Transcript Document

Introduction
to
Symbolic Logic
(part 2)
San Diego Math Circle
David W. Brown
Theorems, Tautologies,
Proofs
Oh my!
We’ve just encountered some statements that are,
in fact, “theorems” of propositional logic
Laws of contraposition
( p → q ) ↔ ( ~q → ~p )
( q → p ) ↔ ( ~p → ~q )
Biconditional introduction/elimination
(p → q)  (p ← q) ↔ (p ↔ q)
We demonstrated the truth of these equivalences by
comparing the truth tables for each side of the
equivalence symbol.
Since the truth values of each side (either T or F) were the
same in every row, we “proved” the equivalence to be
true for all possible truth values of propositions p and q.
Tautology, Contradiction,
Contingency
A tautology is a statement that is true for all
possible truth values of its atomic substatements;
i.e., is equivalent to the constant-true function T
and thus expresses a universal truth.
A contradiction is a statement that is false for all
possible truth values of its atomic substatements;
i.e., is equivalent to the constant-false function F
and thus expresses a universal falsehood.
A contingency is a statement that is true for some
possible truth values of its atomic substatements,
and false for others; i.e., is not equivalent to
either constant-true function T, nor the constantfalse function F. Most propositions are
contingencies.
Contradiction Example
Definitions:



b = “The boy did it.”
d = “The dog did it.”
c = “The cat did it.”
Givens:



The boy says:
The dog says:
The cat says:
(~b  d)
(~d  c)
(~c  b)
All givens in a proof are presumed
true. Here, this means that the
boy, the dog, and the cat are
given to be speaking the truth.
Proof:
(~b  d)  (~d  c)  (~c  b)
(~b  b)  (~d  d)  (~c  c)
FFF
F
Contradiction
Conjunction of givens
Commutative & associative laws
Law of noncontradiction
Idempotency
→ Somebody’s lying.
Tautology
The equivalences we just proved are tautologies:
( p → q ) ↔ ( ~q → ~p )
( q → p ) ↔ ( ~p → ~q )
(p → q)  (p ← q) ↔ (p ↔ q)
Tautologies of equivalence allow us to “substitute”
one side of the equivalence for the other in any
logical expression.
Other tautologies (e.g., of implication) allow us to
form inferences.
This is essential for developing a “calculus” for
proving theorems.
“Unary” Tautologies
Laws of negation:
 p  ~p ↔ F
 p  ~p ↔ T
Laws of idempotency
 p  p ↔ p
 p  p ↔ p
Laws of identity
 p  T ↔ p
 p  F ↔ p
Laws of reflexivity
 p ↔ p
 p → p
Laws of domination
 p  F ↔ F
 p  T ↔ T
Vacuous truth / Double negation


F→p
~~p ↔ p
Spotlight on
Laws of negation
p
p

~p ↔
F
p
p
 ~p ↔
T
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
T
T

p  ~p ↔ F
Law of noncontradiction
No wff can be both T and F

p  ~p ↔ T
Law of the excluded middle
Every wff must be T or F – no “maybe”
Some Basic “Binary” Tautologies

Laws of simplification (conjunction elimination)

(p  q ) → p
(p  q ) → q

Laws of addition (disjunction introduction)


p → (p  q)
q → (p  q)

Laws of absorption



p  (p  q) ↔ p
p  (p  q) ↔ p
Algebra-like Tautologies
Commutative Laws:


pq↔qp
pq↔qp
“like:”
“commutative law of x”
“commutative law of +”
Associative Laws:


(p  q)  r ↔ p  (q  r)
(p  q)  r ↔ p  (q  r)
“associative law of x”
“associative law of +”
Distributive Laws:


p  (q  r) ↔ (p  q)  (p  r )
p  (q  r) ↔ (p  q)  (p  r )
“distributive law of x over +”
“distributive law of + over x” X
Transitive Laws:


(p ↔ q)  (q ↔ r) → (p ↔ r)
(p → q)  (q → r) → (p → r)
“transitive property of =”
“transitive property of >”
Note:


