Propositional formula
1. Propositional Logic
Propositional formulas
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Propositional formulas are assertions –
they express how things are.
They can be true or false, depending
on the situation.
Propositional formulas:
•x<10
•x<10 → x2<100
•(x=10 ∧ y=12) ! (z=4 ∨
z=5)
•It is raining or snowing.
The lecture
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Propositional formula
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Propositional operations
For a precise treatment, natural
language is replaced by a formal
language.
Propositional formulas are built up from
proposition symbols p0, p1,… by means
of propositional operations that we
learn about in this lecture. (We use also
p,q,r,… for proposition symbols.)
Proposition symbols are the atomic
propositional formulas; they cannot be
broken into parts.
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Not propositional formulas
•x+10
•sin(x)
•I promise that he comes.
•Stop that!
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•The propositional operations are:
¬
”not”
•Conjunction
∧
”and”
•Disjunction
∨
”or”
•Implication
→
”if ... then...”
•Equivalence
”if and only if”
•Parentheses (,) are used for clarity.
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Propositional formulas
•Negation
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Convention about parantheses
The propositional formulas are of the form
pn
!
¬A
(A∧B)
!
(A∨B)
!
(A→B)
(A B)
Parentheses are left out unless
necessary for unambiguous reading.
A∧B∧C means either ((A∧B)∧C) or
(A∧(B∧C))
Similarly A∨B∨C
where A and B are again propositional
formulas.
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Examples
Proposition symbols explained
"
p0∨p1
"
(p1∨p2)→p3
!
"
¬(p1∨p2)
!
"
p0∧(p1∨p2)
"
¬(p3 → (p2→p1))
"
(p0∧p2∧p5)∨(¬p1∧¬p3∧¬p4)∨
(¬p1∧¬p4∧p5)∨(p0∧¬p3∧p4)
"
(p0∨¬p∨p2)∧(p1∨¬p5∨¬p6)∧
(¬p1∨¬p4∨p5)∧(p0∨p3∨p4)
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Implication explained
A∨B: A or B or both.
I am in Rome or I have lost my way.
4<x or y<3.
The train is at a station or the train is
moving.
A and B are called disjuncts.
!
!
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A∧B: Both A and B.
It rains and it blows.
4<10 and 7<3.
The door is closed and the train is
moving.
A and B are called conjuncts.
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Disjunction explained
!
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Conjunction explained
Negation: ¬A is the denial of A
¬It rains: It is not the case that it
rains; it does not rain.
¬4>10: 4 is not greater than 10.
¬The door is closed: The door is not
closed.
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Denote basic states of affairs which can be true
or false
" It is raining
" The lamp is lit
" 4<10
" x<10
" The door is closed
" The train is moving
" The switch is on
" I am in Rome
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Negation explained
Denoted p0, p1,…
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A→B: If A then B.
If it rains then the streets are wet.
If x<3 then x<10.
If the train is moving then the door is
closed.
A= the antecedent of A→B
B= the consequent of A→B
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An example of implication
Equivalence explained
!
p0 ”The train is moving”
!
!
p1 ”The door is closed”
!
!
!
p0→p1 ”If the train is moving, then the
door is closed”
Note that p0→p1 makes no claim about
situations in which the train is not
moving.
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!
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Avoiding ambiguity
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!
!
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Note that p0→p1→p2 would be ambiguous.
This can be prevented by proper use of
parentheses, e.g. p0→(p1→p2) or
!
Other ambiguous formulas are A∧B∨C,
A∨¬B∧C, A→B∧C, A∨B→C.
Unambiguous are for example (A∧B)∨C,
(¬A∨B)∧C, A! ¬(B∧C), ¬A∨(B→C).
We consider also A∧B∧C and A∨B∨C
unambiguous.
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Main connective
(p0→p1)→p2.
!
A䊽B: A if and only if B.
The lamp is lit if and only if the switch
is on.
x<10 if and only if x+5<15.
The door is locked if and only if the
train is moving.
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The main connective is a useful technical
notion related to formulas.
The main connective of
¬A
is
negation
¬
A∧B is
conjunction
∧
A∨B is
disjunction
∨
A→B is
implication
→
A B is
equivalence
Example: the main connective of
(A∨¬B)!(C∨D) is implication.
!
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Main connective
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Subformula
Another useful technical notion related to formulas is
that of a subformula.
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The
Logical analysis of a formula usually
starts by identifying the main
connective.
subformulas of
¬A are ¬A and the subformulas of A
A∧B are A∧B and the subformulas of A and B
A∨B are A∨B and the subformulas of A and B
A→B are A→B and the subformulas of A and B
A䊽B are A B and the subformulas of A and B
Example: The subformulas of (p0∨¬p1)→(p2∧p1) are:
(p0∨¬p1)→(p2∧p1), p0∨¬p1, p2∧p1, p0, ¬p1, p2, p1.
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Immediate subformula
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The importance of subformulas
Special among subformulas are immediate
subformulas.
The only immediate subformula of
¬A
is
A
The immediate subformulas of
A∧B
are
A, B
A∨B
are
A, B
A→B are
A, B
A B
are
A, B
!
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All fundamental methods in logic are
based on analyzing formulas in terms
of their immediate subformulas.
We learn several of them:
"
"
"
Example: The immediate subformulas of (p0∨¬p1)→
Evaluating truth
Natural deduction
Semantic tree
(p2∧p1) are: (p0∨¬p1) and (p2∧p1).
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