“Necessarily p” is not “p and certainly p”. In a possible-worlds model it is true at a world when p holds at every world that world can see, and “possibly p” when p holds at some such world. Compute both from a list of worlds, arcs and atoms, and watch one added arc change the verdict.
Core ideas
Modal logic adds two words to the logic of Arguments and Validity, necessarily and possibly, and gives them a meaning by quantifying over worlds. Everything in this course that mentions necessity uses this one rule.
A world is a complete way things could be, and the model says which simple sentences are true at each. An arc from one world to another says the second is possible as seen from the first. Nothing else is in the model.
At a world, □p is true when p is true at every world it sees, and ◇p when p is true at some world it sees. The first is a universal claim and the second an existential one, over the seen worlds only.
Hold the atoms fixed and change who sees whom, and □p can change from true to false. So necessity is never a property of p alone; it is a property of p and a model.
The material
The rule, a model in which it gives a surprising answer, and the case where a world sees nothing.
A model is a set of worlds, an accessibility relation saying which worlds each world can see, and a valuation saying which sentence letters are true at which world. A formula is evaluated at a world; the same formula can be true at one world and false at another.
The connectives of “Truth Values and the Connectives” work world by world, exactly as before. The two new operators are the only part that looks beyond the world being evaluated, and they look only at the worlds it sees.
Take a model with two worlds. World w1 sees w2 and nothing else, and the letter p is true at w2 only. At w1 the box asks about the one world w1 sees, w2, and p is true there, so □p is true at w1. But p is not true at w1 itself. That is the surprise the lesson is built around: necessity without truth. In this model p is necessary as seen from w1 and false at w1.
The reading “necessarily p means p is true and certain” fails here for both of its halves. Certainty is a fact about someone’s evidence and the model has no believers in it. Truth at the world itself is not guaranteed either, because w1 does not see itself. Give w1 an arc to w1 and the guarantee returns: the box now includes the world being evaluated, p is false there, and □p is false. This is the first appearance of a fact the next lesson turns into a table: whether necessity implies truth depends on whether worlds see themselves.
The rule for the box is a universal claim over the worlds seen. A universal claim over no worlds has nothing to fail it, so it is true. If w1 sees no world at all, then □p is true at w1 for every p, including a p that is false everywhere, and ◇p is false for every p, since there is no seen world at which to find it. The two operators are linked by ◇p ≡ ¬□¬p, and at a world that sees nothing the pair behaves as the lab shows: necessarily true and possibly false together.
Two worlds, p true at w2 only. If w1 sees only w2, then □p is true at w1. If w1 also sees itself, then □p is false at w1. The valuation did not change. One arc did.
The lab quantifies over the worlds you give it, and the arcs are your choice. Whether the arcs for logical necessity, for what is known, or for what is permitted should look one way or another is a question about those subjects, and the lab leaves it open. What it settles is the consequence: given these arcs, this is the verdict.
Interactive
Modal formulas at every world, and the axioms the frame validates
Method
Five moves, in this order. The third is where most mistakes happen.
Write every world with the sentence letters it makes true. A letter not listed at a world is false there.
For each world write the worlds it sees. Check whether a world sees itself, and whether any world sees nothing.
Only these are consulted. A world the evaluated world cannot see is irrelevant to its box, however true or false p is there.
For a box, p must hold at every collected world, and at a world that sees nothing it holds vacuously. For a diamond, p must hold at one collected world, and at a world that sees nothing it fails.
Read p at the evaluated world. If the box is true and p is false, necessity has come apart from truth, and the cause is an arc that is missing.
Two worlds. The arc runs from w1 to w2, and p is true at w2 only.
The first verdict is the one a reader finds odd, and the lab prints it. The second shows what repaired it. When a world sees itself, the box includes the world being evaluated, and the odd case disappears.
Given worlds, arcs and a valuation, you should be able to state the value of □p and ◇p at a named world, including a world that sees nothing, and to say which single added or removed arc turns a true box into a false one. A verdict stated without the list of worlds it consulted has not yet been computed.
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