1
The argument has one modal premise
The premise is ◇□G: it is possible that G is necessary, where G says that a maximally great being exists. The conclusion is □G, and by T, if the frame is reflexive, G itself.
Lesson · Identity, Modality and Freedom · Modality
The Ontological Argument in S5
If it is possible that a maximally great being necessarily exists, then, in S5, it necessarily exists. The lab evaluates the conditional on a frame where it holds and on one where it fails, and shows that the missing ingredient is a property of the arcs. What remains for the reader is whether to grant the possibility premise and whether S5 is the right logic for God.
G: a maximally great being exists
◇□G → □G valid on euclidean frames
S4 + B = S5: the conditional appears
valid in S5; the premise ◇□G is disputed
A valid argument whose premise carries the weight
Core ideas
What this lesson establishes
The previous lessons gave the tools: a box and a diamond read off the arcs, and the axioms the arcs force. This lesson applies them to one old argument.
2
The step from premise to conclusion is a property of the frame
The conditional ◇□G → □G is valid on a frame exactly when the frame is euclidean: whenever a world sees two worlds, the first sees the second.
3
Validity is not the same as acceptance
A valid argument with a doubtful premise gives you a choice. You can accept the conclusion, or deny the premise, and the second has an equally good argument available.
The material
Where S5 does the work
The argument, the frame property it needs, the frame that lacks it, and the shape of the exit.
The argument runs in three lines. A being is maximally great if it is as excellent as a being can be in every possible world. A being like that, if there is one, exists necessarily, since otherwise there would be a world in which it was missing, and it would not be maximally great there. So the claim that there could be such a being is the claim that it is possible that something exists necessarily. Write G for “a maximally great being exists”. The first premise is ◇□G. The second is a fact about the logic of necessity, and the conclusion is □G.
premise 1: ◇□G
logic: ◇□G → □G
conclusion: □G
The second line is valid on a frame exactly when the frame is euclidean, meaning that if a world sees two worlds, the first of them sees the second. The reasoning is short. Suppose w sees v, where □G holds, and w sees u. Euclidean says v sees u, and □G at v gives G at u. So every world w sees has G, and □G holds at w. In a frame that is reflexive and euclidean, which is the same as an equivalence relation, every world can see every other in its group, and that is S5. The lab shows the equivalence frame validating the conditional.
A frame where it fails
Weaken S5 by one property. Let w1 see itself and w2, and let w2 see only itself. This frame is reflexive and transitive, the system S4, but not symmetric. Put G at w2 only. At w2 the box holds, since the only world w2 sees has G. At w1, ◇□G is true because w1 sees w2. But □G is false at w1, because w1 sees itself and G is false there. The premise is true and the conclusion false, so the conditional fails on this frame.
The missing arc is the one from w2 back to w1. Add it, and w2 now sees a world where G is false, so □G fails at w2 and the antecedent fails at w1 as well. The conditional is true, and the frame is the equivalence relation of S5, the S4 frame plus the symmetry that B demands. The argument therefore needs the frame to be euclidean, which on a reflexive frame means symmetric as well as transitive, and it gets no help from S4.
Theorem · What the lab establishes
On a euclidean frame the conditional ◇□G → □G holds at every world for every valuation of G. On the frame without the arc from w2 to w1 there is a valuation and a world where it fails. The argument is valid in S5.
What the argument costs
Two exits are open, and each is an argument. The first is to deny ◇□G. The same logic proves ◇□¬G → □¬G, so whoever grants that it is possible that God necessarily exists must also refuse the parallel premise, that it is possible that God necessarily does not exist, and in S5 both premises cannot be true at one world. Granting the first is therefore not a modest concession. It carries the conclusion with it, and the modest-sounding word “possible” hides that. The second exit is to reject S5 as the logic of the necessity in question, for example because metaphysical necessity may not be symmetric. Then the argument has no valid form to stand on.
The lab can say none of this for you. It computes that the argument is valid on one frame and invalid on another. Which frame suits God, or necessity, or the word “possible” in the first premise, is the philosophical question, and it comes after the logic, not instead of it.
Interactive
Possible worlds
Modal formulas at every world, and the axioms the frame validates
Method
Testing a modal argument on a frame
Five moves. The fourth is the one a reader tends to skip.
1
State the argument as premises, a logic and a conclusion
Mark which line is a claim about the world and which is a principle about necessity. Here the first is ◇□G and the second is the conditional.
2
Test the logical line on a frame the logic is meant to describe
Build the frame, type the conditional, and read the value at a world. On an equivalence relation it holds.
3
Weaken the frame by one property
Remove the arc that symmetry requires, and read the value again. The change in the tile is the property the argument needs.
4
Check the parallel premise
Replace G with ¬G in the premise and the conclusion. If the logic proves both, the two premises cannot both be granted, and the choice is no longer neutral.
5
Decide where to push
Either the premise, or the claim that the frame is the right one. Say which you pick and what it costs.
The argument on a frame
Score 0 / 0Choose the answer you can justify.
The conditional on a frame without symmetry
Two worlds. World w1 sees w1 and w2, w2 sees only w2, and G is true at w2 only.
box of G at w2: w2 sees w2, G there: true
diamond of box of G at w1: w1 sees w2: true
box of G at w1: w1 sees w1, G false there: false
the conditional at w1: true then false, so false
add the arc from w2 to w1
box of G at w2: now w2 sees w1, G false there: false
diamond of box of G at w1: false, so the conditional: true
On the frame without symmetry the antecedent is true and the consequent false. Restoring the arc does not make the consequent true. It makes the antecedent false, which is how the conditional becomes true on that frame, and it is the sense in which S5 does the work.
Common mistakes
1
Believing that the modal ontological argument is invalidOn the equivalence frame the lab finds the conditional true, and the proof for euclidean frames shows why it holds at every world for every valuation. The argument is valid in S5. What the reader may doubt is the first premise, or the claim that the necessity in question has the arcs of S5, and each of these is a reason to reject the conclusion without calling the reasoning a mistake.
2
Treating the conditional as a claim about God◇□G → □G mentions G only as an example of a letter. The same conditional is valid for ¬G, for p and for any sentence at all, so it carries no theological content. All the content is in ◇□G, which is why a reader who grants it has granted nearly everything.
3
Taking a valid argument to be a proof of its conclusionA valid argument can be run backwards. If □G is false at a world, the conditional gives that ◇□G is false there as well. A reader who finds the conclusion more doubtful than the premise is entitled to deny the premise, and nothing in the lab ranks the two doubts.
Check your understanding
✓
Finish when you can say which property of the frame the argument needs and which premise is left to dispute.
You should be able to evaluate the conditional on an equivalence relation and on a reflexive transitive frame without symmetry, state the frame property that separates them, and name two replies open to someone who finds the conclusion hard to accept. A reply that says only “the argument is a trick” has not located the premise.
Note: The frame tile and the axioms tile describe the frame, not the formula. A formula that is true at w1 on one valuation does not show validity, and the proof for euclidean frames is what shows it for every valuation.