Modal Propositional Logic Flashcards
What does ‘☐’ mean?
the necessity connective
What does ‘◇’ mean?
the possibility connective
what is special about modal propositional logic?
it models a set of worlds and includes the connectives ◻︎ and ◇
How many different MPL systems are there? Name them.
6
K, D, T, B, S4, S5
There are different systems within MPL. What differentiates them?
Each system is characterised by Type and a characteristic axiom.
Order the MPL systems in order from least restrictive to most restrictive.
K, D, T, B/S4 (the same restrictiveness), S5
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.
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.
Vm(𝛼,w) = ?
I(𝛼,w)
Vm(◻︎𝜙, w) iff …
for all v in W such that wRv, Vm(𝜙,v)=1
Vm(◇𝜙, w) iff …
for some v in W such that wRv, Vm(𝜙,v)=1
Vm(𝜙,w)=1
Translate this into ‘ordinary english’.
𝜙 is true at world w in model M
When is a MPL model a K model?
no matter what
When is a MPL model a D model?
iff R is serial: for every w∈W there is some v∈W such that wRv
When is a MPL model a T model?
iff R is reflexive: for every w∈W, wRw
when is a MPL model a B model?
iff R is reflexive and symmetric: for every w,v∈W, if wRv then vRw
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
When is a MPL model a S5 model?
iff R is reflexive, symmetric, and transitive
define a possible world
A possible world is a complete and possible scenario
Define this flavour of modality: Physical necessity
True in all possible worlds where the (actual) laws of nature are obeyed
Define this flavour of modality: Metaphysical necessity
True in all possible worlds whatsoever
Define this flavour of modality: Deontic necessity
True in all possible worlds where ones (moral, prudential) duties are fulfilled
Define this flavour of modality: epistemic necessity
True in all possible worlds compatible with what i know
Define the accessibility relation
☐𝜙 is true at a world w iff it is true at all worlds accessible from w
what is the characteristic schema for the K system?
☐(𝜙→𝜓)→(☐𝜙→☐𝜓)
what is the characteristic schema for the D system?
☐𝜙→¬☐¬𝜙
what is the characteristic schema for the T system?
☐𝜙→𝜙
what is the characteristic schema for the B system?
𝜙→☐◇𝜙
what is the characteristic schema for the S4 system?
☐𝜙→☐☐𝜙
what is the characteristic schema for the S5 system?
◇𝜙→☐◇𝜙
If an argument is valid in a system T, what does this say about its validity in other systems?
It is also valid in B, S4 and S5,
nothing can be said about the validity of the argument in K or T.
What does a top asterisk mean?
A top asterisk indicates commitments about EVERY possible world
What does a bottom asterisk mean?
a bottom asterisk indicates commitments about SOME possible worlds