In ordinary algebra, commutative laws,
associative laws, distributive laws, etc. are
axioms – primitive statements that stand
without proof.
In logic, the similar statements are proven
from more primitive truths; this is a
reflection of the fact that logic is more
fundamental than the higher mathematics
built upon it.
De Morgan’s Laws
De Morgan’s Laws tell us how to negate
conjunctions and disjunctions; something
frequently necessary in practice
~ (p  q) ↔ ~p  ~q
~ (p  q) ↔ ~p  ~q
Note that this is quite different from the
relation between negation (-) and the
binary operations of algebra, (+) and (x).
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
~(p  q) ~p  ~q
The
equivalence
of the
rightmost
columns
proves the
tautologies
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Negate p
~(p  q) ~p  ~q
Negate p
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Negate q
~(p  q) ~p  ~q
Negate q
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Form
disjunction of
p and q
~(p  q) ~p  ~q
Form
conjunction of
p and q
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Negate
disjunction of
p and q
~(p  q) ~p  ~q
Negate
conjunction of
p and q
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Form
conjunction of
~p and ~q
~(p  q) ~p  ~q
Form
disjunction of
~p and ~q
Truth Tables for De Morgan’s Laws
p
q
~p
~q
(p  q)
~(p  q) ~p  ~q
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
p
q
~p
~q
(p  q)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
~(p  q) ~p  ~q
The
equivalence
of the
rightmost
columns
proves the
tautologies
Rules of Inference
“The proof of proof”
Classical vs. Propositional
Notations for Inference
p→q Major Premise
p
Minor Premise
q
Conclusion
p→q Major Premise




~q
Minor Premise
~p
Conclusion
Modus Ponens
(p → q )  p → q
Modus Tollens
(p → q )  ~q → ~p
Some Common Inference rules






Modus Ponens
Modus Tollens
Syllogisms:
• Disjunctive
• Hypothetical
Reductio ad Absurdam
Proof by Contradiction
Indirect Proof


Dilemmas:
• Constructive
• Destructive
Proof by Cases
Modus Ponens
Latin: “modus ponendo ponens” – the mode
that affirms by affirming.
That is, the consequent is affirmed (proven
true) by affirming (asserting the truth of)
the antecedant.
If p and p → q, then q.
-or(p  (p → q )) → q
(truth flows forward)
Truth Table Proof of
Modus Ponens
p
q
p→q
p(p→q)
[p(p→q)]→q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Truth Table Proof of
Modus Ponens
p
q
p→q
p(p→q)
[p(p→q)]→q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Truth Table Proof of
Modus Ponens
p
q
p→q
p(p→q)
[p(p→q)]→q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Truth Table Proof of
Modus Ponens
p
q
p→q
p(p→q)
[p(p→q)]→q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Condensed Truth Table Proof of
Modus Ponens
[(p → q)  p] → q
p
q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Modus Tollens
Latin: “modus tollendo tollens” - mode that
denies by denying
That is, the antecedant is denied (proven
false) by denying (asserting the falseness
of) the consequent.
If p → q and ~q, then ~p.
-or((p → q )  ~q) → ~p
(falsehood flows backward)
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Truth Table Proof of
Modus Tollens
~p ~q p→q (p→q)~q [(p→q)~q]→~p
p
q
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
Condensed Truth Table Proof of
Modus Tollens
[(p → q) 
p
q
~ q] → ~
T
T
T
F
F
T
F
T
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
T
T
T
p
Disjunctive Syllogism
The principle of the disjunctive syllogism is that
since at least one of the two propositions
comprising a true disjunction must be true, if one
of them is known to be false, the other can be
concluded to be true:
If ~p and p  q, then q.
-or(~p  (p  q )) → q
Truth Table Proof of
Disjunctive Syllogism
p  q ~p(p  q) [~p(p  q)]→q
p
q
~p
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
T
Truth Table Proof of
Disjunctive Syllogism
p  q ~p(p  q) [~p(p  q)]→q
p
q
~p
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
T
Truth Table Proof of
Disjunctive Syllogism
p  q ~p(p  q) [~p(p  q)]→q
p
q
~p
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
T
Truth Table Proof of
Disjunctive Syllogism
p  q ~p(p  q) [~p(p  q)]→q
p
q
~p
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
T
Truth Table Proof of
Disjunctive Syllogism
p  q ~p(p  q) [~p(p  q)]→q
p
q
~p
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
T
Condensed Truth Table Proof of
Disjunctive Syllogism
~p  (p  q) → q
p
q
T
T
F
F
T
T
T
F
F
F
T
T
F
T
T
T
T
T
F
F
T
F
F
T
Hypothetical Syllogism
The hypothetical syllogism expresses the
transitive property of logical implication:
If p → q and q → r, then p → r.
-or((p → q )  (q → r )) → (p → r )
Note that this relates three propositions, which
involves a domain of ordered triples (p,q,r)
requiring a truth table of 23 rows.
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Truth Table Proof of
Hypothetical Syllogism
p→q
q→r
(p→q)(q→r)
p→r
[(p→q)(q→r)]→(p→r)
T T T
T
T
T
T
T
T T F
T
F
F
F
T
T F T
F
T
F
T
T
T F F
F
T
F
F
T
F T T
T
T
T
T
T
F T F
T
F
F
T
T
F F T
T
T
T
T
T
F F F
T
T
T
T
T
p
q
r
Condensed Truth Table Proof of
Hypothetical Syllogism
p q r [(p → q)  (q → r)] → (p → r)
T
T
T
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
T
F
F
T
T
T
T
F
F
F
F
T
T
F
F
T
T
T
T
T
T
T
F
T
F
T
F
F
T
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
T
T
ἡ εἰς ἄτοπον ἀπαγωγή
Reductio ad Absurdam
q ~p ~q (p →
T
T
F
F
T
F
F
T
T
F
F
T
F
F
T
T
F
T
T
F
T
T
T
T
F
F
T
T
T
T
T
T
p→q

