Modal Propositional Logic

Question 1

What does ‘☐’ mean?

Answer 1

the necessity connective

Question 2

What does ‘◇’ mean?

Answer 2

the possibility connective

Question 3

what is special about modal propositional logic?

Answer 3

it models a set of worlds and includes the connectives ◻︎ and ◇

Question 4

How many different MPL systems are there? Name them.

Answer 4

6

K, D, T, B, S4, S5

Question 5

There are different systems within MPL. What differentiates them?

Answer 5

Each system is characterised by Type and a characteristic axiom.

Question 6

Order the MPL systems in order from least restrictive to most restrictive.

Answer 6

K, D, T, B/S4 (the same restrictiveness), S5

Question 7

What is the definition of a wff in MPL?

Answer 7

(i) Every sentence letter is an MPL-wff
(ii) If 𝜙, 𝜓 are wff then ¬𝜙, (𝜙→𝜓), and so on for other propositional connectives, ◻︎𝜙 and ◇𝜙 are also MPLs-wff.
(iii) only what can be shown to be MPLs-wff using (i) and (ii) are MPL-wffs.

Question 8

Define a MPL model

Answer 8

A MPL-model is a triplet ⟨W,R,I⟩ where:
W is a non-empty set of objects
R is a relation over W
I is a function that assigns 0 or 1 to each sentence letter at each member of W.

Question 9

Vm(𝛼,w) = ?

Answer 9

I(𝛼,w)

Question 10

Vm(◻︎𝜙, w) iff …

Answer 10

for all v in W such that wRv, Vm(𝜙,v)=1

Question 11

Vm(◇𝜙, w) iff …

Answer 11

for some v in W such that wRv, Vm(𝜙,v)=1

Question 12

Vm(𝜙,w)=1

Translate this into ‘ordinary english’.

Answer 12

𝜙 is true at world w in model M

Question 13

When is a MPL model a K model?

Answer 13

no matter what

Question 14

When is a MPL model a D model?

Answer 14

iff R is serial: for every w∈W there is some v∈W such that wRv

Question 15

When is a MPL model a T model?

Answer 15

iff R is reflexive: for every w∈W, wRw

Question 16

when is a MPL model a B model?

Answer 16

iff R is reflexive and symmetric: for every w,v∈W, if wRv then vRw

Question 17

When is a MPL model a S4 model?

Answer 17

iff R is reflexive and transitive: for every w,v,u∈W, if wRv and vRu then wRu

Question 18

When is a MPL model a S5 model?

Answer 18

iff R is reflexive, symmetric, and transitive

Question 19

define a possible world

Answer 19

A possible world is a complete and possible scenario

Question 20

Define this flavour of modality: Physical necessity

Answer 20

True in all possible worlds where the (actual) laws of nature are obeyed

Question 21

Define this flavour of modality: Metaphysical necessity

Answer 21

True in all possible worlds whatsoever

Question 22

Define this flavour of modality: Deontic necessity

Answer 22

True in all possible worlds where ones (moral, prudential) duties are fulfilled

Question 23

Define this flavour of modality: epistemic necessity

Answer 23

True in all possible worlds compatible with what i know

Question 24

Define the accessibility relation

Answer 24

☐𝜙 is true at a world w iff it is true at all worlds accessible from w

Question 25

what is the characteristic schema for the K system?

Answer 25

☐(𝜙→𝜓)→(☐𝜙→☐𝜓)

Question 26

what is the characteristic schema for the D system?

Answer 26

☐𝜙→¬☐¬𝜙

Question 27

what is the characteristic schema for the T system?

Answer 27

☐𝜙→𝜙

Question 28

what is the characteristic schema for the B system?

Answer 28

𝜙→☐◇𝜙

Question 29

what is the characteristic schema for the S4 system?

Answer 29

☐𝜙→☐☐𝜙

Question 30

what is the characteristic schema for the S5 system?

Answer 30

◇𝜙→☐◇𝜙

Question 31

If an argument is valid in a system T, what does this say about its validity in other systems?

Answer 31

It is also valid in B, S4 and S5,

nothing can be said about the validity of the argument in K or T.

Question 32

What does a top asterisk mean?

Answer 32

A top asterisk indicates commitments about EVERY possible world

Question 33

What does a bottom asterisk mean?

Answer 33

a bottom asterisk indicates commitments about SOME possible worlds