1. Introduction to Logic Logical Form: general rules
All logical comparisons must be done with complete statements
A statement is an expression that is true or false but not both
If p or q then r
If I arrive early or I work hard then I will be promoted
Tautologies and Contradictions
A Tautology (t) is a statement that is always true
A Contradiction (c) is a statement that is always false

2. Symbolism The use of symbols ~ denotes negation (Not)
If p = true, ~p = false
^ denotes conjunction (And)
p^q = true iff (if and only if) p = true and q = true
v denotes disjunction (Or)
p vq = true iff p = true or q = true or p^q = true
XOR: exclusive or
P XOR q = (p vq) ^ ~(p^q), “p or q but not both”
Order of operations
~ is first, ^ and v are co-equal
P^q v r is ambiguous, so parenthesis need to be used: (p^q) v r
~p^q = (~p) ^ q

3. Connection to Mathematics
Inequalities
x ≤ a means x < a or x = a: (x < a) v (x = a)
Same for x ≥ a
a ≤ x ≤ b means (a ≤ x) ^ (x ≤ b)
a (NOT)> x = a ≤ x
Same for opposite
a (NOT) ≤ x = a > x

4. Truth Tables Truth Tables Every expression has a truth table
This table represents all the possible evaluations of the expression
To build a truth table, construct a table with cells corresponding to every possible value of the variables and the resulting value of the expression

5. Equivalence Logical equivalence Showing non-equivalence
Two statement forms are logically equivalent iff their truth tables are entirely the same
Ex: p^q = q^p
P = ~(~p)
Showing non-equivalence
Two methods:
Use truth tables: this takes a long time
Use an example statement like “0 < 1”

6. Common Logical Forms
The following are known as axioms. Use these to simplify logical forms easily
Commutative Laws: p^q = q^p , pvq = qvp
Associative Laws: (p^q)^r = p^(q^r), (pvq)vr = pv(qvr)
Distributive Laws: p^(qvr) = (p^q)v(p^r)
p v(q^r) = (pvq)^(pvr)
_ Identity Laws: p^t = p, pvc = p
_ Negation Laws: pv~p = t, p^~p = c
_ Double Negative Law: ~(~p) = p
_ Idempotent Laws: p^p = p, pvp = p
_ Universal Bound Laws: pvt = t, p^c = c
_ De Morgan’s Laws: ~(p^q) = ~pv~q, ~(pvq) = ~p^~q
_ Absorption Laws: p√(p^q) = p, p^(pvq) = p
_ Negations of t and c: ~t = c, ~c = t

7. Conditional Statements
If Structures
Statement form: “if p then q”
Noted: p→q, p is Hypothesis, q is conclusion
Truth Values: p→q is false iff p = true and q = false
In statement forms, “→” is evaluated last
Division Into Cases: Show pvq→r=(p→r)^(q→r)
Build truth table and evaluate each term separately
Then fill in each side of the equation and compare the values

8. Equivalence of If An If statement can be translated into an Or
p→q = ~pvq
People often use this equivalence in everyday language.
By De Morgan’s Law
~(p →q) = p^~q
Caution: The negation of an If does not start with “if”

9. Transformations of If The Contrapositive of an If
The contrapositive of p →q is ~q →~p
A contrapositive is always logically equivalent to the original statement, so it can be used to solve equations
A contrapositive is both the converse and the inverse of a statement
The Converse and Inverse
The Converse of p →q is q →p
The Inverse of p →q is ~p →~q
Neither is logically equivalent to the original statement
If tomorrow is Easter then tomorrow is Sunday
If tomorrow is Sunday then tomorrow is Easter?

10. Other Forms of If Only If
“p only if q” means that p may occur only if q occurs
Equivalent to: ~q →~p
Equivalent to: p →q
This does not mean “p if q”, which says that if q is true, p must be true

11. Valid and Invalid Arguments
An argument is a sequence of statements and an argument form is a sequence of statement forms.
A basic argument is: p→q p :q
_ All statements except the final one are the premises
_ The final is the conclusion
_ This is read: “If p then q; p occurs, therefore q follows
_ The argument is valid iff the conclusion is true when all of the premises are true

12. Testing an Argument Testing an argument for validity
Identify the premises and conclusion
Construct a truth table showing the possible truth values for each statement and statement form
If a situation exists in which all of the premises are true but the conclusion is false, the argument form is invalid
To simplify, fill in all rows where all premises are true

13. Common Argument Forms Modus Ponens: A famous argument form
p→q: p:: q
If p occurs then q occurs: p occurs:: therefore q occurs
Modus Tollens
p →q: ~q:: ~p
If q doesn’t occur, p can’t occur
A rule of inference is an argument form that is valid.
There are infinitely many of them
Modus Ponens and Tollens are rules of inference

14. More Common Forms Generalization Specialization Elimination
p::pvq and q::pvq
p occurs, therefore either p or q occurred
Used to classify events into larger groups
Specialization
p^q::p and p^q::q
Both p and q occur, therefore p occurred
Used to put events into smaller groups
Elimination
Pvq: ~q::p and pvq:~p::q
P or Q can occur: Q doesn’t:: p must
you can choose one by ruling the other out
Transitivity
p →q:q →r::p →r
If p then q: if q then r:: therefore if p then r
Contradiction Rule:
~p →c::p
If the negation of p leads to a contradiction, p must be true.

15. A Simple Proof Proof by Division Into Cases pvq: p →r:q →r:: r
p or q will occur: if p then r: if q then r:: r occurs
You may only know one thing or another. You must simply show that in either case, the result is the same

16. Iff Defined The Biconditional (iff) This is: “p if, and only if q”
Denoted: p↔q and is coequal with →
p iff q = (p→q) ^ (q→p)
If p has the same truth value as q, p↔q is true

17. Fallacies An error in reasoning that results in an invalid argument
Three kinds
Using ambiguous premises (Statements that are not T/F)
Begging the Question: assuming the conclusion without deriving it from the premises
Jumping to a Conclusion: verifying the conclusion without adequate grounds

18. Common Errors Converse Error: Inverse Error p →q: q:: p – FALSE
If p then q: q occurs:: p must occur – FALSE
Inverse Error
p →q: ~p:: ~q - FALSE
If p then q: p doesn’t occur:: q can’t occur - FALSE