Predicate Logic

Question 1

What is the name for specific objects (a,b,c)?

Answer 1

Constants

Question 2

What are variables (xyz)?

Answer 2

Placeholders for arbitrary objects

Question 3

What can Fx be thought as?

Answer 3

It [x] is F

Question 4

What do predicates do?

Answer 4

Establish relationships

Question 5

Give the semantics for the universal indicator in tableaux

Answer 5

ν(∀xA)=1 iff ν(Ax(kd))=1 for all objects d in the domain

Question 6

Give the semantics for existential indicator in tableaux

Answer 6

ν(∃xA)=1 iff ν(Ax(kd))=1 for some object d in the domain

Question 7

What is required in an interpretation for predicate logic?

Answer 7

{D, v}

Question 8

What do we say about objects in the domain?

Answer 8

For each object d∈D, there is a constant kd that is guaranteed to refer to that object, i.e. ν(kd)=d

Question 9

Give the semantics for atomics

Answer 9

For atomics, ν(Pnt1…tn)=1 iff the sequence ⟨ν(t1),…,ν(tn)⟩ is in ν(Pn)
(thus, ν(t1=t2)=1 iff ⟨ν(t1),ν(t2)⟩ is in ν(=) iff ν(t1) and ν(t2) are the same object)

Question 10

State the tableaux rules for universal indicator

Answer 10

∀-rule ¬∀-rule
∀xA __ ¬∀xA ✓
_ ↓ _____ ↓
Ax(t) __ ∃x¬A

Apply to any term t free for x that is on the branch (if no eligible terms, you can introduce new constant).Don’t check off.

Question 11

State the tableaux rules for the existential indicator

Answer 11

∃-rule ____ ¬∃-rule
∃xA ✓ ____ ¬∃xA ✓
_ ↓ _________ ↓
Ax(t) ______ ∀x¬A

Must use a new constant t that does not already occur elsewhere on the branch.

Question 12

What are the identity rules for tableaux?

Answer 12

=-rule  \_\_ / =-rule
 A x(t) \_\_\_\_ t/ =t ✓
_s =t \_\_\_\_\_\_ x 
\_\_↓
 A x(s) 

Don’t check off either node.

Question 13

Regarding substitution, when is y free for x in A?

Answer 13

If x is not within the scope of a quantifier over y

Question 14

What is a tactic for substitution if y isn’t free for x?

Answer 14

Pick another variable (eg z)

Question 15

What does v do in an interpretation?

Answer 15

Takes a name and spits out an object

Question 16

What is Kd?

Answer 16

A special name for each object in the domain

Question 17

What is not permitted in predicate logic?

Answer 17

An empty domain

Otherwise vacuous

Question 18

In Natural deduction, when can we apply the identity rule (t = t)?

Answer 18

Anytime, it is not an assumption sequence

Question 19

What is important about the order of letters in =E?

Answer 19

Second letter must be subbed by the first, not the other way around

eg
n (1) Ax(t) …
m (2) s=t …
n,m (3) Ax(s) 1,2 =E

Question 20

Describe the criteria and process for VI

Answer 20

1. Prove that the context A is satisfied by an arbitrary object (to derive that Ax(y) is true for some appropriate variable y
2. To prove its arbitrary, we cant have y free in any of the assumptions Ax(y) depends on
3. Also cant have y show up free in the universal VxA (eg Fy –> Gy)

Question 21

How does VE work?

Answer 21

Sub in some term
eg
n (1) VxA …
n (2) Ax(t) 1 VE

^If everything is A then t is A

Question 22

What is the macro thought for EE?

Answer 22

If you can show arbitrarily that y . ‘ . C, the ExFx . ‘ . C

Question 23

How do you know y is arbitrary?

Answer 23

y is not free in any other assumptions that B/C depends on

Question 24

What do we need in the counter-model of Modal Predicate logic?

Answer 24

{D, W, R, v}

Question 25

What don't you need to specify in countermodels?

Answer 25

=

Question 26

How might you write a countermodel that has Rab?

Answer 26

``` D = {d1, d2} v(a) = d1 v(b) = d2 v(R) = {} ```

Question 27

How do Pn predicates work?

Answer 27

``` P1 = (_) P2 = ( _ , _ ) P3 = ( _ , _ , _ ) etc. ```

Question 28

What can things have?

Answer 28

Multiple names (eg v(a) = d1, v(b) = d1)

Question 29

What is the countermodel structure for predicate logic and how does it differ in Modal Predicate?

Answer 29

``` D = Set of objects v = Constants --> objects in domain v = Predicates --> a set of objects in domain v = Predicates (P2) --> A set of pairs of objects ``` Modal same but with accessibility relation, worlds and v of these ^ in each of those worlds (where applicable)