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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
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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)
vr
 ~p^q = (~p) ^ q
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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
 Same for opposite
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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
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Logical 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”

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)
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_
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
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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
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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

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”

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?
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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

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
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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

Modus Ponens: A famous argument form
◦ p→q: p:: q
◦ If p occurs then q occurs: p occurs:: therefore q
occurs
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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
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Generalization
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Specialization
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Elimination
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Transitivity
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Contradiction Rule:
◦ p::pvq and q::pvq
◦ p occurs, therefore either p or q occurred
◦ Used to classify events into larger groups
◦ p^q::p and p^q::q
◦ Both p and q occur, therefore p occurred
◦ Used to put events into smaller groups
◦ 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
◦ p →q:q →r::p →r
◦ If p then q: if q then r:: therefore if p then r
◦ ~p →c::p
◦ If the negation of p leads to a contradiction, p must be true.

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
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The Biconditional (iff)
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◦
◦
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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


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
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Converse 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