The document covers key concepts in symbolic logic such as logical operations, truth tables, tautologies, valid and invalid arguments, and special valid argument forms. Logical equivalence and valid argument forms like modus ponens, modus tollens, disjunctive syllogism and hypothetical syllogism are explained. Examples are provided to illustrate logical statements, truth tables, and how to determine if an argument is valid or invalid.

1. Symbolic Logic
• Logical Form and Equivalence
• Conditional Statements
• Valid and Invalid Arguments
• Digital Logic Circuits (Boolean Polynomials)
1.1.1

2. Logic of Compound Statements
• A statement (or proposition) is a sentence that is
true (T) or false (F), but not both or neither.
• Examples:
Today is Monday.
x is even and x > 7.
If x2 = 4, then x = 2 or x = -2.
1.1.2

3. Counterexamples
• If a sentence cannot be judged to be T or F or is
not even a sentence, it cannot be a statement.
• Examples:
Open the door! (imperative)
Did you open the door? (interrogative)
If x2 = 4. (fragment)
1.1.3

4. Compound Statements
• Denote statements using the symbols p, q, r, ...
• Denote the operations ~,  (to be defined
shortly), where:
p  q - conjunction of p and q (p and q);
p  q - disjunction of p and q (p or q);
~ p - negation of p (not p);
p  q - implication of p and q (p implies q);
1.1.4

5. Compound Statements (cont’d.)
• A Compound statement (or statement form) is a
statement which includes at least one operation
and one other “atomic” statement.
• For example, “x = 7 and y = 2” is a compound
statement based on the “atomic” statements
p = “x = 7” and q = “y = 2”.
• In this instance, we can symbolize the compound
statement as r = p  q.
1.1.5

6. Compound Statements (cont’d.)
• The Truth Table of a compound statement is the
collection of all the output truth values
corresponding to all possible combinations of
input truth values of the atomic statements.
• Since each atomic statement can take on 1 of 2
values, 2 inputs have 4 combinations, 3 inputs
have 8, 4 inputs have 16, 5 inputs have 32, etc.
1.1.6

7. Logical Operations
• Negation: p ~p
T F
F T
• Conjunction: • Disjunction:
p q (p  q) p q (p  q)
T T T T T T
T F F T F T
F T F F T T
F F F F F F
1.1.7

8. Example: (p  q)  ~r
• Proceed from left to right:
p q r (p  q) ~r (p  q)  ~r
T T T T F F
T T F T T T
T F T T F F
T F F T T T
F T T T F F
F T F T T T
F F T F F F
F F F F T F
1.1.8

9. Logical Equivalence
• Two compound statements are logically
equivalent if they have the same truth table. We
denote this as p  q.
• p ~p ~(~p)
T F T
F T F hence p  ~(~p).
• ~(p  q)  ~p  ~q ?
No, since ~(T  F)  T, but (~T  ~F)  F.
1.1.9

10. Tautology & Contradiction
• A statement whose truth table is all “T” is called
a tautology, denoted as p  t.
• A statement whose truth table is all “F” is called
a contradiction, denoted as p  c.
• Clearly, ~t  c and ~c  t.
• Are all logical statements either tautology or
contradiction?
1.1.10

11. Algebra of Symbolic Logic
• Commutative Laws:
p  q  q  p
p  q  q  p
• Associative Laws:
(p q)  r  p  (q r)
(p q)  r  p  (q r)
• Distributive Laws:
p (q r)  (p q) (p r)
p (q r)  (p q) (p r)
1.1.11

12. Algebra of Symbolic Logic
• Identity Laws:
p  t  p
p  c  p
• Negation Laws:
p  ~p  c
p  ~p  t
• Double Negative Laws: ~(~p)  p
• Negations of t and c: ~t  c ~c  t
1.1.12

13. Algebra of Symbolic Logic
• Idempotent Laws: p  p  p p  p  p
• DeMorgan’s Laws:
~(p q)  ~p  ~q
~(p q)  ~p  ~q
• Universal Bound Laws: p  c  c p  t  t
• Absorption Laws:
p (p q)  p
p (p q)  p
1.1.13

14. Section 1.2
• Conditional Statements
• Logical Equivalences Involving Conditionals
• Converses, Inverses, and Contrapositives
• Biconditional Statements
1.2.14

15. Conditional Statements
• If p and q are statement variables, the conditional
or implication of q by p is “If p then q” or
“p implies q” and is denoted by p  q.
• The truth table of the implication operator is:
p q p  q
T T T
T F F
F T T
F F T
• Example: If you mow my lawn, I’ll pay $20.
1.2.15

