Chapter 1. Basic Modal Notions

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 13-21.

 

(9) Basic modal notions

① In modal logic, there are a number of non-truth-functional concepts.

② Modal Operator L (□)

L: necessity operator
"necessarily," "must be," "is bound to"
Instead of writing "A is bound to be B," we can write "It is bound to be the case that A is B."
Lp= Necessarily p 

③ The Interpretation of Lp

1. Whenever Lp is true, so is p itself.
2. express what morally ought to be so.
※ The latter allows the possiblity that Lp may be true but p itself false, since people do not always do what they ought to do.
※※ If Lp then p is a certain principle; some systems (e.g., T, S4, B, S5) contain it and others (e.g., K, D) not.

④ In any of the interpretations we have referred to, the necessity operator is not a truth-functional one: that is, the truth-value of p itself is not always sufficient to determine the truth-value of Lp. Hence we cannot define L in terms of any combination of the PC operators, and we therefore introduce it as a new primitive symbol.

⑤ Modal Operator M (◇) 

M: "possibly," "can be," "may be"
Mp = "possibly p"
Mα = for any α, ~L~α
Lp: p is necessarily true / Mp: p is possibly true
Lp: it is morally obligatory that p / Mp: it is morally permissible that p
Lp: it is always be the case that p / Mp: It will sometime be the case p

⑥ Impossiblity: ~M or L~

⑦ Propositions which are neither necessary nor impossible are called contingent.

⑧ Necessary (or, strict) implication

L(p⊃q) : "If p then q" is a necessary truth, q follows logically from p
p⊃Lq : if p is true then q is a necessary truth.

⑨  If it rains throughout December it is bound to rain on Christmas Day.

L(p⊃q) : "It will rain on Christmas Day" follows from "it will rain throughout December."
p⊃Lq : "If it rains throughout December, then it is a necessary truth that [It will rain on Christmas Day]
※ [ ] → contingent proposition  

⑩ Someone knows that p then p is true.

L(p⊃q)
p⊃Lq : The only necessary truths can ever be known.

 

(10) The language of propositional modal logic 

① Primitive Symbols

②

Formation Rules

※ every wff of PC is also a wff of modal logic.

※ ※ Lp ⊃ p, MLp ⊃ p, L(p v q) ⊃ Mq), (Lp ∧ Mq) ⊃ L(Lp v Mq), and (MLMp ∧ p) ≡ Lp are wff of modal logic which are not wff of PC.

 

(11) Validity in propositional modal logic

① A wff is valid iff it comes out true for every uniform replacement of its variables by propositions (as same as PC).

② In PC, because of the truth-functional nature of all the operators, this initial account led directly to a quite simple formal definition of validity. In modal logic, however, things are not as straightforward; for modal operators are not truth-functional, and it is not at all clear at the outset under what conditions propositions containing them are to count as true or false.

• (a) Whereas determining the truth-value of a non-modal proposition involves only a consideration of how things actually are, determining the truth-value of a proposition of the form ‘Necessarily p’ or ‘Possibly p’ involves a consideration of how things might have been, of the nature of conceivable states of affairs alternative to the actual one.
• (b) For each conceivable state of affairs there is a range of states of affairs which are possible relative to that one. (This reflects the idea we sometimes express by saying that if things were different a new range of possibilities might be opened up, so that things that are not even possible as things stand might be possible then.)
• (c) In any given conceivable state of affairs, ‘Possibly p’ counts as true iff p itself would be true in at least one state of affairs which is possible relative to that one, and ‘Necessarily p’ counts as true iff p itself would be true in every such state of affairs.

 

(12) Modal Game

① PC game and Modal game

	PC Game	Modal Game
Player	1	n ≥ 1
Sheet	1	n ≥ 1

② Modal Game (pre-playing)

1. determining which players, if any, each player is to be able to see during the course of the game.
2. Being able to see someone means no more than taking note of that person's response.
3. It will be sufficient to specify, for each player, which players are to be watched and which ignored.
4. We may decide that some players shall be able to see themselves while others shall not; if player A can see player B, B may or may not be allowed to see A; and so forth.
5. before the game begins, each player is provided, as the single player in the PC game was, with a sheet of letters.
6. a seating arrangement: the set of players together with the specification of who is to be able to see whom.
7. a seating arrangement, and this together with the players’ sheets a setting for the modal game, or simply a setting.

③ Modal Game (playing) 

• The game proceeds by calling, to the whole set of players at once, any wff of modal logic we choose, provided that, as in the PC game, its well-formed parts, beginning with the variables, are called first.
• We can again assume that the wff are written in primitive notation, with all defined operators eliminated, though we shall, for clarity, state the rule for wff containing M explicitly.

④ Modal Game (Implication) 

• (a) In each setting each wff of modal logic (when appropriately prepared for) will get, from each player, a unique response.
• (b) In a given setting the call of a formula may of course lead some players but not others to raise their hands, but if it leads every player without exception to raise his or her hand we shall say that that formula is successful in that setting.
• (c) a wff should count as valid iff it is true for all values of its variables.
• (d) In the case of the PC game what this means is that the wff must be successful no matter what sheet is given to the player.
• (e) the PC game is simply the modal game played in a seating arrangement with just one player and with only PC wff being called.

One Player 
1. see himself 
2. cannot see himself
affect on a wff containing L, M.
So, it is meaningless when only PC wff are called. 

⑤ Modal Game (validity) 

an appropriate generalization of our notion of validity to make it cover modal wff is that of being valid in a seating arrangement, in this sense.

a wff α is valid in a given seating arrangement iff in that seating arrangement all the players would raise their hands for α, no matter what sheets were distributed to them.

∴ There will be as many different kinds of validity for modal formulae as there are different seating arrangements, and hence that we can have no unique account of validity in modal logic.

At first sight this may seem undesirable; yet on reflection a plurality of criteria of validity is just what our earlier discussion of modal notions would lead us to expect. If ‘necessarily’ and ‘possibly’ can be used in a variety of different senses, then it is quite reasonable to suppose that corresponding to each of these senses there will be a different range of acceptable seating arrangements. In fact the possibility of having different kinds of seating arrangements is part of what gives modal logic its richness. (p. 19)

⑥ The case of Lp ⊃ p

The case of Lp ⊃ p illustrates some of the richness of modal logic. For it is not difficult to see that this wff is valid not only in the seating arrangement described two paragraphs back, where A and B can see themselves and each other, but also in any seating arrangement in which all players can see themselves. And this means that any sense of ‘necessary’ in which whatever is necessary is true can be reflected by restricting the seating arrangements to those in which all players can at least see themselves.

⑦ K-valid wff

There are, however, some wff which are valid in every seating arrangement.

1. all PC-valid wff are K-valid: for in responding to a PC wff a player in the modal game takes no notice of any other players, and a PC-valid wff is precisely one which any sheet of letters whatsoever would lead a player to raise his or her hand.
2. An example of a specifically modal wff which is K-valid

⑧ We can think of the modal game in this way:

	What it represents
The players	conceivable states of affairs, possible worlds
The players each player is allowed to see	the states of affairs which are possible relative to the state of affairs which that player represents
The letters on a player’s sheet	the propositions that are true in that state of affairs
Raising a hand and keeping it down	truth and falsity in the state of affairs the player represents
The K-validity of a wff	that wff would turn out to be true in every conceivable state of affairs, no matter what propositions we were to replace its variables by, no matter what was true or false in that state of affairs, and no matter what states of affairs were possible relative to that one.