1. Logic of Informatics
Introduction

2. Historical view Philosophical Logic Symbolic Logic Mathematical Logic
500 BC to 19th Century
Symbolic Logic
Mid to late 19th Century
Mathematical Logic
Late 19th to mid 20th Century
Logic in Computer Science
Logic became popular in the early 20th century among philosophers and mathematicians

3. Logic is called the CALCULUS of Computer Science!
LOGIC is a term for CS CALCULUS is a term for Physical sciences & Engineering Disciplines
Many CS areas use LOGIC, such as:
Architecture (logic gates)
Programming Languages ( Semantics & Logic Programming) AI
Databases (SQL)

4. Philosophical Logic 500 B.C – 19th Century
Logic dealt with arguments in the natural language used by humans.
Example
All men are mortal.
Socrates is a man
Therefore, Socrates is mortal.

5. Philosophical Logic [2]
Natural languages are very ambiguous.
Eric does not believe that Mary can pass any test.
...does not believe that she can pass some test, or
...does not believe that she can pass all tests
I only borrowed your car.
And not ‘borrowed and used’, or
And not ‘car and coat’
Tom hates Jim and he likes Mary.
Tom likes Mary, or
Jim likes Mary

6. Symbolic Logic Mid to late 19th Century.
The modern development of symbolic logic begin with George Boole in the 19th century.
Attempted to formulate logic in terms of a mathematical language
Rules of inference were modeled after various laws for manipulating algebraic expressions.

7. Mathematical Logic Late 19th to mid 20th Century
Frege proposed logic as a language for mathematics in 1879.
With the rigor of this new foundation, Cantor was able to analyze the notion of infinity in ways that were previously impossible. (2N is strictly larger than N)
Russell’s Paradox
T = { S | S ∉ S}

8. Logic in Computer Science
In computer science, we design and study systems through the use of formal languages that can themselves be interpreted by a formal system.
Boolean circuits
Programming languages
Design Validation and verification
AI, Security. Etc.

9. Logic in Computer Science
Propositional Logic
First Order Logic
Higher Order Logic
Theory of Construction
Real-time Logic, Temporal Logic
Process Algebras
Linear Logic

10. Propositions Logic
A proposition is a statement that is either true or false
2+2=4 (true)
New York City is in Oregon (false)
Today is Friday (its either true or false)
Do your homework! (not a proposition)

11. Compound and Primitive Propositions
Compound Propositions are propositions that are composed of sub-propositions connected together in various ways
roses are red and violets are blue
Mark is smart or he studies a lot
A (atomic, elementary) proposition is the underlying meaning of a simple declarative sentence, which is either true or false

12. Predicate logic Extension of propositional logic
A ‘predicate’ is just a property
Predicates define relationships between any number of entities using qualifiers:
 “for all”, “for every”
 “there exists”

13. Example Let P(x) be the property
‘if x is a triangle then the sum of its internal angles is 180o”
In predicate logic:
 x P(x)
“For every x such that x is a triangle, the sum of the internal angles of x is 180o”

14. Another example Let P(x) be the property ‘x is an integer and x2 = 4’
Then
 x P(x)
“There exists x such that x is an integer and x2 = 4”

15. Newton’s second law of motion
“for every x”
“of type called object”
“or”
 x: Object  stationary(x)  in-uniform-motion (x)   f : Force  x is-acted-upon-by f
“there exists an f”
In English: “for every x of a certain type referred to as an Object, x is stationary, x is in uniform motion, or there is an f of type Force such that x is acted upon by f”

16.  and  Remember:  x ‘for every x’, or ‘for All x’
 x ‘there is an x’ or ‘there Exists an x’
Tip:
Think of  as an upside down ‘A’ (‘for All’)
Think of  as a backwards ‘E’ (‘there Exists’)

17. Propositional connectives
These are the words that we use to join atomic propositions together to form compound propositions. E.G:
In 1938 Hitler seized Austria, (and) in 1939 he seized former Czechoslovakia and in 1941 he attacked the former USSR while still having a non-aggression pact with it

18. Propositional connectives
Propositional logic has four connectives
Name
Read as
Symbol
negation
‘not’ 
conjunction
‘and’ 
disjunction
‘or’ 
implication
‘if…then…’ 

19. Interpretation of connectives
negation p is true if and only if p is false 
A conjunction pq is true if and only if both p and q are true 
A disjunction pq is true if and only if p is true or q is true. 
An implication p  q is false if and only if p is true and q is false 
The biconditional “p if and only if q” is written p  q

20. Some more terminology…
Expressions either side of a conjunction are called conjuncts (pq)
Expressions either side of a disjunction are called disjuncts (pq)
In the implication p  q, p is called the antecedent and q is the consequence

21. Precedence of connectives
In complex propositions, brackets may be used to remove ambiguity.
(p q) r versus p  (q  r)
By convention, the order of precedence
Brackets, Negation, Conjunction, Disjunction, Implication

22. Mathematical Logic Statement Formulas Definitions
Symbols p ,q ,r ,...,called statement variables
Symbols ~, ∧, ∨, →,and ↔ are called logical connectives
A statement variable is a statement formula
If A and B are statement formulas, then the expressions (~A ), (A ∧ B) , (A ∨ B ), (A → B ) and (A ↔ B ) are statement formulas
Expressions are statement formulas that are constructed only by using 1) and 2) above
Discrete Mathematical Structures: Theory and Applications

23. Formalisation
Although both Stanley and Gordon are not young, Stanley has a better chance of winning the next bowling tournament, despite Gordon’s considerable experience

24. Formalisation (continued)
Atomic propositions:
p – Stanley is young
q – Gordon is young
r – Stanley has a better chance of winning the next bowling tournament
s – Gordon has considerable experience in bowling
Formalisation in Propositional Logic:
(p)  (q)  r  s

25. Truth Table
A truth table for a compound statement is a list of the truth or falsity of the statement for every possible combination of truth and falsity of its components.
In other words, a truth table helps to show whether a statement is true or false.

26. Rows
To find the number of rows used in a truth table, take the number 2 raised to the power of the number of variables.
For example, if there was a p statement and a q statement, there would be 2 variables, 2^2 is 4.
If there were three statements, it would be 2^3, or 8 rows.

27. Columns
The columns under the connectives /\, and \/, stand for the conjunction, and disjunction of the expression on the two sides of that connective.

28. Negation Truth Table p ~ p The opposite of p is ~p T F
“Not true” is “false”
“Not false” is “true”

29. Conjunction Truth Tablep q
p /\ q
p and q T
True only if both are true. F

30. Disjunction Truth Tablep q
p \/ q
p or q T
True if either on is true F
False only if both are false

31. Logical implication: IMPLIES
Is a proposition “if p, then q”.
Produces all true values except when p is true and q is false.
Keywords: if-then, if, then, whenever, only if, etc.
We often use p q
The Truth table of p → q
All true except when p is true and q is false.
5/20/2019

32. Biimplication
Let p and q be statements. The statement “p if and only if q” is called the biimplication or biconditional of p and q
The biconditional “p if and only if q” is written p  q
p  q is read:
“p if and only if q”
“p is necessary and sufficient for q”
“q if and only if p”
“q when and only when p”
Discrete Mathematical Structures: Theory and Applications

33. Biconditional [2] Truth Table for the Biconditional:
Discrete Mathematical Structures: Theory and Applications

34. Construct truth-tables for each of the following formulae:
P & (Q v ¬P)
¬P → Q
Q v ¬(P & Q)
P → ¬(P & Q)
P → (P v Q)
Q → (P → Q)
P & ¬P
(Q v P) & ¬P