16. Hypotheses & Conclusions
• In the form “If p then q” the statement p is called
the hypothesis and the statement q is the
conclusion.
• Conditionals form the basis of “deductive”
reasoning. (Aristotilean Logic)
• In looking at the truth table, we consider the
cases where the hypothesis is false to yield
vacuous results. The interesting cases are when
the hypothesis is true.
1.2.16

17. Logical Equivalences
• In the framework of symbolic logic, the
implication operator would seem to be a new and
distinct process.
• However, this is not the case!
• Theorem: p  q  ~p  q.
• Thus, we can always rewrite an implication as a
disjunction.
• Corollary: ~(p  q)  p  ~q.
1.2.17

18. Negation of a Conditional
• From the previous corollary, the negation of
p  q is p  ~q.
• For example, the negation of
If today is Sunday, then I wash my car.
is:
Today is Sunday and I do not wash my car.
1.2.18

19. Converse of a Conditional
• Given the statement p  q, we define its
converse to be the statement q  p.
• For example, the converse of
If today is Sunday, then I wash my car.
is:
If I wash my car, then today is Sunday.
1.2.19

20. Contrapositive of a Conditional
• Given the statement p  q, we define its
contrapositive to be the statement ~q  ~p.
• For example, the contrapositive of
If today is Sunday, then I wash my car.
is:
If I do not wash my car, then today is not Sunday.
1.2.20

21. Inverse of a Conditional
• Given the statement p  q, we define its inverse
to be the statement ~p  ~q.
• For example, the inverse of
If today is Sunday, then I wash my car.
is:
If today is not Sunday, then I do not wash my car.
1.2.21

22. Equivalent Forms
• Theorem: Given the statement p  q, we have
that p  q  ~q  ~p.
• Corollary: Given the statement p  q, we have
that q  p  ~p  ~q.
• Therefore from the above, we see that a
conditional and its contrapositive are logically
equivalent.
• Moreover, the statement’s converse and inverse
forms are logically equivalent to each other.
1.2.22

23. Biconditional Statements
• Definition: Given the statement variables p and
q, the biconditional of p and q is read, “p if and
only if q,” denoted p  q and means that both
p  q and q  p .
• By direct calculation: p q p q
T T T
T F F
F T F
F F T
1.2.23

24. Using the Biconditional
• Looking closely at the truth table, we see that
p  q is T whenever p and q have the same truth
value.
• Theorem: p  q is a tautology implies p  q and
conversely, p  q implies p q is a tautology.
• This gives us a systematic way to calculate
logical equivalence, rather then just scan the
matches of truth values by eye.
1.2.24

25. Section 1.3
• Valid and invalid argument forms.
• Special valid argument forms.
• Dilemmas
• Fallacies.
• Contradictions and valid arguments.
1.3.25

26. Valid and Invalid Arguments
• An argument (or argument form) is a sequence of
statements.
• All statements but the final one are called
premises, assumptions, or hypotheses.
• The final statement is called the conclusion.
• An argument form is valid provided its
conclusion is always true whenever all of its
premises are true.
1.3.26

27. Valid and Invalid Arguments
• An argument (or argument form) is a sequence of
statements.
• All statements but the final one are called
premises, assumptions, or hypotheses.
• The final statement is called the conclusion.
• An argument form is valid provided its
conclusion is always true whenever all of its
premises are true.
• The truth of the conclusion follows inescapably
from the truth of the hypotheses.
1.3.27

28. Testing for Validity
• Identify the premises and conclusion.
• Construct a truth table for the premises and
conclusion.
• Find the critical rows, where all premises are T.
• For each critical row, if the conclusion is also T,
then the argument is valid.
• If at least one critical row leads to a conclusion
being F, the argument is invalid.
• If there are no critical rows, the argument is
vacuously valid.
1.3.28

29. A Valid Argument
p (q  r)
~r
(p  q)
Truth Table: p q r [p (q  r)] ~r (p  q)
T T T T F T
T T F T T T
T F T T F T
T F F T T T
F T T T F T
F T F T T T
F F T T F F
F F F F T F
1.3.29

30. An Invalid Argument
p (q  r)
~r
(p  r)
Truth Table: p q r [p (q  r)] ~r (p  r)
T T T T F T
T T F T T T
T F T T F T
T F F T T T
F T T T F T
F T F T T F
F F T T F T
F F F F T F
1.3.30

31. Special Argument Forms
Modus Ponens: p q
p
q
Truth Table: p q p q p q
T T T T T
T F F T F
F T T F T
F F T F F
Premises: If today is Sunday, then I wash my car.
Today is Sunday.
Conclusion: I wash my car.
1.3.31

32. Modus Tollens
Modus Tollens: p q
~q
~p
Truth Table: p q p q ~q ~p
T T T F F
T F F T F
F T T F T
F F T T T
Premises: If today is Sunday, then I wash my car.
I do not wash my car.
Conclusion: Today is not Sunday.
1.3.32

33. Disjunctive Addition
Disjunctive Addition: p
p  q
Truth Table: p q p q
T T T
T F T
F T T
F F F
Premise: Today is Sunday.
Conclusion: Today is Sunday or I wash my car.
1.3.33

34. Conjunctive Simplification
Conjunctive Simplification: p  q
p
also q
Truth Table: p q p q
T T T
T F F
F T F
F F F
Premise: Today is Sunday and I wash my car.
Conclusion 1: Today is Sunday.
Conclusion 2: I wash my car.
1.3.34

35. Disjunctive Syllogism
Disjunctive Syllogism: p  q p  q
~p ~q
q p
Truth Table: p q p q ~p
T T T F
T F T F
F T T T
F F F T
Premises: Today is Sunday or Saturday.
Today is not Sunday.
Conclusion: Today is Saturday.
1.3.35

36. Hypothetical Syllogism
Hypothetical Syllogism: p  q
q  r
p  r
Premises: If x is an integer, then x is a rational.
If x is a rational, then x is a real.
Conclusion: If x is an integer, then x is real.
1.3.36

37. Dilemma: Division Into Cases
Dilemma: p  q
p  r
q  r
r
Premises: x is positive or x is negative.
If x is positive , then x2 is positive.
If x is negative, then x2 is positive.
Conclusion: x2 is positive.
1.3.37

38. Application: Find My Glasses
1. If my glasses are on the kitchen table, then I saw them at
breakfast.
2. I was reading in the kitchen or I was reading in the
living room.
3. If I was reading in the living room, then my glasses are
on the coffee table.
4. I did not see my glasses at breakfast.
5. If I was reading in bed, then my glasses are on the bed
table.
6. If I was reading in the kitchen, then my glasses are on
the kitchen table.
1.3.38

39. Find My Glasses (cont’d.)
Let: p = My glasses are on the kitchen table.
q = I saw my glasses at breakfast.
r = I was reading in the living room.
s = I was reading in the kitchen.
t = My glasses are on the coffee table.
u = I was reading in bed.
v = My glasses are on the bed table.
1.3.39

40. Find My Glasses (cont’d.)
Then the original statements become:
1. p  q 2. r  s 3. r  t
4. ~q 5. u  v 6. s  p
and we can deduce :
1. p  q 2. s  p 3. r  s 4. r  t
~q ~p ~s r
~p ~s r t
Hence the glasses are on the coffee table!
1.3.40

41. Fallacies
• A fallacy is an error in reasoning that
results in an invalid argument.
• Three common fallacies:
– Using vague or ambiguous premises;
– Begging the question;
– Jumping to a conclusion.
• Two dangerous fallacies:
– Converse error;
– Inverse error.
1.3.41

42. Converse Error
If Zeke cheats, then he sits in the back row.
Zeke sits in the back row.
Zeke cheats.
• The fallacy here is caused by replacing the
impication (Zeke cheats  sits in back)
with its biconditional form (Zeke cheats 
sits in back), implying the converse (sits in
back  Zeke cheats).
1.3.42

43. Inverse Error
If Zeke cheats, then he sits in the back row.
Zeke does not cheat.
Zeke does not sit in the back row.
• The fallacy here is caused by replacing the
impication (Zeke cheats  sits in back)
with its inverse form (Zeke does not cheat
 does not sit in back), instead of the
contrapositive (does not sit in back  Zeke
does not cheat).
1.3.43

44. Contradiction Rule
• If you can show that assuming statement p
is false leads logically to a contradiction,
then you can conclude that p is true.
• In argument form: ~p  c
p
• This is the logical heart of the proof method
called Proof by Contradiction.
1.3.44

45. Section 1.4
• Digital Logic Circuits
• Boolean Polynomials
• Normal Forms (Disjunctive/Conjunctive)
• Designing Circuits with Specified
Conditions
• Showing Two Circuits Are Equivalent
1.4.45

