1
A frame is worlds and arcs
A frame has no valuation. An axiom is valid on a frame when it is true at every world of the frame for every way of assigning p to the worlds. The lab tests every assignment.
Lesson · Identity, Modality and Freedom · Modality
Frames, Axioms and What Necessity Obeys
Drop the valuation and keep only the worlds and arcs: that is a frame. An axiom is valid on a frame when it is true at every world under every valuation. The lab computes which of five axioms a frame validates and which of five properties it has, so the match between them is something you read off a tile and not something you are told.
T □p → p every world sees itself
D □p → ◇p every world sees a world
B p → □◇p seeing goes both ways
4 □p → □□p seeing passes through
5 ◇p → □◇p worlds seen share a view
Five axioms, five properties of the arcs
Core ideas
What this lesson establishes
The previous lesson fixed the meaning of the box for one model. This one asks what the box obeys across all the models that share a set of arcs.
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Each axiom asks for one property of the arcs
T asks that every world see itself. D asks that every world see some world. B asks that seeing be symmetric, 4 that it be transitive, and 5 that it be euclidean. The lab lists the properties a frame has beside the axioms it validates.
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The axioms are chosen by choosing the frame
No axiom is a discovery about necessity in general. To say that necessity obeys T is to say that the arcs of the kind of necessity you mean are reflexive, which is a claim about that kind of necessity.
The material
From arcs to axioms
The five axioms, the property each corresponds to, and the one that is valid on every frame.
Definition · Frame and validity on a frame
A frame is a set of worlds and an accessibility relation. A formula is valid on the frame if it is true at every world under every valuation. The formula □(p → q) → (□p → □q), called K, is valid on every frame and is not listed in the lab.
axiom formula what the arcs must do
D □p → ◇p serial: every world sees some world
T □p → p reflexive: every world sees itself
B p → □◇p symmetric: if w sees v, then v sees w
4 □p → □□p transitive: if w sees v and v sees u, then w sees u
5 ◇p → □◇p euclidean: if w sees v and u, then v sees u
The table is not a list of facts to memorise, because each row can be reasoned out. Suppose a world w does not see itself. Make p true at every world w sees and false at w. Then □p is true at w and p is false at w, so T fails at w. The other direction is as direct: if every world sees itself, then □p includes p at the world itself, so T holds. Transitivity is the same argument one step longer. If w sees v, v sees u and w does not see u, make p true at everything w sees and false at u. Then □p is true at w, and □□p is false at w, because v sees u.
B is easy to misread. It does not say that what is true is necessarily true, which would be p → □p. It says that what is true is necessarily possible: at a world w where p holds, every world v that w sees must itself see a world with p, and symmetry supplies one, since v sees w.
Which logic is which
The frames people reach for have names. A frame that is reflexive and transitive validates the system called S4. A frame that is an equivalence relation, reflexive, symmetric and transitive, validates S5, and in S5 every world in a group sees every other. Each step up the list adds a property, so each adds axioms. What the lab does is compute the list for the frame you build, so you can add one arc and see the first axiom that appears.
That is also its limit. A frame that validates all five axioms is a frame, and whether it is the right model of logical necessity, of metaphysical necessity or of what a person can know is not computed. The next lessons use S5 for one argument and ask whether it was a fair choice.
Interactive
Possible worlds
Modal formulas at every world, and the axioms the frame validates
Method
Reading the axioms off a frame
Five moves. The first four need only the arcs.
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Write the arcs and nothing else
Forget the valuation. A frame is judged by who sees whom.
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Check each property by its definition
Reflexive: every world has an arc to itself. Serial: every world has an arc out. Symmetric: every arc has its reverse. Transitive: every two-step path has a shortcut. Euclidean: every two arcs leaving the same world are joined by an arc from the first target to the second.
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Pair each property with its axiom
Serial with D, reflexive with T, symmetric with B, transitive with 4, euclidean with 5. A frame validates an axiom exactly when it has the matching property.
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Name the logic if there is one
Reflexive and transitive is S4. Reflexive, symmetric and transitive is S5.
5
Test one valuation if you want a case
To see an axiom fail, type it as the formula, choose the valuation that breaks it, and read the value at the world the proof used.
From arcs to axioms
Score 0 / 0Choose the answer you can justify.
Growing a frame one arc at a time
Four frames on three worlds. The second drops the arc from 3 to 1 and adds the loops; the third and fourth only add arcs.
cycle, w1 sees w2 sees w3 sees w1: D
loops, w1 sees w2, w2 sees w3: D, T
add: w1 sees w3, the shortcut: D, T, 4
add: every reverse arc: D, T, B, 4, 5
Each line was read off the tile, and each new axiom appeared when its property did. Nothing about necessity was consulted: the axioms followed from the arcs.
Common mistakes
1
Treating the axioms as truths about necessity that every frame obeysOn the serial-only frame the lab lists D and nothing else: T fails at w1, 4 fails because w1 does not see w3, and so on. Which of the axioms hold is a fact about the arcs, and the arcs are what you chose. To claim that necessity obeys T is to claim that the right frame for it is reflexive, which can be argued for or against.
2
Reading B as “what is true is necessarily true”B is p → □◇p, which says what is true is necessarily possible. On the frame where every world sees every world, B is valid, yet p → □p is false at a world where p is true and another seen world lacks it. Symmetry brings the world back into view; it does not fix its truth values.
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Assuming the frame that validates more axioms is the better model of necessityThe s5 frame validates all five and the serial-only frame only one. That is a difference in the arcs and not a difference in correctness. Whether the necessity at issue, in a particular argument, has arcs that are symmetric and transitive is the question the next lessons ask.
Check your understanding
✓
Finish when you can read the axiom list off a set of arcs, and add one arc to make a chosen axiom appear.
Given three or four worlds and their arcs, you should be able to state which of reflexive, serial, symmetric, transitive and euclidean the frame has, name the axioms it therefore validates, and say which single arc must be added for a named axiom to be valid. A match stated without the pair of arcs that produces it has not been checked.
Note: The axiom tile checks every assignment of p to the worlds, which is why a frame of six worlds is the limit of the lab. The formula tile shows one assignment, so a formula that is true at w1 is not thereby valid on the frame, and the tiles can differ.