p → ~q
~p

q)

p
(p
→ ~q) → ~p
Reductio ad Absurdam involves concluding
both a consequent and its negation from the
same antecedent, thus proving the negation
of the antecedent.
This is only technically distinct from proof by
contradiction, since by the law of
noncontradiction we have that q~q↔F.
Proof by Contradiction

p
~p
F
(~p
T
F
F
T
T
F
T
F
F
T
→
F)
→
p
Proof by contradiction proceeds by
negating the proposition representing the
desired proof goal, and from this
concluding any universal falsehood.
Indirect Proof
p
q ~p ~q (~q
→
T
T
F
F
T
T
T
T
F
F
T
F
T
F
F
T
T
F
T
T
T
F
F
T
T
T
T
T

~p)
→
(p
→
q)
Indirect proof also follows as a special case of the
equivalence of any implication and its
contrapositive.
(p → q) ↔ (~q → ~p)
Dilemmas
Constructive Dilemma
(p  q)  (p → r)  (q → s)→ (r  s)
related to modus ponens
truth feeds forward
Destructive Dilemma
(p → r)  (q → s)  (~r  ~s) → (~p  ~q)
related to modus tollens
falsehood flows backward
Constructive Dilemma
(p → r)
(p  q)
⇗
or
⇘
modus ponens
modus ponens
⇘
or
⇗
(r  s)
(q → s)
(p  q)  (p → r)  (q → s) → (r  s)
Note that four propositions requires a truth table of 24 rows.
“Dilemma” generalizes to “multi-lemma”, with more disjuncts.
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Condensed Truth Table Proof of
Constructive Dilemma

q) → (r

T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
T
T
T
T
F
F
T
T
F
F
F
F
T
T
T
F
F
T
F
r)

p q
r
s [(p →
(q → s)]
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
F
T
F
T
T
F
T

(p
s)
Destructive Dilemma
(p → r)
(~r  ~s)
⇗
or
⇘
modus tollens
modus tollens
⇘
or
⇗
(~p  ~q)
(q → s)
(p → r)  (q → s)  (~r  ~s) → (~p  ~q)
Note that four propositions requires a truth table of 24 rows.
“Dilemma” generalizes to “multi-lemma”, with more disjuncts.
Condensed Truth Table Proof of
Destructive Dilemma
p q
r
s
T
T
T
T
T
T
T
F
F
T
F
T
T
T
F
T
F
F
F
T
T
F
T
T
F
T
F
F
T
F
T
T
F
T
T
F
F
F
F
F
F
T
T
F
T
F
T
T
T
T
T
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
F
T
F
F
T
F
T
T
T
T
F
F
F
F
F
T
F
T
T
T
F
T
T
T
T
T
T
T
F
T
T
F
T
T
F
T
F
F
F
T
T
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
T
T
F
F
T
T
T
T
T
F
F
T
T
F
F
T
F
T
T
T
T
T
T
T
F
F
F
T
T
T
T
T
T
T
T
F
F
F
F
T
T
T
T
T
T
T
(p
→
r)

