Introduction to Modal Logics

Introduction

Logic is supposed to be a way of talking about the sort of arguments that are valid because of the 'form' of the arguments.

In Propositional Logic we learned how to deal with arguments with very simple forms that could be understood in terms of propositions that were connected to each other by truth functions. The truth functions of interest there were things like negation, conjunction, disjunction, material implication, etc.

We made up a formal language using logical symbols: ~, &, v, x, É, º, (, );

and we used them thus

       If Either A or not B  then C and D

=    (A v ~B) É (C & D)

We found that that was not adequate for many important arguments that involved quantifiers such as 'most', 'many', 'some', 'a few', 'all', 'any', ... We therefore introduced quantifiers and subject-predicate structures into propositional logic to get the Predicate Calculus. 

We used the quantifier symbols:

" - which is the universal quantifier symbol to translate all, every,  etc.

$ - which is the existential quantifier symbol to translate some, a,  etc.

and we use them thus:

       Every x is such that Rx 

=    ("x) Rx

       There is an x such that Rx 

=    ($x) Rx

Now we find that there's another class of arguments that look as if they can be formalized involving claims of necessity and possibility - what are called modalities.

If I am to go to the Olympics I must buy a ticket, to buy a ticket I must get a passport. But I can't get a passport so it's not possible for me to go to the Olympics.

That argument looks valid in the same way that PL or QL arguments can be valid: if the premisses are true then the conclusion must be true. And you can replace all the non-logical and non-modal words in the argument and come up with another valid argument - so the validity is formal. Yet it is not an argument that can be translated into PL or QL without doing an injustice to the clear meaning or the formal validity.

Possible Worlds

I mentioned the 'clear meaning' of the modal argument above. But it really isn't that clear.

There's an intuitive notion that we have of possibly true and necessarily true as being related to normally true  in possible worlds. We can look back to Leibniz to see these things being used in philosophy, and David Lewis made a lot of hay by saying that if we have to understand modality in that way, and if modal statements can really be true or false, then there must really be such things as possible worlds (not just the actual world.)

We understand A is possible as meaning A is true in some possible world.

                              A is necessary means A is true in all possible worlds.

We need to ask ourselves are such translations reliable? Do they really capture the modal notions involved?

(Recall that the material conditional, for example, doesn't exactly capture all we mean when we say if ... then ....)

Talk about this more later after we've seen some modal systems.)

Formalizations

The first thing that we need to do is to introduce some formal notation. Here are the modal operators.

[] - which translates 'it is necessary that' (Some also use L)

<> - for 'it is possible that'. (Some use M)

and we use them thus:

       A is necessarily true 

=    []A

       B is possibly true 

=    <>B

In such a formalization we have the following intuitions about modalities and the sentences and terms they feature in.

Modal Negation (MN) equivalences

1.    ~<>p = []~p 

2.    <>~p = ~[]p

3.    []p = ~<>~p

4.    <>p = ~[]~p

Translating Modal Terms

p is contingent = (<>p & <>~p) (= Ñp)

p is analytic = []p (according to one intuition)

p is analytically true = []p

p is analytically false = []~p

p is analytic = ([]p v []~p) (according to another intuition)

p is not contingent = ~(<>p & <>~p) (= []~p v []p = analytic)

p is contradictory = []~p

p is consistent/compatible with q = <>(p & q) (= pOq)

p is incompatible with q = ~pOq (= pÆq)

When translating modal sentences be careful that the modality is put in the right place.

'If A then necessarily B' should be translated as [](A É B), not (A É []B)

Translate the example sentence above to get 

    [](O É T)

    [](T É P)

    ~<>P
    _______

    ~<>O

And sometimes the modal operator should be thought of as outside the formal argument. Thus If T then C, and T, so it must be C. In this case the 'must' refers to logical consequence and attaches, if to anything, to the 'therefore' sign of the argument. Certainly we can't derive a necessity by an argument like that.

A Sample Logic

You get different modal logics when you accept different intuitions about what modal arguments can tell you, or how you should understand modality, or what modal statements are always true. 

C.I. Lewis defined a series of logics that we'll be looking at. One of them is called S5. It is the one which most closely matches our general intuitions, especially if we think of the modality as being logical possibility or logical necessity.

We'll start with a Modal Propositional Logic and later extend it to a quantified form.

