statements, truth values, and logical connectives. It also differentiates between propositions and non-propositions, and statements and non-statements. Examples are given of propositions and non-propositions. The basics of propositional logic and how to determine the truth values of expressions are explained.

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FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

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Lesson 1
Introduction to Logic

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LESSON OBJECTIVES
❑ Use the formal symbols for logic
❑ Define logic and mathematical logic
❑ Differentiate a proposition from non-proposition; a
statement from not a statement;
❑ Determine the truth values of an expression in propositional
logic

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PRE-TEST ACTIVITY 1
❑ Examine whether the following are propositions or not
proposition
1. Earth is the only planet in the universe, that has life.
2. 12 /4 = 3.
3. Who are you talking to?
4. Read this sentence carefully.
5. X + 4 = 1
6. 3 is on odd integer.
7. V + V =W
8. Quezon City is the capital of the Philippines.
9. 2+4=6
10. The World is Bat?
11. What time is it?
12. Read this carefully?
13. x+y=z
14. x is greater than 2
15. Close the door.

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LOGIC
❑ the science of the correctness or incorrectness of
reasoning, or the study of the evaluation of arguments.
❑ the study of the principles and methods that distinguishes
between a valid and an invalid arguments
❑ technically defined as “the science or study of how to
evaluate arguments and reasoning.”
❑ methods of reasoning
❑ focus on relation

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LOGICAL REASONING
❑ used on mathematics to prove theorems
❑ computer science – to verify correctness of programs and
to prove theorems.
Aristotle (382-322 BC) is generally
regarded as the Father of Logic.

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MATHEMATICAL LOGIC
❑ symbolic logic
❑ a branch of mathematics with close connections to
computer science.
❑ the discipline that mathematicians invented in the late 19th
and early 20th centuries to stop talking nonsense

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Note:
❑ logic and mathematical reasoning, has numerous
applications in computer science as well as in Information
Technology
❑ these rules are used in the design of computer circuits, the
construction of computer programs, the verification of the
correctness of programs

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STATEMENT
❑ proposition
❑ basic building block of logic
❑ a declarative sentence which is either true (T) or false (F),
but not both
❑ a statement is atomic if it cannot be divided into smaller
statements, otherwise it is called molecular.
❑ i.e. atomic statements
❑ Telephone numbers in the USA have 10 digits.
❑ The moon is made of cheese.
❑ 42 is a perfect square.

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STATEMENT
❑ i.e. atomic statements
❑ Every even number greater than 2 can be expressed as
the sum of two primes.
❑ 3 + 7 = 12
❑ i.e. not statements
❑ Would you like some cake?
❑ The sum of two squares.
❑ 1 + 3 + 5 + 7 + · · · + 2n + 1.
❑ Go to your room!
❑ 3 + x = 12

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STATEMENT
❑ There are also statements (or propositions) which are
considered ambiguous such as
❑ Mathematics is fun.
❑ Calculus is more interesting than Trigonometry.
❑ It was hot in Manila.
❑ Street vendors are poor.

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STATEMENT
❑ we can build more complicated (molecular) statements out
of simpler (atomic or molecular) ones using logical
connectives (“and”, “or”, “if…then”, “if and only if”, “not”)
❑ binary connectives (connect two statements)
❑ unary connective (applies to single statement)
❑ logical connectives
❑ “and” (∧)
❑ “or” (∨)
❑ “if…then” (→)
❑ “if and only if” (↔)
❑ “not” (¬)

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STATEMENT
❑ i.e. molecular statement
❑ Sam is a man and Chris is a woman
❑ Sam is a man or Chris is a woman
❑ if Sam is a man, then Chris is a woman
❑ Sam is a man if and only if Chris is a woman
❑ Sam is not a man
❑ axioms that are true statements about the model
❑ a list of inference rules that let us derive new true
statements from the axioms
❑ a theory that consists of all statements that can be
constructed from the axioms by applying the inference rules

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STATEMENT
❑ i.e.
❑ All fish are green (axiom).
❑ George Washington is a fish (axiom).
❑ From “all X are Y ” and “Z is X”, we can derive “Z is Y ”
(inference rule).
❑ Thus George Washington is green (theorem).
❑ Theories are attempts to describe models.
❑ A model is typically a collection of objects and relations
between them.
❑ A theory is consistent if it can’t prove both P and not -P for
any P.

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STATEMENT
❑ Note:
❑ If we throw in too many axioms, you can get an
inconsistency:
❑ i.e. “All fish are green; all sharks are not green; all
sharks are fish; George Washington is a shark”
❑ If we don’t throw in enough axioms, we under constrain
the model.

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THE LANGUAGE OF LOGIC
❑ the basis of mathematical logic is propositional logic
❑ here the model is a collection of statements that are either
true or false
❑ “George Washington is a fish”
❑ “George Washington is a fish or 2+2=5”
❑ Predicate logic adds both constants (stand-ins for objects in
the model like “George Washington”) and predicates
(stand-ins for properties like “is a fish”)

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PROPOSITION
❑ a proposition (simple statement) may be denoted by a
variable like P, Q, R, …., called a proposition (statement)
variable or sentential variable.
❑ the value of a proposition called its “Truth Value” (the truth
and falsity of the statement); denoted by
❑ T or 1 – if it is TRUE
❑ F or 0 – if it is FALSE
❑ opinion, interrogative, and imperative are not propositions
❑ Truth Table
P
0
1

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PROPOSITION
❑ Examples that are propositions
❑ “Beijing is the capital of China.”
❑ “1+2=3”
❑ 1+0=1
❑ 1+2=3
❑ Every cow has four legs.
❑ Grass is green.
❑ There are four finger in a hand.

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PROPOSITION
❑ Examples that are not propositions
❑ Sit down! (imperative, command)
❑ X+1=2 (not clear)
❑ Who’s there? (interrogative, question)
❑ “1+2” (expressions with a non-true / false value)
❑ X+2=3

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POST-TEST ACTIVITY 1
❑ Examine whether the following are propositions or not
proposition
1. Earth is a planet in the solar system, that has one moon.
2. 16 / 4 = 3.
3. Where are you going?
4. Listen carefully.
5. Y + 2 = 3
6. 4 is on even integer.
7. F + V + S = FVS
8. Sta. Cruz is the capital of the Laguna.
9. 10 + 4 = 14
10. Who is your favorite Marvel Hero?
11. When is your birthday?
12. Read this carefully.
13. x + y = z
14. y is lesser than 2
15. Open the door.

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• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science
University of Colorado.
• Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
• Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES