1. Outline Logic Propositional Logic Well formed formula Truth table
Tautology & Contradiction
Proof System for Propositional Logic
Deduction method
Formalizing English arguments
Text book chapters 1.1 and 1.2

2. To define a logic, answer three questions: What are the models?
Logics
To define a logic, answer three questions:
What are the models?
What are the formulas?
Which formulas are true in which models?
A logic is a formal system relating syntax (formulas) and semantics (models of the world).
Proposition logic is by Aristotle in a set of books called the Organon in 40 BC.

3. Propositional Logic: Models
A statement or a proposition is a sentence that is either true or false. Represented as upper cap letters of the alphabet.
“She is very talented”
“There are life forms on other planets in the universe”
Statement letters can be combined into new statements using logical connectives of
conjunction (AND,)
disjunction (OR, )
implication ()
equivalence ()
negation ()
The first sentence is not a statement since “she” is not defined.
The second sentence is a statement since it will take a true or a false value, though we do not have enough knowledge currently to answer which.

4. Propositional Logic: Formulas
Truth tables define how each of the connectives operate on truth values.
Truth table for implication ()
Equivalence connective A  B is shorthand for (A  B)  (B  A)
Truth table for equivalence ()
Binary & unary connective
A->B: A is precedent and B is consequent.
A->B means B is a necessary condition for A. A is sufficient condition for B.
Qs. How many rows in the truth table of a ternary connective?

5. When is a formula true in a model?
Each Boolean connective has a truth table
e.g. P Q
P  Q T F P Q
P  Q T F
He is intelligent but lazy. I  L.
He likes Dante and Dan Brown. D  B. P P T F

6. What about the other connectives?P Q
P  Q T F P Q
P  Q T F T F T F
Tricky cases
“P is the same as Q”
Let’s say I make the statement: If I ace my Discrete Math exam, then I will go to the movies.
Consider the following cases: I ace my DM exam and I go to the movies. Then my statement was true.
I ace my DM exam but I do not go to the movies. Then my statement was false.
I do not ace my DM exam and I still go to the movies. Then my statement was true.
I do not ace my DM exam and I do not go to the movies. Then my statement was true.
By convention AB is considered true if A is false, regardless of the truth value of B.

7. Example formulas and non-formulas
“If the rain continues, then the river will flood.” Express by implication.
“A good diet is a necessary condition for a healthy cat.” Express by implication.
“Julie likes butter but hates cream.” Negate this.
“Fire is a necessary condition for smoke” Same as “If there is smoke, then there is fire”

8. Main connective: Last connective to be applied in a wff
Formulas & Precedence
Well-formed formula: A valid string with statement letters, connectives, and parentheses
Examples
Precedence
ABC  (AB)C OR A(BC)?
Main connective: Last connective to be applied in a wff
Example

9. Example of Truth Table for a WFF
Compound WFF (AB) (CA)
# rows in a truth table with n statement letters = ?

10. Tautologies and Contradictions
A tautology is a formula that is true in every model. (also called a theorem)
for example, (A  A) is a tautology
What about (AB)(AB)?
Look at tautological equivalences on pg. 8 of text
A contradiction is a formula that is false in every model.
for example, (A  A) is a contradiction
Not a tautology:
Students with a high GPA (G) means they will become millionaires (M). G -> M. This may be true for the class of 1950 but may not be true for all graduating batches.
Students with a high score in EE 369 (S) means they will become millionaires (M). S -> M is a tautology.
For the former statement, you will have to add qualifiers.
(x class of 1950) G(x) -> M(x)

11. Negate “The couch is either new or comfortable.”
De Morgan’s Laws
(AB) = AB
Negate “The couch is either new or comfortable.”
(AB) = AB
Negate “The book is thick and boring.”
The couch is old and uncomfortable.
The book is either thin or interesting.

