Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docx

Chapter 1: Logic and Proof
Propositional Logic Semantics
Propositional variables: p, q, r, s, ... (stand for simple
sentences)
T: any proposition that is always true
F: any proposition that is always false
Compound propositions: formed from propositional variables
and logical operators (all binary except negation):
Negation ¬
Conjunction ∧
Disjunction ∨
Implication →
Biconditional ↔
Exclusive Or ⊕
Truth Tables: assign all possible T, F to all possible variables,
and determines all possible T, F of compound propositions; with
n variables there are 2n rows in the table
Negation changes T to F and vice versa
Conjunction is only T if both conjuncts are T
Disjunction is only F is both disjuncts are F
Implication is only F is the antecedent is T and the consequent
is F
Biconditional is only true if they have the same tvalue
Exclusive Or is only T if they differ in tvalue
Two (compound) propositions are equivalent (≡) iff they always
have the same tvalue (see also below)
English translations:
Conjunction: and, but, although, yet, still, ...
Disjunctions: or, unless
Implication: if, if ... then, only if, when, implies, entails,
follows from, is sufficient, is necessary, when, whenever
Biconditional: if and only if, just in case, is necessary and
sufficient

A set of propositions is consistent iff there is some assignment
of tvalue that makes all T
A set of propositions is inconsistent iff there is no assignment
of tvalue that makes all T
A tautology is a compound propositions that is always T
A contradiction is a compound propositions that is always F
A contingency is a compound propositions that is sometimes T,
sometimes F
A compound proposition is satisfiable iff some assignment of
tvalues make it T
A compound proposition is unsatisfiable iff no assignment of
tvalues make it T
Two compound propositions p and q are logically equivalent iff
p ↔ q is a tautology
Common equivalences:
DeMorgan’s Laws (Dem)
¬(p ∨ q)≡¬p ∧ ¬q
¬(p ∧ q)≡¬p ∨ ¬q
Identity Laws (Id)
p ∧ T ≡p
p ∨ F ≡p
Domination Laws (Dom)
p ∨ T ≡T
p ∧ F ≡F
Idempotent Laws (Idem)
p ∨ T ≡T
p ∧ p ≡p
Double Negation Law (DN)
¬(¬p) ≡ p
Negation Laws (Neg)
p ∨ ¬p ≡T
p ∧ ¬p ≡F
Commutative Laws (Comm)
p ∨ q ≡q ∨ p
p ∧ q ≡q ∧ p

Associative Laws (Assoc)
(p ∨ q) ∨ r ≡p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡p ∧ (q ∧ r)
Distributive Laws (Dist)
p ∨ (q ∧ r) ≡
(p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡
(p ∧ q) ∨ (p ∧ r)
Absorption Laws (Abs)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Conditional Laws (Cond)
p →q≡ ¬p ∨ q
¬(p →q)≡ p ∧ ¬q
Biconditional Law (Bicond)
p ↔ q ≡ (p →q) ∧ (q →p)
Quantifier Negation (QNeg)
¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x )
¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x )
Predicate and Relational Logic (Quantificational Logic, First
Order Logic): Semantics
Variables: x, y, z, ...
Predicates/Relations, Propositional Functions: P(x), M(x),
Q(x,y), S(x,y,z), ...
Constants: a, b, c, 0, -1, 4, Socrates, ...
Domain (U): set of things the variables range over
Propositional functions are neither T nor F; however, if all the
variables are replaced by constants they become propositions
and therefore T or F
Universal Quantifier, “For all x”, symbol: x
Existential Quantifier, “There exists an x”, “For some x”,
symbol: x
Some Quantifier Equivalences (DeMorgan’s):

An assertion involving predicates and quantifiers is valid iff it
is true: 1) for all domains, and 2) for all propositional functions
An assertion involving predicates and quantifiers is satisfiable
iff it is true: 1) for some domains, and 2) for some propositional
functions
Proofs: Syntactic Rules
An argument is valid iff it has a valid argument form
An argument form is valid iff it is impossible for the premises
to be true and the conclusion false
A argument form isinvalid iff it IS possible for the premises to
be true and the conclusion false (a counterexample exists)
Inferences rules are simple valid argument forms that can be
used to construct more complex arguments.
Each propositional inference rule has associated with it a
tautology: the conjunction of the premises implies the
conclusion
Modus Ponens (MP)
Modus Tollens (MT)
Hypothetical Syllogism (Hyp)
Disjunctive Syllogism (Disj)
Addition (Add)
Simplification (Simp)
Conjunction (Conj)

