The document discusses evaluating whether logical formulas are tautologies, contradictions, or contingent statements. It provides an example of analyzing the formula "∃xTxh" and constructing two models - one where the formula is false, showing it is not a tautology, and one where it is true, showing it is not a contradiction and is therefore contingent. It also discusses how to show sentences are not logically equivalent by providing a model where one is true and the other false. Finally, it addresses consistency, which requires finding a model where all sentences are true.

1. MODELS
PART 3

2. HW ANSWERS

3. PART B #8-10

4. PART B #5-6

5. 1. ∃xTxh
Step 1. Let’s prove prove that this formula is not a tautology.
We need a model where this senetnece is false.
This formula is false in this model:
Step 2. Let’s set the UD:
UD = {Abby, Billy}
Step 3. Now let’s set the referent for the singular terms:
In the formula we have ‘h’ as a singular term.
What will be there referent for ‘h’?
We can set up Abby as the referent for h.
Referent (h) = Abby
Step 4. Now let’s set the extension for T in the way that would make this formula false:
Extension (T) = {0}

6. ∃xTxh
Step 5. Let’s show now that this formula is not a contradiction.
It’s not a contradiction if there is a model where this formula is true.
Let’s construct a model where this formula is true,
This formula is true in this model:
UD= {Abby, Billy}
Extension (T)= {<Billy, Abby>}
Referent (h)= Abby
We have shown that this formula is not a tautology – because have shown that there is a
model where it’s false.
And we have shown that this formula is not a contradiction by providing the model where it’s
true.
It’s not a contradiction and it’s not a tautology – therefore, it’s contingent.

7. LOGICAL EQUIVALENCY
• A SET OF SENTENCES IS LOGICALLY EQUIVALENT WHEN A |= B
and B |= A
A |= B means that there is no model where A is true and B is false
B |= A means that there is no model where B is true and A is false.
A SET OF SENTENCES IS NOT LOGICALLY EQUIVALENT WHEN THERE IS A MODEL WHERE
EITHER A IS TRUE AND B IS FALSE, OR A MODEL WHERE B IS TRUE AND A IS FALSE

8. TO SHOW THAT SENTENCES ARE NOT
LOGICALLY EQUIVALENT ….
• To show that a set of sentences is not logically equivalent we need to
show a model where one sentence is true and another is false.

9. PRACTICE: PART F
• 1. Ja, Ka
To show that they are not logically equivalent we need to construct a model where one of the sentences is
true and another is false. For example, Ja is true and Ka is false.
Step 1. Ja, Ka
1 0
UD: {Abby, Billy}
Extension (J):
Extension (K):
Referent (a): Abby
UD: {Abby, Billy}
Extension (J): {Abby}
Extension (K): {0}
Referent (a): Abby

10. • ∃xJx, Jm
1 0
UD: {Marry, Austin}
Ext (J) = {Austin}
Ref (m) = Marry
• ∀xRxx, ∃xRxx
0 1
UD: {Marry, Austin}
Ext (R) = { <Marry, Marry>, <Marry, Austin>}

11. • ∃xPx → Qc, ∃x (Px → Qc)
0 1
UD = {Marry, Austin}
Ext (P) = {Austin}
Ext (Q) = {0}
Ref (c) = Marry

12. CONSISTENCY
• A SET OF SENTENCES IS CONSISTENT WHEN THERE IS A MODEL
WHERE ALL OF THEM ARE TRUE