The Syntax and Semantics of PL Languages in General All languages have a set of symbols rules for constructing compound constructions out of atomic constructions and meanings assigned to the significant units ID: 328308
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Slide1
The Language of Propositional Logic
The Syntax and Semantics of PLSlide2
Languages in General
All languages have a set of symbols, rules for constructing compound constructions out of atomic constructions, and meanings assigned to the significant units.
For example, the letter ‘A’ is part of English, but not part of Hindi.
For example, English is a Subject-Verb-Object language, while Arabic is Subject-Object-Verb.
For example, ‘snow’ means snow in English, but ‘schnee’ means snow in German.
The syntax of a language is the grammar of the language.
The semantics of a language is the meaning of the significant parts. Slide3
The Language of Propositional Logic
Propositional Logic, PL, is a formal language, which has a set of symbols, a syntax, and a semantics. It is not a natural language, like English.
It is possible to translate sentences of most natural languages, such as Greek, English, German, French, etc… into PL.
PL is a language that focuses on a small set of expressions. These expressions are the words used to connect propositions (sentences) to one another: ‘and’, ‘or’, ‘if…,then’, ‘not’, ‘if and only if’, and combinations of them. Slide4
The Syntax of PL
Symbols:
Propositional Letters:
P, Q, R, S, T, U, V, W, X, Y, and Z
Logical Operators:
‘
’ arrow
‘
’
broken arrow
‘
’ triple bar
‘’ carrot
‘’ wedge
Grouping Symbols:
‘(, )’ parentheses, and ‘[,]’ bracketsSlide5
Object Language vs. Meta-language
The language of PL described previously is the object-language. The object language is the actual language that is used for communicating in the language. For example, just as the word ‘simple’ is part of the object language of English, the formula ‘(P
Q)’ is part of the object language of PL.
The object language of PL must be distinguished from the meta-language for PL. The meta-language for PL is the language used for talking about PL. It is not part of PL, and is primarily used to describe the grammar and meaning of formulas at a level of generality.
The meta-language variables are the lower case English letters:
p
,
q
,
r
,…
z
. Slide6
Rules for Well-Formed Formulas
All propositional letters P….Z are atomic well-formed formulas.
If
p
and
q
are well formed, then so are the following:
p
(
p
q
)
(
p
q
)
(
p
q
)
(
p
q
)
Nothing is a well-formed formula, unless it follows from (1) and (2).Slide7
Examples
Not well-formed
P
QP
R
A
R)
Well-formed
(P
(Q R))
(V
(R S))
(Q (R S))
(S (R T))
(P (R
S))Slide8
The Semantics of PL
PL is a language that only focuses on propositional connectives and operators. In English the main propositional connectives are ‘and’, ‘or’, ‘not’, ‘if…, then..’, and ‘if and only if’.
Since PL is only focused on these terms it only has a semantics for these terms.
The semantics for PL is binary and exclusive. There are only two truth-values:
T
and
F
, and no statement is both
T
and
F.
It is important to note that the semantics of PL is for the logical operators of its language: ‘
’,
‘
’,
‘
’, ‘’, and ‘’. Slide9
Logical Meaning vs. Non-Logical Meaning
L
ogicians
try to give definitions of
the propositional operators of PL that match
perfectly with
the
logical
meaning of ‘and’, ‘or’, ‘not’, ‘if…then…’, and ‘if and only if
’.
However,
there
are many cases in which the English use of, for example ‘and’ or ‘if…,then…’, do not match the definition given to carrot and arrow.
In philosophy of logic one studies what the correct definition of the logical operators should be, and other questions about whether logic is binary and exclusive. Slide10
Truth-Tables
In order to define the logical symbols of PL, one needs to use a truth-table. A truth-table is a table for visually displaying the distribution of truth and falsity across a compound formula given the basic inputs from the atomic letters.
p
q
T
T
T
F
F
T
F
FSlide11
Broken Arrow, Negation
The definition of broken arrow is intended to capture the logical meaning of the word ‘not’, and the function of negation.
The core idea is that the output is the opposite of the input.
p
p
T
F
F
TSlide12
Carrot, Conjunction
The definition of carrot is intended to capture the logical meaning of the word ‘and’, and the function of conjunction.
The core idea is that the output is true only if both inputs are true.
p
q
(
p
q
)
T
T
T
T
F
F
F
T
F
F
F
FSlide13
Wedge, Disjunction
The definition of wedge is intended to capture the logical meaning of the word ‘or’, and the function of disjunction.
The core idea is that the output is true as long at least one input is true.
p
q
(
p
q
)
T
T
T
T
F
T
F
T
T
F
F
FSlide14
Arrow, Material Conditional
The definition of the arrow is intended to capture the logical meaning of the phrase ‘if…., then…’, and the function of material conditional.
The core idea is that the output is false only when the first input is true, and the second input is false.
p
q
(
p
q
)
T
T
T
T
F
F
F
T
T
F
F
TSlide15
Triple Bar, Biconditional
The definition of triple bar is intended to capture the logical meaning of the phrase ‘if and only if’, and the function of biconditional.
The core idea is that the output is true just in case the inputs are the same.
p
q
(
p
q
)
T
T
T
T
F
F
F
T
F
F
F
T