An Introduction to Propositional Symbols and Translation

An Introduction to
Propositional Logic
Translations: Ordinary Language to
Propositional Form
What is a Proposition?
Propositions are the meanings of
I have no money
Ich habe kein geld.
 Meanings are the thoughts, concepts,
ideas we are trying to convey through
speech and writing.
Simple Propositions
Fast foods tend to be unhealthy.
 Parakeets are colorful birds.
(p. 290)
Simple propositions are grammatically
independent expressions of
Compound Propositions
If fast foods tend to be unhealthy, then you
shouldn’t eat them.
 Parakeets are colorful birds, and colorful
birds are good to have at home.
 People are free, if and only if they can
choose their actions and there are
no forces compelling those actions.
The Focus of Propositional Logic
Propositional truth is determined by
consulting typical sources of information.
Propositional logic is determined by
examining how various propositions are
Types of relations between
1 proposition is offered in support of
another (simple argument)
 1 proposition expressed the condition
under which a 2nd proposition is true
(conditional statement)
 1 sentence offers two proposed
alternatives, and a 2nd proposition negates
one of these alternatives (disjunctive
Propositional Symbols
Symbolizing propositions allows us to focus on
the relations between propositions (logic) rather
than the content of those propositions (truth).
P1: UCLA will either raise fees or P1: R or C
cut back on services.
P2: UCLA will not raise fees.
P2: Not R
Con: UCLA will cut back on
Con: C
Module Objectives - 1
Learn how to symbolize complex
 Simple propositions, which express
grammatically independent units of information,
are easy to symbolize: “It is cold” = “C”
 Complex propositions are sentences which
contain 2 or more simple propositions. These can
be more difficult to symbolize.
Module Objectives - 2
Learn how to determine the truth-value of
complex propositions
Remember, an argument can only establish
the truth of it’s conclusion if all its premises
are true (and its reasoning is valid)
• “Fees are rising at UCLA” is either true or false
• “Either fees are rising or services are being cut
back” could be true or false – depending on the
actual situation regarding fees and services at UCLA.
Module Objectives - 3
Create and interpret truth tables for both
propositions and arguments (series of
Truth tables allow us to see all possible
conditions under which a statement could
be true and could be false.
Module Objectives – 4 & 5
Learn to recognize common argument
forms, and know when an argument form
is valid or invalid
Prove that an argument is valid or invalid
when it doesn’t fit a common argument
Basics of Propositional Logic
All arguments are reducible to symbols, which represent either
elements of an argument or ways these elements are put together.
1. All arguments contain statements, by definition. Each
statement is represented by a “propositional variable” –
p, q, r, s
2. All arguments also contain connections, or ways in which
individual propositions are related. Each of these connections
are represented by one of five “operators”:
Putting propositional variables together with operators creates a
“statement form,” or a symbolic blueprint identifying typical structures
of English expressions.
Propositional Operators
~ (“tilde,” negation)
• (“dot,” conjunction)
v (“wedge,” either-or)
Not, it is false that, conjunctions like “don’t”
And, also, but, in addition, moreover
Or, unless
> (“implication” or “conditional,” if,then).
Is a sufficient (or necessary) condition of, ifthen, implies, given that, only if
Ξ (“biconditional,” if and only if)
If and only if, is equivalent to, is a sufficient
and necessary condition of
Note on the “Tilde”
1. All operators except the tilde must
relate at least two propositions.
2. The tilde negates either a proposition
directly, or an operator relating to
propositions (by standing directly
before a parentheses/bracket/etc.) .
Examples of “Tilde” Functions
~ p = not p; p is not true, etc
~ p ● ~ q = p is false and q is false; p
and q are both false
~ ( p ● q ) = not both p and q (maybe one
is true and one false)
Rules for Operator Types - 1
If there is more than one operator (excluding tildes), then
some portion of the statement must be included in
Well-formulated formula Not a well-formulated
formula (non-wff)
Rules for Operator Types - 2
The tilde negates either a proposition directly, or an
operator relating to propositions (by standing directly before
a parentheses/bracket/etc.).
Well-formulated formula Not a well-formulated
formula (non-wff)
~ (p ≡ q) ● r
(p ~ ≡ q) ● r
Working with Propositional Symbols - 1
If I get 80 points on the test, I’ll
get a B on the test.
Partial symbolization
of the expression:
If P, then B (material implication;
conditional statement)
Statement form:
p > q (in variables)
Working with Propositional Symbols - 2
p. 290, #5
Psychologists and psychiatrists
do not both prescribe
prescription drugs.
Partial symbolization
of the expression:
Not both G and T → conjunction
Statement form:
Working with Propositional Symbols - 3
p. 290, #11
If Internet use continues to grow, then
more people will become cyberaddicts
and normal human relations will
Partial symbolization
of the expression:
If I, then both C and D → mixed
Statement form:
p > (q●r)
Tips for Translation
Use “clue words”:
 “If,
then”; “on the condition that”: >
 Both; and; also; etc: ●
 Either, or; or maybe both: v
 If one, then the other; if and only if;
always occur together:
 Negation;
it is not true that, not: ~