Download `A`, `B`

Sentential Variables (Sentential Schematic Letters,
Sentence Letters):
‘A’, ‘B’, ‘C’, ‘D’, ‘A′′ ’, ‘B′′ ’, ‘C′′ ’, ‘D′′ ’,… etc.
Symbols standing for unspecified sentences.
Compare:
For every two numbers x and y , there exists a number, x ⋅ y ,
which is their product.
For every two sentences, A and B, there exists a sentence, A∧
∧ B,
which is their conjunction.
Note: x⋅⋅ y = y⋅⋅ x, but, in general, A∧
∧ B ≠ B∧
∧ A.
For example:
‘Jack went to the movie and Jill went home’
≠
‘Jill went home and Jack went to the movie’
A∧
∧ B and B∧
∧ A are logically equivalent,
but, in general, they are different.
A∧
∧ B = B∧
∧ A iff (if and only if) A = B.
I.1
Sentential Expressions:
Either sentential variables, or expressions obtained by
combining sentential variables (in the appropriate way) with
connective signs.
For example:
‘A∧
∧ B’
I.2
Negation:
· For every sentence A there is a sentence that is the negation
of A.
The negation of A is written as:
¬A
‘¬
¬ ’ is the negation sign. It denotes ¬ : the negation operation.
Note: ‘Negation’ refers to the operation and also to the resulting
sentence. The sentence ØA is called the negation of A.
Negation can be applied to any sentence. Applying it to ¬ A, we
get:
¬ ¬ A.
Reapplications of negation yield:
¬ A, ¬ ¬ A, ¬ ¬ ¬ A, ¬ ¬ ¬ ¬ A, …etc. ad infinitum.
I.3
Compare: For every number, x , there is a number, –x, that is the
negative of x. But:
– – x = x , while ØØ A ¹ A (always!)
All the sentences ØA, ØØA, ØØØA, ØØØØA, …
are different.
The Interpretation (semantics) of ¬ :
· If the value of A is T (true), the value of ¬ A is F (false).
· If the value of A is F, the value of ¬ A is T.
Regarding T and F as opposite values, we can say that the effect of
Ø is to toggle (reverse) the truth-value.
I.4
Conjunction:
· For every sentences A and B, there is a sentence
that is the conjunction of A and B.
The conjunction of A and B is written as:
A∧
∧B
‘∧
∧ ’ is the conjunction sign; it denotes ∧ : the conjunction operation.
The Interpretation (semantics) of ∧ :
· If the value of A is T and the value of B is T the value of
A ∧ B is T.
· In all other cases, the value of A∧
∧ B is F.
Note: ‘All other cases’ covers three cases:
A gets T B gets F
A gets F B gets T
A gets F B gets F
I.5
Repeated Applications and Grouping
A ^ :B
is the conjunction of A and :B.
:A ^ B can be read in two ways:
(i) The conjunction of :A and B: (:A) ^ B
(ii) The negation of A ^ B: :(A ^ B)
(i) and (ii) are di®erent sentences. Can also di®er in truth-value:
If B gets F, then, independently of the value of A, (:A) ^ B gets
F, but :(A ^ B) gets T.
(:A) ^ B: Scope of the negation is A.
:(A ^ B): Scope of the negation is A ^ B.
(:A) ^ (:B):
Scope of the ¯rst application of :' is A .
Scope of the second application of : is B
:((:A) ^ (:B)):
Scope of the ¯rst : is (:A) ^ (:B).
Scope of the second : is A
Scope of the third : is B
I.6
6
Scopes of conjunctions:
Each application of ^ has two scopes, a left scope and a right scope.
(:A) ^ B:
Left scope of ^ is :A; right scope of ^ is B.
:(A ^ B):
Left scope of ^ is A; right scope of ^ is B.
Parentheses are used, whenever needed to determine the way of
reading the expression. For convenience we use also square and
curly brackets.
We can rewrite
:((A ^ B) ^ :(C ^ D))
as:
:[(A ^ B) ^ :(C ^ D)]
I.7
7I.7
6
Grouping Conventions
In algebra:
`¡3 + 4' is read as `(¡3) + 4', not as `¡(3 + 4).
`3 ¢ 4 + 5' is read as `(3 ¢ 4) + 5', not as `3 ¢ (4 + 5)'
The negative sign, `¡', and the multiplication sign, `¢', bind stronger
than the addition sign, `+'.
In logic: `:' binds stronger than any of the other connective signs.
When some parentheses are missing, ¯x the scopes
of the negation signs to be the smallest scopes that
are consistent with the given expression.
`:A ^ B' is read as: `(:A) ^ B' .
The scope of : is A
`::A ^ :(B ^ C)'
is read as: `(::A) ^ :(B ^ C)' .
The scope of the ¯rst (leftmost) negation is :A; the
scope of the second is A the scope of the third is
B ^ C.
`:(:A ^ :B) ^ C' is read as: [:((:A) ^ :B)] ^ C .
The scope of the ¯rst negation is (:A) ^ :B; the
scope of the second is A; the scope of the third is B.
I.8
I
I.8
Homework 2.1
Insert parentheses in the following expressions according to the
grouping convention, so as to ensure unique readability. (Do not
add parentheses if there is no danger of ambiguity.)
Having done this, write down the scopes of all occurrences of negations in 4, and all the left and right scopes of the occurrences of
conjunctions in 2. (In each case start from the leftmost occurrence.)
1. :(:A ^ :(B ^ A))
2. :(:A ^ :(B ^ C)) ^ (A ^ B)
3. :(A ^ (:A ^ :B))
4. :(A ^ (:A ^ :C)) ^ ::B
5. C ^ :(C ^ :(A ^ C))
6. A ^ :(C ^ (:C ^ B))
I.9
9
Truth-Tables
The truth-table for negation:
A
T
F
:A
F
T
The truth-table for conjunction:
A
T
T
F
F
B A^B
T
T
F
F
T
F
F
F
Note: The use of `A' and `B' is of no particular signi¯cance. We
could have used any other sentence letters:
C A C ^A
T T
T
T F
F
F T
F
F F
F
I.10
Note: The order of rows is not essential; they can be rearranged.
The columns can be rearranged, provided that they retain their
headings. For example:
A
F
T
A
T
F
F
T
:A
T
F
B A^ B
F
F
F
F
T
F
T
T
B A^B
T
T
F
F
F
F
T
F
A
T
F
T
F
Truth-table showing the values of several sentences:
A
T
T
F
F
B :B
T F
F T
T F
F T
A ^ :B
F
T
F
F
Truth table showing values of iterated negation sentences:
A :A
T F
F T
::A :::A
T
F
F
T
. . .
. . .
. . .
I.11
A truth table can contain a column for every sentence in some
arbitrary collection It should have a column for every sentential
variable occuring in any of these sentences ! The truth-values
of sentential variables that do not occur in a sentence have no
e®ect on the value of the sentence.
A
T
T
T
T
F
F
F
F
B
T
T
F
F
T
T
F
F
C C ^ :B
T
F
F
F
T
T
F
F
T
F
F
F
T
T
F
F
:(C ^ :B) :A ^ :(C ^ :B) A ^ (C ^ :B) :(A ^ (C ^ :B))
T
F
F
T
T
F
F
T
F
F
T
F
T
F
F
T
T
T
F
T
T
T
F
T
F
F
F
T
T
T
F
T
Note: The number of rows in a truth-table with n sentential
variables is 2 n.
Homework 2.2
Write down the truth-tables for the sentences of Homework 2.1,
after inserting the required parentheses.
I.12