06
VALIDITY IN PROPOSITIONAL LOGIC
Paal Antonsen
[email protected]
https://sites.google.com/site/paalantonsen/teaching/logic
a
Formal Logic
The story so far. . .
I We have introduced the grammar of propositional logic, and learned
how to identify the general (propositional) argument form of particular
arguments.
The story so far. . .
I We have introduced the grammar of propositional logic, and learned
how to identify the general (propositional) argument form of particular
arguments.
Bigby solves the crime if he finds the photo. He doesn’t find the photo
if and only if he doesn’t inspect the hotel room. Bigby does inspect
the hotel room. Therefore, Bigby solves the crime.
p
q
r
=
=
=
Bigby solves the crime
Bigby finds the photo
Bigby inspects the hotel room
q⊃p
∼q ≡ ∼r
r
—————–
p
The story so far. . .
I We’ve defined truth on an evaluation for formulas of propositional logic.
Truth on an evaluation
For every atomic formula p, v (p) = 1 or v (p) = 0.
1
if v (A) = 0
v (∼A) =
0
otherwise
1
if v (A) = 1 and v (B) = 1
v (A & B) =
0
otherwise
0
if v (A) = 0 and v (B) = 0
v (A ∨ B) =
1
otherwise
0
if v (A) = 1 and v (B) = 0
v (A ⊃ B) =
1
otherwise

if v (A) = 1 and v (B) = 1
 1
1
if v (A) = 0 and v (B) = 0
v (A ≡ B) =

0
otherwise
The story so far. . .
I We’ve represented the meaning of the connectives using truth tables.
A
0
0
1
1
B
0
1
0
1
∼A
1
1
0
0
A&B
0
0
0
1
A∨B
0
1
1
1
A⊃B
1
1
0
1
A≡B
1
0
0
1
I Today we’ll use truth tables to check validity of arguments forms.
Validity in propositional logic
I Before we can do that we need to define validity. We do so by employing
the already understood notion of truth on an evaluation.
Validity in propositional logic
I Before we can do that we need to define validity. We do so by employing
the already understood notion of truth on an evaluation.
Validity: generic
An argument hX , Ai is valid iff in every case, if all the premises X are true
then the conclusion A is also true.
Validity in propositional logic
I Before we can do that we need to define validity. We do so by employing
the already understood notion of truth on an evaluation.
Validity: generic
An argument hX , Ai is valid iff in every case, if all the premises X are true
then the conclusion A is also true.
Validity: propositional logic
An argument hX , Ai is valid in propositional logic (X |= A) iff on every
evaluation, if all the premises X are true then A is also true.
Validity in propositional logic
I Before we can do that we need to define validity. We do so by employing
the already understood notion of truth on an evaluation.
Validity: generic
An argument hX , Ai is valid iff in every case, if all the premises X are true
then the conclusion A is also true.
Validity: propositional logic
An argument hX , Ai is valid in propositional logic (X |= A) iff on every
evaluation, if all the premises X are true then A is also true.
I In the brief and more formal version:
Validity: propositional logic
X |= A iff for every evaluation v and every B ∈ X , if v (B) = 1 then v (A) = 1.
How do we check validity?
HOW TO : CHECK THE VALIDITY OF ARGUMENTS
(a) Set up a truth table where each premise and the conclusion is given its
own column.
(b) Fill in the truth values for each column, following
TABLES FOR COMPLEX FORMULAS .
HOW TO : MAKE TRUTH
(c) Note every row where each premise has the value 1 (true). Check if the
conclusion has the value 1 (true) in each of those rows:
YES
NO
⇒
⇒
argument is valid
argument is invalid
How do we check validity?
HOW TO : CHECK THE VALIDITY OF ARGUMENTS
(a) Set up a truth table where each premise and the conclusion is given its
own column.
(b) Fill in the truth values for each column, following
TABLES FOR COMPLEX FORMULAS .
