Logic as a Tool
Chapter 1: Understanding Propositional Logic
1.2 Propositional logical consequence
Logically correct inferences
Valentin Goranko
Stockholm University
September 2016
Goranko
Propositional logical consequence
A propositional formula C is a logical consequence from the
propositional formulae A1 , . . . , An , denoted
A1 , . . . , An |= C ,
if C is true whenever all A1 , . . . , An are true,
i.e., every assignment of truth-values to the variables occurring in
A1 , . . . , An , C which renders the formulae A1 , . . . , An true, renders the
formula C true, too.
If A1 , . . . , An |= C , we also say that C follows logically from A1 , . . . , An ,
and that A1 , . . . , An logically imply C .
Logical consequence is reducible to validity:
A1 , . . . , An |= C iff A1 ∧ . . . ∧ An |= C iff |= (A1 ∧ . . . ∧ An ) → C .
Goranko
Propositional logical consequence is reducible to validity
Proposition
For any propositional formulae A1 , . . . , An , B, the following are
equivalent:
1. A1 , . . . , An |= B
2. A1 ∧ . . . ∧ An |= B
3. |= (A1 ∧ . . . ∧ An ) → B
4. |= A1 → (. . . → (An → B) . . .)
Goranko
Testing propositional consequence with truth tables
Example 1
?
p, p → q |= q
p
T
T
F
F
q
T
F
T
F
p p→q
T
T
T
F
F
T
F
T
q
T
F
T
F
Yes.
Goranko
Testing propositional consequence with truth tables
Example 2
?
p → q |= q → p
p
T
T
F
F
q p→q q→p
T
T
T
F
F
T
T
T
F
F
...
...
No.
Goranko
Testing propositional consequence with truth tables
Example 3
?
p → r , q → r |= (p ∨ q) → r
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r p→r
T
T
F
F
T
T
F
F
T
T
F
T
T
T
F
T
q→r
T
F
T
T
T
F
T
T
p ∨ q (p ∨ q) → r
T
T
T
F
T
T
T
F
T
T
T
F
F
T
F
T
Yes.
Goranko
Sound rules of propositional inference
A rule of propositional inference (for short, inference rule) is a scheme:
P1 , . . . , Pn
,
C
where P1 , . . . , Pn , C are propositional formulae. The formulae P1 , . . . , Pn
are called premises of the inference rule, and C is its conclusion.
An inference rule is (logically) sound if its conclusion logically follows
from the premises.
A propositional inference is an instance of a rule, where propositions are
uniformly replaced by the propositional variables.
A propositional inference is (logically) correct if it is an instance of a
sound inference rule.
Goranko
Propositional inference: example 1
Consider the propositional inference:
Alexis is singing.
If Alexis is singing, then Alexis is happy.
Alexis is happy.
It is obtained from the following rule, called Modus Ponens:
p, p → q
q
This rule is sound, therefore, the inference is logically correct.
Goranko
Propositional inference: example 2
Now consider the propositional inference:
2 plus 2 equals 4.
If 5 is greater than 3, then 2 plus 2 equals 4.
5 is greater than 3.
It is based on the rule
p, q → p
q
which is not sound. Therefore, the inference is not logically correct.
Goranko