Semantics: Handout 4
Propositional Logic
Yılmaz Kılıçaslan
October 08, 2009
Introduction
Formal logic requires a precisely specified artificial/formal language which then serves as a
well defined object about which particular claims can be asserted and proved. This is a step of
idealization necessary to obtain general mathematical results. Relevant fragments of natural
language can then be translated into the artificial language to obtain precise results for that
particular fragment. Therefore a primary goal is to construct increasingly rich formal
languages that can represent larger and larger fragments of natural language (though logic per
se is concerned with natural language structure only insofar as it affects logical validity). We
will embark on this enterprise starting with propositional logic, which is the simplest and
most basic logical system there is. However, before giving the full details of a formal
language for propositional logic, we will have a look at some issues to get some initial insight
into the syntax and semantics of such a language.
Truth-Functional Connectives
Propositional logic has connectives and negation as its logical constants. Propositional logic
recruits its connectives from among connectives of natural language that link two sentences
into a new one. However, not every connective from natural language can serve as a logical
connective. The principle of compositionality (Handout 2) requires that the meaning of a
composite sentence depends only on the meanings of the sentences of which it is composed.
As a starting assumption, we consider the meaning of an indicative sentence to be its truth
value, i.e. either true or false. With these restrictions, and can serve as a logical connective
but because cannot. Consider the following sentences:
(1)
John has bumped his head and he is crying.
(2)
John is crying because he has bumped his head.
(3)
John is crying.
(4)
John has bumped his head.
Sentence (1) is true if both (3) and (4) are true, and false if either is false. This is a
manifestation of the fact that the truth value of an and sentence depends only on the truth
values of the two parts of which it is composed. But this does not apply to sentence (2): it
might be the case that John has in fact bumped his head and is indeed crying, but this need not
require (2) to be true (maybe John is crying because Mary doesn’t love him); and conversely,
if (2) is true then John is crying because it is raining is false even if it is raining. Thus, the
truth value of a because sentence depends on more than just the truth values of the two parts
constituting it. Connectives which give rise to sentences whose truth value depends only on
the truth values of the connected sentences are said to be truth-functional. So and is, and
because is not, a truth-functional connective.
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Since we here restrict the meaning of sentences to their truth values, compositionality requires
that we consider only truth-functional connectives. A choice for truth-conditional connectives
for a propositional language is in a sense arbitrary. Our propositional language will include
the two-place connectives ,, , and , which correspond to the English connectives and,
or, if (…then), if and only if, respectively. These connectives together with the one-place
negation operator ¬ constitute the set of logical constants of our language. As logical
variables there are symbols which stand for propositions. These symbols are called
propositional letters or propositional variables. In general we shall designate them by the
letters p, q, and r, where necessary with subscripts as in p1, r2, q3, etc. It is usual to use
different letters for different propositional symbols. The propositional letters and the
composite expressions which are formed from them by means of the connectives are grouped
together as sentences or formulas. We will designate these by means of the letters φ and ψ, etc.
For these metavariables, p, q, and r, there is no convention that different letters must designate
different formulas.
Truth Tables
How the truth value of a composite sentence is formed from the truth values of its
constituents, i.e. their structural semantics, is prescribed in a truth table. The truth tables for
our logical constants are as follows:
Negation:
φ
0
1
¬φ
1
0
Conjunction:
φ
0
0
1
1
ψ
0
1
0
1
(φ  ψ)
0
0
0
1
Disjunction:
φ
0
0
1
1
ψ
0
1
0
1
(φ  ψ)
0
1
1
1
2
Implication:
φ
0
0
1
1
ψ
0
1
0
1
(φ  ψ)
1
1
0
1
Equivalence:
φ
0
0
1
1
ψ
0
1
0
1
(φ  ψ)
1
0
0
1
Each row in a truth table corresponds to a particular model in which we assign particular
interpretation to the logical variables.
The Language L0
Let us now specify the syntax of an artificial propositional language L0, and define the
semantics of this language.
Syntax of L0
A. Basic Expressions
1. Logical Constants: ¬,, , , 
2. Propositional Variables: p, q, r (with or without subscripts)
B. Formation Rules
1. any propositional variable of L0 is a formula of L0.
2. If φ is a formula of L0, then so is ¬φ.
3. If φ and ψ are formulas of L0, then so are (φ  ψ), (φ  ψ), (φ  ψ), (φ  ψ)
4. Nothing else is a formula of L0.
Note on the distinction between object language and metalanguage: in the above specification
of the syntax of L0, symbols in the vocabulary of L0, such as ‘¬’, ‘’, ‘p’, are used to construct
object language expressions, while other symbols, such as ‘φ’, ‘ψ’ are used as variables in the
metalanguage that range over the formulas of L0 as values. In the recursive clause of the
definition of a formula of L0, the two types of symbols appear together, e.g. in the expression
‘¬φ’. In such contexts, we adopt the convention that the object language symbols such as ‘¬’
are used in the metalanguage to name themselves. This autonomous interpretation of object
language symbols eliminates the need to worry about fussy quotation conventions when doing
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logic. Alternatively, it is possible to adopt a slightly more abstract approach, and treat the
symbols ‘¬’, ‘’, ‘p’, etc. as terms in the metalanguage that name particular object language
expressions, but perhaps the expressions named are different from the names. Under this
construal, we never have direct access to the object language symbols themselves, while we
do have terms in the metalanguage that refer to them. In this manner, it is not even necessary
that the object language consist of written symbols. Instead, the object language could be
comprised of, for example, musical notes or patterns of light.
Semantics of L0
A model (or interpretation or structure) for L0 is a function F such that for all propositional
variables  of L0,
F()  {1, 0}
Relative to a particular interpretation F of the propositional variables of L0, we can
inductively define semantic value [φ]F for every formula φ of L0:
1. []F = F(), for all propositional constants 
2. [¬φ]F = 1 iff [φ]F = 0 (and [¬φ]F = 0 otherwise)
3. [φ  ψ]F = 1 iff [φ]F = 1 or [ψ]F = 1
4. [φ  ψ]F = 1 iff [φ]F = 1 and [ψ]F = 1
5. [φ  ψ]F = 1 iff [φ]F = 0 or [ψ]F = 1
6. [φ  ψ]F = 1 iff both [φ]F = 1 and [ψ]F = 1 or both [φ]F = 0 and [ψ]F = 0
Some important logical properties of formulas of L0:
1. φ is a logical truth (tautology), written ╞ φ, iff for all models F, [φ]F = 1. Eg. (p 
¬p)
2. φ is a logical falsehood (contradiction) iff for all models F, [φ]F = 0 Eg. (p  ¬p)
3. φ is contingent iff there is a model F such that [φ]F = 1, and a model F’ such that
[φ]F’ = 0 Eg. p, (p  q)
4. φ and ψ are logically equivalent iff for all models F, [φ]F = [ψ]F . If φ and ψ are
logically equivalent, then ╞ (φ  ψ)
5. ψ is a logical consequence of φ, written φ╞ ψ, iff for all models F, if [φ]F = 1, then
[ψ]F = 1. So given our formal definition of a model F for the language L0, the
foregoing yields a mathematically precise characterization of truth functionally valid
arguments with φ as premise and ψ as conclusion. The original, intuitive idea that it is
‘not possible’ for the conclusion to be false and the premises true can now be
expressed in terms of the nonexistence of formal counter-models.
6. φ╞ ψ iff ╞ (φ ψ). Accordingly φ and ψ are logically equivalent iff φ╞ ψ and ψ╞ φ.
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