Resolution (Res)
Universal Instantiation (UI – unrestricted)
Universal Generalization (UG – restricted: c must be new to the
proof)
Existential Instantiation (EI – restricted: c must be new to the
proof)
Existential Generalization (EI – unrestricted)
Proofs
Many proofs involve conditionals: to prove if p, then q, assume
p is true and deduce q using the above rules.
A theoremis shown to be true by using: 1) definitions, 2) other
theorems, 3) axioms (which are assumed to be true), and 4) the
rules of inference.
A lemma is an intermediate theorem proven to aid tin the proof
of the final theorem.
A corollaryis a theorem that follows quickly from a theorem.
A conjecture is a non-theorem that one thinks is true.
A direct proof uses the 4 tools used for proving theorems.
An indirect proof is:
A proof that proves the contrapositive instead, or
A proof by contradiction, which assumes the negation of that to
be proved and deduces a contradiction
Disproof by counterexample shows the invalidity of an
argument by finding a domain that makes all of the premises

true and makes the conclusion (conjecture) false.
Existence proofs prove the existence of an entity with a certain
property (usually mathematical)
Uniqueness proofs prove BOTH the existence and uniqueness of
an entity with a certain property (usually mathematical): show if
there are two with the property, then they are really the same
(i.e., suppose there are 2, x and y, and show x=y)
Number Theory
n is an even integer iff there exists an integer k, k ≠ 0, and n=2k
n is an odd integer iff there exists an integer k, such that
n=2k+1 or n = 2k-1
4
Why we are looking at the ‘value’ of college all wrong
By Valerie Strauss November 1, 2014
St. John’s College in Annapolis. (Photo by Mark Gail/The
Washington Post)
There is a national debate about whether going to college is
worth the increasingly hefty price tag. The argument against it
is that many students come out four — or five or six — years
later and can’t find a job that pays a lot, or they can’t find a job
at all. But in this post, St. John’s College President Christopher
B. Nelson argues that “education and economics are essentially
incompatible” and that the economic lens is the wrong way to
judge education. Nelson has been president of St. John’s, in
Annapolis, Maryland, since June 1991. Before that he, practiced
law in Chicago for 18 years and was chairman of his law firm.
As university president, he has become a national spokesman
for the liberal arts. St. John’s, with a campus in Annapolis and

in Santa Fe, N.M., has an unusual liberal arts curriculum, one
based on discussion of works from the Western Canon.
By Christopher B. Nelson
As college admission deadlines loom, new lists and rankings
proliferate along with reports questioning the “value” of a
college education. The obsession with quantification is rooted
in a habit of applying economic categories to everything. Yet
education and economics are essentially incompatible. The lens
of economics distorts our judgment about the true worth of
higher education.
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The incompatibility rests on a fundamental difference between
economics and education. Begin with the idea of economics as
the science of scarcity. The price of a commodity is largely
dependent on its relative scarcity. Economic value increases
when a commodity becomes scarce, and a commodity that is not
scarce will become scarce if it is distributed widely and used up
indiscriminately. Scarcity is basic to the world view of
economics—so much so that the language of economics speaks
as though scarcity and value are inseparable.
The things that matter most in education, though, do not fit this
paradigm. They are not scarce, and yet they are extremely
valuable—indeed they are among the most valuable in human
life. They do not become scarce by being shared. Instead, they
expand and grow the more they are shared.
One of these things is knowledge. Knowledge has never been
exhausted by spreading it to more and more people. Today, it is
more abundant than at any time in the past, and it reproduces
more prolifically as it is shared. Moreover, technology has made
it possible to store knowledge efficiently and to access it
cheaply. No wonder that the economic paradigm is having
difficulty capturing and domesticating it into a well-behaved
economic commodity.
This is disconcerting for institutions that think of themselves