46. Digital Logic Circuits
• Developed by Claude Shannon in 1938 to
model telephone switching circuits:
x y
Series Switch
x AND y
x
y
Parallel Switch
x OR y
1.4.46

47. Logical Gates
• Instead of working with switches, we model
digital circuits using gates: AND-gates, OR-
gates, and NOT-gates.
• We draw these as:
x
y
x + y
OR
x
y
xy
AND
x x’
NOT
1.4.47

48. Notation
• Modeling digital circuits leads to the
equivalent analysis of symbolic logic.
• Symbolic Logic Digital Circuits
T, t 1, 1
F, c 0, 0
p, q, r, ... x, y, z, ...
~p x’
p  q xy
p  q x + y
1.4.48

49. Boolean Polynomials
• When modeling, we use Boolean
polynomials to describe algebraically the
function of a combinatorial circuit.
• A combinatorial circuit is one in which the
output at any time depends on the inputs at
the previous time. (i.e. no feedback loops)
• A Boolean polynomial is a function which
takes 0,1 inputs and outputs a 0 or 1 using
the operations AND, OR, and NOT.
1.4.49

50. Examples of Boolean Polynomials
• When working with Boolean polynomials,
we must first know the specific input
variables.
• Examples:
f(x,y,z) = x + y + z
f(x,y) = x’ + xy
f(x,y,z) = x(y + z’)
1.4.50

51. Evaluating Boolean Polynomials
• Using: x y x’ y’ (x + y) xy
1 1 0 0 1 1
1 0 0 1 1 0
0 1 1 0 1 0
0 0 1 1 0 1
• Examples: Find f(x,y) = x’+ xy
x y x’ xy (x’+ xy)
1 1 0 1 1
1 0 0 0 0
0 1 1 0 1
0 0 1 0 1
1.4.51

52. Normal Forms
• Expressing a Boolean Polynomial in its
normal form provides an easy method to
calculate its truth table.
• We can create two different normal forms
for Boolean Polynomials: the disjunctive
and the conjunctive normal form.
• These forms are made up of special terms
called minterms or maxterms.
1.4.52

53. Disjunctive Normal Form
• A minterm is a Boolean polynomial that is
only the product of each variable or its
negation (but not both).
• Examples: f(x,y) = xy’
f(x,y,z) = x’yz’
f(w,x,y,z) = wx’y’z
• The disjunctive normal form (DNF) is a
Boolean polynomial that is the sum of
minterms (sum of products).
1.4.53

54. Disjunctive Normal Form (cont’d.)
• Express f(x,y,z) = x + x’z in its DNF.
f(x,y,z) = x + x’z
= x(y + y’)(z + z’) + x’(y + y’)z
= (xy + xy’)(z + z’) + (x’y + x’y’)z
= xyz + xyz’+ xy’z + xy’z’+ x’yz + x’y’z
• The thing to note here is that each minterm
has an output of 1 at only a single,
particular line of the truth table.
• i.e. xy’z = 1 at 101 and = 0 elsewhere.
1.4.54

55. Disjunctive Normal Form (cont’d.)
• We can now think of the inputs, in fact, as
their associated minterms to get outputs:
x y z x y z f(x,y,z)
1 1 1 x y z 1
1 1 0 x y z’ 1
1 0 1 x y’z 1
1 0 0 x y’z’ 1
0 1 1 x’y z 1
0 1 0 x’y z’ 0
0 0 1 x’y’z 1
0 0 0 x’y’z’ 0
1.4.55

56. Designing Circuits
with Specified Conditions
• In the other direction:
x y z f(x,y,z)
1 1 1 0 0 0 0 0
1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0
1 0 0 0  0 + 0 + 0 + 0
0 1 1 1 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 1 0 0 0 1
0 0 0 0 0 0 0 0
f(x,y,z) = xy’z + x’yz + x’yz’+ x’y’z
1.4.56

57. Conjunctive Normal Form
• In a similar fashion, we can analyze
functions using the conjunctive normal
form - the product of sums.
• In this case, we look for the 0’s in the
function’s output and associate each with a
maxterm, whose output is 0 at that row.
1.4.57

58. Equivalent Circuits
• Two logical circuits are equivalent if and
only if they have the same truth table.
• This can be thought similarly as holding
when the two circuits have the same
disjunctive (conjunctive) normal form.
1.4.58