12. The sequence “proves” the last formula
A Proof System
A proof system is a syntactic system for finding formulas implied by the hypotheses
“syntactic” means manipulating syntax
i.e. manipulating formulas rather than models.
P1P2 … PnQ is a tautology
A proof is a sequence of formulas, where each formulas in the sequence is either
a hypothesis
a formula justified by previous formulas and a proof rule
The sequence “proves” the last formula
Stress on the word syntactic in the definition of proof. These proofs are going to be true independent of the model.

13. Example proof rule P P  Q ________________ Q
Given the formulas above the line, we can add the formula below the line to the proof.
modus ponens (mp)
So what can we deduce from PQ and Q’? P’
If the diet is healthy, then the cat is agile.
This translates to H -> A. The standard form of translating an “if … then” sentence.
Also if I say “A is a necessary condition for B” then it means “B is a sufficient condition for A” that is B -> A.

14. Example proof
Hypotheses: C, B, B  (C  A)
Conclusion: A
B premise
B  (C  A) premise
C  A 1, 2, mp
C premise
A 3, 4, mp

15. Proof rules: Equivalence rules (Table 1.12)
Expression
Equivalent
Rule name
Abbr.
P  Q P  Q
Q  P Q  P
Commutative
comm
(P  Q)  R (P  Q)  R
P  (Q  R ) P  (Q  R )
Associative
ass
(P  Q) (P  Q)
P  Q P  Q
De Morgan dm
P  Q
P  Q
Implication
imp
( P) P
Double Neg. dn
P  Q
(P  Q)  (Q  P)
Equivalence
equ
New symbol for negation.

16. Proof Rules: Inference Rules (Table 1.13)
From
Can derive
Rule name
Abbr.
P, P  Q Q
Modus ponens mp
P  Q, Q P
Modus tollens mt
P, Q
P  Q
Conjunction
con
Simplification
sim P
P  Q
Addition
add
Formal system is correct and complete.

17. An Example Proof
[(AB)C]  (CD)  A  D

18. Deduction Method in Proofs
When proving P Q…
add P to premises and prove Q.
Repeat to prove P (Q  R)
add P and Q to premises and prove R.
To prove: P1  P2 … Pn (R S), prove: P1  P2 … Pn  R  S

19. Formalizing English Arguments
To formalize an English argument:
Find the minimal statements in the argument and symbolize them with propositional letters A, B, …
Convert English connectives to propositional ones.
Give a proof of the conclusion using the premises.

20. Examples
Jack went to fetch a pail of water. Jack fetches a pail of water only if Jack works hard. Jack works hard only when Jack is paid. Therefore Jack was paid.

21. Minimal True/False Statements
Jack went to fetch a pail of water. Jack fetches a pail of water only if Jack works hard. Jack works hard only if Jack is paid. Therefore Jack was paid.
A = Jack fetches a pail of water
B = Jack works hard
C = Jack is paid
A. A only if B. B only if C. Therefore C.

22. Eliminating connectives
A. A only if B. B only if C. Therefore C.
becomes
Hypotheses: A, A  B, B  C.
Conclusion: C.
Easy to prove that these hypotheses give this conclusion using rule mp.

23. Examples — a bad argument
Jack went to fetch a pail of water. Jack fetches a pail of water if Jack works hard. Jack works hard if Jack is paid. Therefore Jack was paid.
Hypotheses: A, B  A, C  B.
Conclusion: C.
Hypotheses do not entail conclusion…to show this, give a model that makes the conclusion false.
A=T, B=T, C=F
But if it was given that Jack was paid, then we could have concluded that Jack went to fetch a pail of water.

24. Another example
Fish can walk. Fish can walk only if elephants can fly. Elephants can fly only if eggplants can talk. Therefore, eggplants can talk.
Is this a valid argument?

25. Examples where propositional logic fails
Every positive number is greater than zero. Five is a positive number. Therefore, five is greater than zero.
Minimal statements?
A = Every positive number is greater than zero.
B = Five is a positive number.
C = Five is greater than zero.
Hypotheses: A, B. Conclusion: C.
Conclusion not entailed (consider A = B = T, C = F)
Our logic does not model the internal structure of the propositions