primarily as providers of information. If the knowledge is out
there, freely accessible, why then should anyone pay large sums
of money to a knowledge gatekeeper—let alone go into debt?
Today, the confrontation between free technological access and
proprietary gate-keeping is leading to turmoil about new models
of delivery in higher education.
But the idea that a college or university is a purveyor of
information is a misplaced economic metaphor. Education is not
information transfer. The educated college graduate is not
simply the same person who matriculated four years earlier with
more information or new skills. The educated graduate is a
different person—one who has developed the innate human
capacity for learning, to the point of controlling it. The
educated graduate is an independent learner, able to seek out
answers to whatever questions arise, and able to direct his or
her own learning in accordance with the challenges that life
presents in the circumstances of his or her own life.
The maturation of the student—not information transfer—is the
real purpose of colleges and universities. Of course, information
transfer occurs during this process. One cannot become a master
of one’s own learning without learning something. But
information transfer is a corollary of the maturation process, not
its primary purpose. This is why assessment procedures that
depend too much on quantitative measures of information
transfer miss the mark. It is entirely possible for an institution
to focus successfully on scoring high in rankings for
information transfer while simultaneously failing to promote the
maturation process that leads to independent learning.
It is, after all, relatively easy to measure the means used in
getting an education, to assess the learning of intermediate
skills that prepare one for a higher purpose—things like
mastering vocabulary and spelling, for instance, which help one
to communicate. It is also easy to measure the handy,
quantifiable by-products of a college education, like post-
graduate earning, either in the short term or long term. But both
of these kinds of measures fail to speak to education’s proper

end—the maturation of the student.
We need to move away from easy assessments that miss the
point to more difficult assessments that try to get at the
maturation process. The Gallup-Purdue Index Report entitled
“Great Jobs, Great Lives” found six crucial factors linking the
college experience to success at work and overall well-being in
the long term:
1. at least one teacher who made learning exciting
2. personal concern of teachers for students
3. finding a mentor
4. working on a long-term project for at least one semester
5. opportunities to put classroom learning into practice through
internships or jobs
6. rich extracurricular activities
We should turn all our ingenuity toward measuring factors like
these, difficult as that task might be, and use these results to
push back against easy assessments based on the categories of
economics.
Unless we stop taking the easy way, unless we get past our habit
of interpreting everything in economic terms, we will never
grasp the true value of a college education.
ESSAY #1—ILLUSTRATION
An author who writes illustrative essays exemplifies a central
idea through examples. Good examples are used to further
clarify and understand the author’s message while helping to
build a connection with readers. In formulating illustrative
essays, a writer either focuses on one specific example or uses

several examples in explaining the central idea.
Genre/Medium:Illustration/Typed essay
Purpose:The writer of an illustration essay uses examples to
reveal the essential characteristics of a topic and/or to reinforce
a thesis. For this assignment you will write an illustration essay
that follows the five-paragraph essay format on one of the essay
topics listed below.
Format:Your five-paragraph essay must contain a concrete
closed (parallel, three-point) thesis statement at the end of the
first paragraph and follow the MLA guidelines.
Audience:This essay will target a scholarly audience.
Therefore, your language and style should meet the intellectual
needs of individuals who read on a collegiate level. Are you
writing to an audience that consists of classmates and the
professor who have experienced similar obstacles in life
towards becoming literate scholars? As you think about your
audience, write to pique the interest of your audience by
considering what your readers already know and what they need
to know.
Stance:What attitude will you convey through illustration? Will
you portray yourself as serious, intellectual, passionate,
desensitized, or sarcastic? Think about your stance and convey
your stance throughout your essay.
Instructions:Write a five-paragraph illustration essay on one of
the topics below.
(A). In the essay “Why We Are Looking at the ‘Value’ of
College All Wrong,” Christopher
Nelson defines the educated graduate as an independent
learner who seeks “out
answers to whatever questions arises” while applying the

lessons he/she learns in
college to tackle life’s challenges.
Write an essay explaining ways you, as a college student,
are evolving into an
“independent learner.” Provide evidence from Nelson’s
essay to support your
assertions.
The Foundations: Logic and Proofs
Chapter 1, Part III: Proofs
Rules of Inference
Section 1.6
Revisiting the Socrates Example
We have the two premises:
“All men are mortal.”
“Socrates is a man.”
And the conclusion:
“Socrates is mortal.”
How do we get the conclusion from the premises?

