Transcript Symbolic Language and Basic Operators

Symbolic Language and Basic
Operators
Kareem Khalifa
Department of Philosophy
Middlebury College
Overview
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Why this matters
Artificial versus natural languages
Conjunction
Negation
Disjunction
Punctuation
Sample Exercises
Why this matters
 Symbolic language allows us to abstract
away the complexities of natural
languages like English so that we can
focus exclusively on ascertaining the
validity of arguments
 Judging the validity of arguments is
an important skill, so symbolic
languages allow us to focus and hone
this skill.
 Symbolic language encourages
precision. This precision can be
reintroduced into natural language.
More on why this matters
 You are learning the conditions
under which a whole host of
statements are true and false.
 This is crucial for criticizing
arguments.
 It is a good critical practice to
think of conditions whereby a
claim would be false.
Artificial versus natural languages
 Symbolic language (logical syntax) is an artificial
language
 It was designed to be as unambiguous as
possible.
 English (French, Chinese, Russian, etc.) are natural
languages
 They weren’t really designed in any strong
sense at all. They emerge and evolve through
very “organic” and (often) unreflective cultural
processes.
 As a result, they have all sorts of ambiguities.
 The tradeoff is between clarity and expressive
richness. Both are desirable, but they’re hard to
combine.
Propositions as letters
 Logical syntax represents individual
propositions as letters.
 When we don’t care what the proposition
actually stands for, we represent it with a
lowercase letter, typically beginning with p.
 When we have a fixed interpretation of a
proposition, we represent it with a capital letter,
typically beginning with P.
 Ex. Let P = “It’s raining.”
 Sometimes, letters are subscripted. Each
subscripted letter should be interpreted as
a different proposition.
Dispensable translation manuals
 Often, the letters are given an
interpretation, i.e., they are mapped onto
specific sentences in English.
 Ex. Let P be “It is raining;” Q be “The streets
are wet,” etc.
 However, this is not necessary. The validity
of an argument doesn’t hinge on the
interpretation.
 If p then q
p
q
Logical connectives: some basics
 A logical connective is a piece of logical
syntax that:
 Operates upon propositions; and
 Forms a larger (compound) proposition out of
the propositions it operates upon, such that the
truth of the compound proposition is a function
of the truth of its component propositions.
 Today, we’ll look at AND, NOT, and OR.
 Khalifa is cunning and cute.
 Khalifa is not cunning.
 Either Khalifa is cunning or he is foolish.
Conjunction
 AND-statements
 Middlebury has a philosophy department
AND it has a neuroscience program.
 Represented either as “” or as “&”
 I recommend “&,” since it’s just
SHIFT+7
 “p & q” will be true when p is true
and q is true; false otherwise.
Truth-tables
 Examine all of the
combinations of
component
propositions, and
define the truth of
the compound
proposition.
p
q
p&q
T
T
T
T
F
F
F
T
F
F
F
F
Subtleties in translating English
conjunctions into symbolic notation
 The “and” does not always appear in
between two propositions.
 Khalifa is handsome and modest.
 Khalifa and Grasswick teach logic.
 Khalifa teaches logic and plays
bass.
More subtleties
 Sometimes “and” in English means “and
subsequently.”
 The truth-conditions for this are the same as “&”,
but the meaning of the English expression is not
fully captured by the formal language.
 Many English words have the same truthconditions as “&” but have additional
meanings.
 Ex. “but,” “yet,” “still,” “although,” “however,”
“moreover,” “nevertheless”
 General lesson: The meaning of a
proposition is not (easily) identifiable with
the truth-functions that define it in logical
notation.
Negation
 Represented by a
“~”
 In English, “not,”
“it’s not the case
that,” “it’s false
that,” “it’s absurd
to think that,” etc.
p
~p
T
F
F
T
Disjunction
 Represented in English by “or.”
 However, there are two senses of
“or” in English:
 Inclusive: when p AND q are true, p OR
q is true
 Exclusive: when p AND q is true, p OR q
is false
 Logical disjunction (represented as
“v”) is an inclusive “or.”
Which is inclusive and which is
exclusive?
 You can take Intro to Logic in the fall
or the spring.
 Exclusive. You can’t take the same
course twice!
 You can take Intro to Logic or
Calculus I.
 Inclusive. You’d then be learned in logic
and in math!
Truth table for disjunction
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
Punctuation
 We can daisy-chain logical connectives
together.
 Either Polly and Quinn or Rita and Sam will not
win the game show.
 If we have no way of grouping propositions
together, it becomes ambiguous
 ~P & Q v R & S
 Logic follows the same conventions as math
{ [ ( ) ] }, though some logicians prefer to
use only ( ( ( ) ) ).
 [(~P&~Q) v (~R&~S)]
A few quirks
 A negation symbol applies to the smallest
statement that the punctuation permits.
 Ex. “~p & q” is equivalent to “(~p) & q”
 It is NOT equivalent to “~(p & q)”
 This reduces the number of ( )
 We can also drop the outermost brackets of
any expression.
 Ex. “[p & (q v r)]” is equivalent to “p & (q v r)”
Lessons about punctuation from
logic
 Make sure, in English, that you
phrase things so that there is no
ambiguity
 Commas are very useful here
 When reading, be especially sensitive
to small subtleties about logical
structure that would change the
meaning of a passage.
Sample Exercise A3 (327)
 ~London is the capital of England &
~Stockholm is the capital of Norway
 ~T & ~F
F&T
F
Sample Exercise A4 (327)
 ~(Rome is the capital of Spain v Paris
the capital of France)
 ~(F v T)
 ~(T)
F
Sample Exercise A9 (327)
 (London is the capital of England v
Stockholm is the capital of Norway) &
(~Rome is the capital of Italy &
~Stockholm is the capital of Norway)
 (T v F) & (~T & ~F)
 (T) & (F & T)
 (T) & (F)
F
Sample Exercise C3 (329)
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Q v ~X
Q v ~F
QvT
T
Exercise C12 (329)
 (P & Q) & (~P v ~Q)
 The first conjunct (P&Q), can only be
true if P = Q = T
 However, this would make the whole
conjunction false. Here’s the ‘proof’:
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(T&T) & (~T v ~T)
(T) & (F v F)
(T) & (F)
F
Exercise D9 (330)
 It is not the case that Egypt’s food
shortage worsens, and Jordan
requests more U.S. aid.
 ~E & J