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Tutorial 2. Propositional Logic: Syntax

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USC Logic Web

on 2 August 2014

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Transcript of Tutorial 2. Propositional Logic: Syntax

SYNTAX
Propositional Logic:
Mention
Use
&
Consider the following pair of sentences:

1. Los Angeles has two words.
2. Los Angeles has nearly four million inhabitants.
In 2, we want to
use
the name of the city to make a claim concerned with
the city of Los Angeles
itself.
In 1, we want to
mention
the name of the city in order to make a claim concerned with
the linguistic expression
that names the city.
In particular, we will often want to explicitly discuss the nature of a linguistic expression regardless of what it may denote or refer to in a language.
In logic, it is important to make a clear distinction between a
linguistic expression
and whatever that expression may
denote.
The difference between the
MENTION
of a term and the
USE
of a term (which is just the difference between the occurrences of
'Los Angeles'
in Sentence 1 and Sentence 2) allows us to make this distinction.
We will use
quotation marks
to make it clear that we intend to mention an expression and not merely use it.
Notice that this is a MENTION, not a USE. The
city of Los Angeles
does not occur in Sentences 1 and 2, but the
linguistic expression
which names it does.
By placing a linguistic expression in quotation marks, we create a name for it. Thus, in this example, 'Los Angeles' is a name for the name of the city of Los Angeles, and can be used to properly express the claim that the name of the city is composed of two words.
3. 'Los Angeles' has two words.
Examples
Mention in English
4. 'Barbara' is not my name.
5. 1 is a number, but '1' is an Arabic numeral.
6. 'Colorless green ideas sleep furiously' is a
grammatical sentence of English.
7. * Buffalo buffalo, but the sentence 'buffalo buffalo
buffalo' is not true.
Curious?
http://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffalo_buffalo_buffalo_Buffalo_buffalo
*
a note
on
Propositional Logic.
We will now introduce a
formal language
that will allow us to study the structure of a certain class of valid arguments. This is the language of
We will proceed in three stages
when we introduce a formal language:
We will specify the
syntax
or
grammar

of the language, which will tell us which strings of symbols count as sentences of the language.
We will specify a
range of interpretations
for the language, which will tell us how to interpret the formal language and what it is for a sentence of the language to be true under some interpretation of the language.
We will offer a
translation manual
, which will allow us to move from sentences of English to sentences of the formal language.
finer
Think of the language of Propositional Logic as an instrument for the study of arguments under a resolution.
When we use this instrument, we will abstract from various aspects of their formulation in English to focus on the formal skeleton of the argument.
Take, for example, one of the arguments we discussed in the previous tutorial.
Example 1
If the battery is dead, then the laptop will not work. The battery is dead. Therefore the laptop will not work.
Example 2
The laptop will not work. This is because the laptop will not work if the battery is dead, and the battery is dead.
Example 3
The battery is dead. That means the laptop will not work, for the laptop will not work if the battery is dead.
All three are stylistic variants of the same argument, which we can express in the language of propositional logic as follows:
1.
2.
3.
In order to present the
syntax (or grammar) of Propositional Logic,
we will proceed in two
stages:
We will specify the
vocabulary
of Propositional Logic, which is the stock of symbols we can use to form well-formed sentences of the language.
We will specify
strict rules for the formulation of sentences
of Propositional Logic out of symbols in the vocabulary of the language.
We will address the first stage in this tutorial and will take up the second and third stages in tutorials 3 and 4, respectively.
( )
The Vocabulary of Propositional Logic
The stock of symbols of the language of propositional logic includes:
Sentence Letters
(with or without subscripts)
Sentence letters will be used to
translate simple sentences of English
. The language of Propositional Logic will allow us to combine
sentence letters
with
connectives
in order to form
more complex sentences
.
Propositional Connectives
Parentheses
( , )
Propositional Connectives
The intended interpretation of each of these symbols ( ) is given by five connectives used in in English to form more complex sentences out of simpler ones:
Propositional Connectives and English Connectives
Name of the connective
English connective
Symbol
negation
disjunction
conjunction
material conditional
material biconditional
'it is not the case that'
'or'
'and'
'if... then...'
'... if and only if ...'
'it is not the case that'
+
'snow is blue'
=
'it is not the case that snow is blue'
'snow is white'
+
'or'
+
'grass is green'
=
'snow is white or grass is green'
'snow is white'
+
'and'
+
'grass is green'
=
'snow is white and grass is green'
'if ... then ...'
+
'2 is prime'
+
'a square root of 4 is prime'
=
'if 2 is prime, then a square root of 4 is prime'
'... if and only if...'
+
'2 is prime'
+
'a square root of 4 is prime'
=
'2 is prime if and only if a square root of 4 is prime'
We now know that there are three kinds of symbols in the vocabulary of Propositional Logic (PL). The next task is to specify strict rules for the formation of sentences of the language.
For example, in English, there are rules which tell us that
'snow is white and and and and'
is
not

