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