HOW TO : MAKE TRUTH
(c) Note every row where each premise has the value 1 (true). Check if the
conclusion has the value 1 (true) in each of those rows:
YES
NO
⇒
⇒
argument is valid
argument is invalid
I If an argument is invalid there is a counter example; i.e. an evaluation
where all the premises are true but the conclusion is false. You can read
off which one from the truth table.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
p
∨
∼
q
∼
(p
&
q)
As a way of convenience, we will no longer write down the truth values
under the atomic formulas in the right hand column(s)
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
p
∨
∼
q
∼
(p
&
q)
As a way of convenience, we will no longer write down the truth values
under the atomic formulas in the right hand column(s)
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises has the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
? ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 1
X ∼p ∨ ∼q |= ∼(p & q)
p
0
0
1
1
q
0
1
0
1
∼
1
1
0
0
p
∨
1
1
1
0
∼
1
0
1
0
q
∼
1
1
1
0
(p
&
0
0
0
1
q)
In every row where all the premises have the value 1 (true) is also a line
where the conclusion has the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion have the value 1 (true). Argument is valid.
Checking validity: example 2
? ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 2
X ∼p ⊃ (q & r ), ∼q |= p
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
∼
1
1
1
1
0
0
0
0
p
⊃
0
0
0
1
1
1
1
1
(q
&
0
0
0
1
0
0
0
1
r)
∼
1
1
0
0
1
1
0
0
q
p
0
0
0
0
1
1
1
1
In every row where all the premises have the value 1 (true) is also a line
where the conclusion has the value 1 (true). Argument is valid.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
? p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
We note down each row where all the premises have the value 1 (true),
and then check the value of the conclusion in those rows.
Checking validity: example 3
X p ⊃ ∼q, p ⊃ r , ∼r |= ∼q
p
0
0
0
0
1
1
1
1
q
0
0
1
1
0
0
1
1
r
0
1
0
1
0
1
0
1
p
⊃
1
1
1
1
1
1
0
0
∼
1
1
0
0
1
1
0
0
q
p
⊃
1
1
1
1
0
1
0
1
r
∼
1
0
1
0
1
0
1
0
r
∼
1
1
0
0
1
1
0
0
q
There is ast least one row where alle the premises has the value 1 (true)
and the conclusion have the value 0 (false). Argument is invalid.
Checking validity: example 4
Kittens are adorable. Therefore,
Alexander was poisoned or he
wasn’t.
p
q
=
=
Kittens are adorable
Alexander was poisoned
p |= q ∨ ∼q
Checking validity: example 4
Kittens are adorable. Therefore,
Alexander was poisoned or he
wasn’t.
p
q
=
=
Kittens are adorable
Alexander was poisoned
p |= q ∨ ∼q
p
0
0
1
1
q
0
1
0
1
p
0
0
1
1
q
0
1
0
1
∨
1
1
1
1
∼
1
0
1
0
q
0
1
0
1
Checking validity: example 4
Kittens are adorable. Therefore,
Alexander was poisoned or he
wasn’t.
p
q
=
=
Kittens are adorable
Alexander was poisoned
p |= q ∨ ∼q
p
0
0
1
1
q
0
1
0
1
p
0
0
1
1
God both is and isn’t a material
being. Therefore, Kenneth gets
eaten by zombies.
p
q
=
=
God is a material being
Kenneth is eaten by zombies
p & ∼p |= q
q
0
1
0
1
∨
1
1
1
1
∼
1
0
1
0
q
0
1
0
1
Checking validity: example 4
Kittens are adorable. Therefore,
Alexander was poisoned or he
wasn’t.
p
q
=
=
Kittens are adorable
Alexander was poisoned
p |= q ∨ ∼q
p
0
0
1
1
q
0
1
0
1
p
0
0
1
1
God both is and isn’t a material
being. Therefore, Kenneth gets
eaten by zombies.
p
q
=
=
God is a material being
Kenneth is eaten by zombies
p & ∼p |= q
q
0
1
0
1
∨
1
1
1
1
∼
1
0
1
0
q
0
1
0
1
p
0
0
1
1
q
0
1
0
1
p
0
0
1
1
&
0
0
0
0
∼
1
1
0
0
p
0
0
1
1
q
0
1
0
1
Some definitions
I Now that we have defined truth on an evaluation, we can use that
concept to define other useful concepts.
A formula A is a tautology iff A is true on every evaluation.
A formula A is a contradiction iff A is false on every evaluation.
A formula A is a contingent iff A is neither tautology nor a contradiction.
A set of formulas S is consistent iff there is at least one evaluation, such
that every member of S is true.