# S5 Modal Trees

Logical System
###### 1/29/20

Rod Girle [2000] Modal Logics and Philosophy Chapter 2

You can make a reasonable start here by thinking about how truth tables might work once there are the modal operators, necessary □ and possible ◊.

To remind ourselves, ordinary truth tables are filled out as follows. [What we are doing here is taking a random ordinary propositional formula, say ~A⊃B then valuing the atomic propositions in it to get, perhaps,  ~True⊃False which is abbreviated ~T⊃F and then you are being asked to fill out the values for the connectives, in this case you will get FTTF.]

#### S5 Modal Trees

[Click to select or unselect. Select one item to extend the Tree, select two to close a branch. [Select two for extending with identity, necessityR etc,]]

You can adjust the layout, to a degree, using the vertical lines at the top.

Here are the Basic Tree Rules for Propositional Logic and the S5 Extension Rules. [If you haven't tried any Trees in this setting, you could look at the video Help with Trees. ]

a) ∴□H⊃H

b) ∴◊H⊃□◊H

c) ∴◊◊H⊃◊H

d) ∴(□(H⊃G)&□(~H ⊃G))≡□G