a sentence, but that the examples considered earlier
are

sentences. The
syntactic
rules for Propositional Logic will be much more straightforward than those for English.
( )
Sentences of Propositional Logic (PL)
All sentence letters are sentences of PL
If and are sentences of PL, then each of the following are sentences of PL:
Nothing else is a sentence of PL.
a.
b.
c.
d.
e.
These rules allow us to associate a
construction tree
with each sentence. These tree for a given sentence describes how the sentence has been formed out of simpler sentences.
Example 1
Notice that if a formula is not merely a sentence letter, then it has a
main connective
, which is the last connective added in the construction. The main connective appears only at the very 'root' of the construction tree for the formula.
Sentence letters are the most basic constituents of any sentence of PL, so every branch of the construction tree for a sentence will end in a sentence letter.
Example 2
Construction Tree
Construction Tree
The main connective in this formula is a
disjunction.
Example 3
Construction Tree
The main connective in this formula is the
conditional
.
Example 4
Construction Tree
The main connective in this formula is a
negation
.
A string of symbols is
not
a sentence of PL if it cannot be constructed out of sentence letters, connectives, and parentheses according to the strict rules of formation detailed so far. We can identify problem
atic strings of

symbols when we set out to provide
a construction tree
for the formula in question.
Example 1
Formation Failures
The question mark indicates that none of our rules of sentence formation can be used to justify the construction of the relevant string of symbols. In this case, the problem is on the first level of the tree: none our rules allow us to construct a sentence by merely placing a sentence letter between two parentheses.
Not a sentence of PL.
(Remember, each node of the construction tree for a sentence must itself be a well-formed sentence.)
Example 3
Formation Failures
Although Rule 2 (Clause b) would allow us to construct a formula from and by inserting an arrow and placing the result between two parentheses, none of the formation rules allow us to construct a formula by inserting an arrow and adding
only a right parenthesis
to the resulting string of symbols.
?
?
Example 3
Formation Failures
?
?
?
?
There are two potential construction trees for this formula, each problematic.
In both cases, the problem arises because the formation rules do not allow us to form a sentence by
merely
placing an arrow between two sentences. Clauses (b)-(e) require that we add left and right parentheses whenever we insert a logical connective between sentences, in order to avoid ambiguity.
( )
Two Notational Conventions
We often leave off parentheses when we write arithmetical equations involving binary operations like
+
and
x
. For example, the arithmetical expression '2 + 3 x 5' may seem ambiguous between (2+(3 x 5) and ((2+3) x 5) depending on which operation is applied first. In practice, however, we are never confused because we take
'2 + 3 x 5' to abbreviate: '2+(3 x 5)'
and
'2+(3 x 5)' to abbreviate: '(2+(3 x 5))'
This is because we take two
notational conventions
for granted in the language of arithmetic.
One may omit the outer parentheses when forming an arithmetical term out of + or x and simpler arithmetical terms.
One may omit the inner parentheses when there is a unique way to restore them on the assumption that multiplication is applied before addition.
'
(
2 + (3 x 5)
)
'
'2 +
(
3 x 5
)
'
Notational Conventions
Arithmetic
With these in mind, we will adopt two parallel conventions for the language of propositional logic.
that the second convention does not allow us to omit the inner parentheses in
Notice
'2+ (2+2)'
Otherwise, the expression ' 2 + 2 + 2 ' would be ambiguous between:
((2+2) + 2)
and
(2+ (2+2))
That is, there would not be a unique way to restore the parentheses.
One may omit the outer parentheses of a complex formula and write (for example) as an abbreviation for the well-formed formula .
One may omit inner parentheses when there is a unique way to restore them on the assumption that conjunction and disjunction take precedence over the conditional and the biconditional.
Notational Conventions
Propositional Logic
So, for example, we will take
to abbreviate
and
to abbreviate
that the second notational convention does not allow us to omit the inner parentheses in expressions like
Notice
As in the arithmetical cases, there is no unique way to restore the parentheses given only the assumption that conjunction and disjunction take precedence over the conditional and biconditional.
(How would you go about finding the construction tree for these formulas? Without the parentheses, we are unable to identify the main connective or distinguish the simpler sentences which compose the more complex one.)
It is important to remember that these
conventions

do not belong to the
official syntax
of the language. It is still true that is
not
a well-formed formula of the language of propositional logic; it is just that we have

agreed to let it abbreviate
a certain well-formed formula of the language.
We have now
specified a formal language
and have offered
strict rules of sentence formation
, which should allow us to determine whether or not a given string of symbols of the language counts as a sentence. In the next tutorial, we will explain how to
interpret
this formalism and how to define important logical concepts such as
consistency
and
validity
.
Formation rules for
Full transcript