(q
→
s)

(~r

~s)
→ (~p

~q)
Proof by Cases
(p  q  r)  (p → s)  (q → s)  (r → s) → s
(p → s)
(p  q  r)
⇗
or
⇘
modus ponens
(q → s)
modus ponens
⇘
or
⇗
s
(r → s)
For the purposes of illustration, we have chosen three cases; the inference is
valid for any number of cases – the truth tables have many rows!
Note that “proof by cases” is a special case of constructive “dilemma” or
“multi-lemma” in which all the consequents are the same proposition.
Beyond Truth Tables

Truth tables are powerful means of proving
“simple” results involving small numbers of
statements; however:
• The number of columns grows in rough proportion to the
complexity of the statements to be proven, and
• The number of rows grows as 2n, where n is the number
of atomic statements in the compound.


For more than the most basic statements, truth
tables can overwhelm manual calculation.
Ultimately, the subject of symbolic logic is about
using a relatively modest set of basic theorems to
prove complex statements through algebra-like
procedures.
Fallacies
Formal vs. Informal Fallacy




a formal fallacy is a deductive argument that
has an invalid form, whereas
an informal fallacy is any other invalid mode of
reasoning whose flaw is not in the form of the
argument; e.g., in flawed premises.
We are concerned only with formal fallacies.
A formal fallacy is not necessarily a
“contradiction” as earlier defined – a universal
falsehood – it is only necessary that the
argument admits an unexpected truth value in
some case(s).
Penguin Logic
Definitions:



p = You are a penguin
o = You are an old TV show
b = You are black & white
Premises:


p → b = If you are a penguin,
then you are black and white
o → b = If you are an old TV
show, then you are black and
white
Penguin conclusion:

p → o = If you are a penguin,
then you are an old TV show
This is the fallacy of the
undistributed middle:
(p → b)  (o → b) → (p → o)
invalid
which can be confused with
hypothetical syllogism in
which the “middle” term is
properly “distributed”:
(p → b)  (b → o) → (p → o)
valid
Affirming the Consequent
Fallacy confused with Modus Ponens
p q
(p → q) 
p → q


T T
T
T
T
T F
F
F
T
F T
T
F
T
F F
T
F
T
p q
(p → q) 
q → p
T T
T
T
T
T F
F
F
T
F T
T
T
F
F F
T
F
T
Modus Ponens
Tautology
M.P. can be viewed as
“affirming the antecedent”


Affirming the
Consequent
Fallacy
One final truth value is “F”,
invalidating the desired
tautology
Denying the Antecedent
Fallacy confused with Modus Tollens
 ~ q → ~
p
q
(p → q)
T
T
T
F
F
T
F
T
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
T
T
T
p
q
(p → q)
T
T
T
F
F
T
F
T
F
F
F
F
T
T
F
T
T
T
T
F
F
F
F
T
T
T
T
T
 ~ p → ~
p


Modus Tollens
Tautology
M.T. can be viewed as
“denying the consequent”
q


Denying the
Antecedent
Fallacy
One final truth value is “F”,
invalidating the desired
tautology
Affirming a Disjunct
Fallacy confused with Disjunctive Syllogism
p
q
(p

q)
 ~ q → p
T
T
T
F
F
T
T
F
T
T
T
T
F
T
T
F
F
T
F
F
F
F
T
T
p
q

p → ~
T
T
T
T
F
F
T
F
T
F
T
T
F
T
T
F
T
F
F
F
F
F
T
T
(p

q)


Disjunctive Syllogism
Tautology
D.S. can be viewed as
“denying a disjunct”
q
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Affirming a Disjunct
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Fallacy
A.D. can be viewed as
confusing “” with “”
LogiCola
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LogiCola is a free downloadable logic tutorial
program that permits students to challenge
themselves with a variety of logical exercises at a
variety of difficulty levels. (Note that Logicola
uses 0 & 1 rather than T & F, and lays out its
truth tables differently than we have done here.)
The program is written by a college professor and
intended as a companion to his own college-level
logic course. It is thus fairly high in content and
low on user frills, but with a little patience and
practice, anyone can get the “knack” of using it.
http://www.jcu.edu/philosophy/gensler/lc/index.htm