The Argument
We can express the premises (above the line) and the conclusion
(below the line) in predicate logic as an argument:
We will see shortly that this is a valid argument.
Valid Arguments
We will show how to construct valid arguments in two stages;
first for propositional logic and then for predicate logic. The
rules of inference are the essential building blocks in the
construction of valid arguments.
Propositional Logic
Inference Rules
Predicate Logic
Inference rules for propositional logic plus additional inference
rules to handle variables, predicates, and quantifiers.
Arguments in Propositional Logic
A argument in propositional logic is a sequence of propositions.
All but the final proposition are called premises. The last
statement is the conclusion.
The argument is valid if the premises imply the conclusion. An
valid argument form is an argument that is valid no matter
what propositions are substituted into its propositional
variables.
If the premises are p1 ,p2, …,pn and the conclusion is q then

(p1 ∧ p2 ∧ … ∧ pn ) → q is a tautology.
Inference rules are all simple valid argument forms that will be
used to construct more complex argument forms.
Rules of Inference for Propositional Logic: Modus Ponens (MP)
Example:
Let p be “It is snowing.”
Let q be “I will study discrete math.”
“If it is snowing, then I will study discrete math.”
“It is snowing.”
“Therefore , I will study discrete math.”
Corresponding Tautology:
(p ∧ (p →q)) → q
Modus Tollens (MT)
Example:
Let p be “it is snowing.”
“If it is snowing, then I will study discrete math.”
“I will not study discrete math.”
“Therefore , it is not snowing.”

(¬q∧ (p →q))→¬p
Hypothetical Syllogism (Hyp)
Example:
Let p be “it snows.”
Let r be “I will get an A.”
“If it snows, then I will study discrete math.”
“If I study discrete math, I will get an A.”
“Therefore , If it snows, I will get an A.”
((p →q) ∧ (q→r)) → (p→ r)
Disjunctive Syllogism (Disj)
Example:
Let p be “I will study discrete math.”
Let q be “I will study English literature.”
“I will study discrete math or I will study English literature.”
“I will not study discrete math.”
“Therefore , I will study English literature.”

(¬p∧ (p ∨ q))→q
Addition (Add)
Example:
Let q be “I will visit Las Vegas.”
“I will study discrete math.”
“Therefore, I will study discrete math or I will visit
Las Vegas.”
p →(p ∨ q)
Simplification (Simp)
Example:
“I will study discrete math and English literature”
“Therefore, I will study discrete math.”
(p∧ q) →p

Conjunction (Conj)
Example:
“I will study discrete math.”
“I will study English literature.”
“Therefore, I will study discrete math and I will study English
literature.”
((p) ∧ (q)) →(p ∧ q)
Resolution (Res)
Example:
Let r be “I will study English literature.”
Let q be “I will study databases.”
“I will not study discrete math or I will study English
literature.”
“I will study discrete math or I will study databases.”
“Therefore, I will study databases or I will study English
literature.”
((¬p ∨ r ) ∧ (p ∨ q)) →(q ∨ r)
Resolution plays an important role in AI and is used in Prolog.

Arguments in Propositional Logic
Each simple inference rule embodies an argument form that is
valid:
It is impossible for the premises to be true and the conclusion
false
Alternatively, in a truth table every row in which ALL of the
premises are T, the conclusion is also T
A propositional argument with a valid form is a valid argument
To show a propositional argument from invalid you need only
show that:
It is possible for the premises to ALL be true and the conclusion
false
Alternatively, in a truth table there is at least one row in which
ALL of the premises are T and the conclusion F
Using the Rules of Inference to Build Valid Arguments
A valid argument is a sequence of statements. Each statement
is either a premise or follows from previous statements by rules
of inference. The last statement is called conclusion.
A valid argument takes the following form:
S1
S2
.
.
.
Sn
C
Valid Arguments

Example 1: From the single proposition
Show that q is a conclusion.
Solution
:
Note: Conjunction should be Simplification
Valid Arguments
Example 2:
With these hypotheses:
“It is not sunny this afternoon and it is colder than yesterday.”
“We will go swimming only if it is sunny.”
“If we do not go swimming, then we will take a canoe trip.”
“If we take a canoe trip, then we will be home by sunset.”
Using the inference rules, construct a valid argument for the
conclusion:
“We will be home by sunset.”

Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docx

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