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Flashcards in Modal Propositional Logic Deck (33)
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1

What does '☐' mean?

the necessity connective

2

What does '◇' mean?

the possibility connective

3

what is special about modal propositional logic?

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

4

How many different MPL systems are there? Name them.

6
K, D, T, B, S4, S5

5

There are different systems within MPL. What differentiates them?

Each system is characterised by Type and a characteristic axiom.

6

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

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

7

What is the definition of a wff in MPL?

(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.

8

Define a MPL model

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.

9

Vm(𝛼,w) = ?

I(𝛼,w)

10

Vm(◻︎𝜙, w) iff ...

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

11

Vm(◇𝜙, w) iff ...

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

12

Vm(𝜙,w)=1
Translate this into 'ordinary english'.

𝜙 is true at world w in model M

13

When is a MPL model a K model?

no matter what

14

When is a MPL model a D model?

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

15

When is a MPL model a T model?

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

16

when is a MPL model a B model?

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

17

When is a MPL model a S4 model?

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

18

When is a MPL model a S5 model?

iff R is reflexive, symmetric, and transitive

19

define a possible world

A possible world is a complete and possible scenario

20

Define this flavour of modality: Physical necessity

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

21

Define this flavour of modality: Metaphysical necessity

True in all possible worlds whatsoever

22

Define this flavour of modality: Deontic necessity

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

23

Define this flavour of modality: epistemic necessity

True in all possible worlds compatible with what i know

24

Define the accessibility relation

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

25

what is the characteristic schema for the K system?

☐(𝜙→𝜓)→(☐𝜙→☐𝜓)

26

what is the characteristic schema for the D system?

☐𝜙→¬☐¬𝜙

27

what is the characteristic schema for the T system?

☐𝜙→𝜙

28

what is the characteristic schema for the B system?

𝜙→☐◇𝜙

29

what is the characteristic schema for the S4 system?

☐𝜙→☐☐𝜙

30

what is the characteristic schema for the S5 system?

◇𝜙→☐◇𝜙