Syntax                                                      

The rules for wffs (WML) are as follows:

A. Define a set of elements

1. Take a defined set of propositional constants: A, B, ...

2. Take a defined set of propositional variables: p, q, ...

(constants and variables together we call propositional letters)

3. Take the eight defined logical symbols (or incomplete symbols): ~, <>, [], &, v, x, É, º, (, ).

B. Set up a recursive definition for the WML.

1. basis clause.

    B1:    Any propositional letter is a WML

2. recursive clause.

    R~:     If X is a WML , then ~X is a WML;

    R<>:   If X is a WML , then <>X is a WML;

    R[]:     If X is a WML , then []X is a WML;

    R*:    If X and Y are WMLs, and * is one of &, v, x, º, É, then (X * Y) is a WML.

3. terminal clause

    T:       Nothing is a WML unless it is a WML because of the preceding clauses.

Semantics                                                 

Assume that there are possible worlds. We'll refer to them with indexes - usually w, v, u, n, m, l, k, j, i. 

The truth of a proposition is relative to the possible world it is in. We say:

    ~X is true in w iff X is false in w.

The application to the other PL truth functions is obvious: here are the rules for the modal operators.

    <>X is true in w iff X is true in v for some world v

    []X is true in w iff X is true in v for every world v

We'll say X(w) = 1/0 for X is true/false in w.

Tree Rules                                               

A wff is a logical truth if it's true in all possible worlds (remember what I said about the interpretation of the modalities for S5) or if its negation is not true in any pw.

Logical truth in S5 is defined as S5-Valid(X) iff ~X(w) = 0 for any w. 

S5 trees for wff of PL are just like PL trees but all indexed to a world. With that proviso, all the PL tree rules are also MPL rules. The special rules for S5 are these:

MN                  ~<>X             w/

                            ...      

                          []~X              w                  

                        ~[]X                w/

                          ...

                         []~X               w

<>S5               <>X                w/v      

                            ...                              

                            X                  v

                         where v is new to this path

[]S5                 []X                  w\v      

                           ...                              

                           X                  v

                         where v is any index

There is a special rule that should be listed here. It appears pointless given []S5, but will be important later in other systems 

[]T                    []X                  w      

                           ...                              

                           X                   w

Try one for     p É (q É p)

Now do           <>~p É ~[]p, 

                          []p É p,

                          ~[]p É <>~p,

                          [](p É q) É ([]p É []q),

Test for validity

                          [](p É q), ~<>q, so ~p (also ~<>p)

Counterexamples                                    

When we construct counterexamples from truth trees we need to remember that propositional letters can have different values in different worlds.

We use a tabular form of presentation 

For example suppose we had letters p, q and worlds n, k, then one possible evaluation (called a system) is given in 

 From a table like this we can also work out these values

We define a counterexample thus:

    For the wff X, a system is a counterexample to validity in S5 iff X is false in any world in that system 

The system given is a c-ex for []p for example.

We can see that it's also a c-ex for p É []p (try it in n)

Try it for [](p É q) É ([]p É []q) 

We can use trees to find counterexamples:

Try it for [](p É q) É (p É []q) 

                          1                         ~([](p É q) É (p É []q))(n)                NTF

                          2                                        [](p É q)(n)                          1, ~É \n,m

                          3                                        ~(p É []q)(n)                        1 ~É

                          4                                                   p(n)                            3 ~É

                          5                                                   ~[]q(n)                       3 ~É

                          6                                                   <>~q(n)                     5 MN /m

                          7                                                   ~q(m)                         6 S5<>

                          8                                            (p É q)(n)                         2 S5[]

                                                                           /         \

                          9                                      ~p(n)           q(n)                     8, É 

                                                                     X,4            (p É q)(m)          2 S5[]

                                                                                       /          \

                          10                                               ~p(m)           q(m)        9, É 

                                                                                    ^             X,7

 So we get the c-ex 

and calculation will show that it is a c-ex. 

The same can be done for arguments.

Try it for the first argument we wrote:

    [](O É T)

    [](T É P)

    ~<>P
    _______

    ~<>O

Read more

Swartz, N. ''The' Modal Fallacy'

In these (1999) lecture notes Swartz gives a number of applications of modal logic, showing how misunderstanding the scope of the modal operator leads to some famous philosophical blunders.