Due to Kripke, we have some examples of sentences which violate the idea that necessary = a priori and contingent = a posteriori.
φ is contingent if its truth-value varies between possible worlds.
φ is a priori if we can know its truth-value without any empirical information
“The morning star is the evening star” seems to be necessary but a posteriori.
Both of them turned out to be Venus, so, the sentence is necessarily true: there’s no possible world where Venus is not identical with itself.
But that identity was a posteriori: it took astronomical observation to discover.
“One metre is the length of the Paris standard bar B” seems to be contingent but a priori.
Someone who understands all the words in this sentence could know a priori that it’s true, i.e. that B is a metre long.
But the statement is contingent: B doesn’t have its length essentially, since it could’ve been bent or shortened.
The formalism of two-dimensionalism can help us explain this.
2D-validity is the formal correlate of a priority
If φ is 2D valid, then any speaker of φ, regardless of which world turns out to be actual, is in a position to recognise its truth just from understanding its meaning.
Superficial contingency is when a sentence’s truth-value varies across possible worlds (holding fixed the actual world).
Deep necessity is when a sentence’s truth makes no substantive demand of reality; if it’s true no matter which world turns out to be actual.
A sentence that is true at every diagonal point is true no matter what, since when you consider each world to be actual, then the sentence is true at that world.
Evans characterises the distinction between superficial and deep necessity as about truth with respect to all worlds, and truth in all worlds.
The former is the semantic notion that □ captures: you hold fixed the actual world and then evaluate at each possible world.
The latter is a more fundamental notion, capturing the idea that “if w were actual, φ would be true”
So, the contingent a priori is superficially contingent but deeply necessary; the necessary a posteriori is superficially necessary but deeply contingent.
The resolution: a sentence can be superficially contingent but a priori. But, it would be “intolerable” for a sentence to be deeply contingent while also a priori.
Superficial contingency doesn’t reflect any genuine dependence on how reality happens to be, so knowing it a priori doesn’t imply you have knowledge without any evidence – the truth is guaranteed.
Deep contingency is one whose truth genuinely depends on a contingent state of affairs. And surely rational reflection alone couldn’t guarantee that some particular contingent feature of reality is in place.
However, Davies concludes that this intolerability is “supported by intuition, rather than by independent argument”: in principle, it might be that a state of affairs S isn’t modally guaranteed (contingency) even if we have an epistemic guarantee (a priori) that it does.
Example: consider the sentence σ = ∀x(@Vx → Vx), where V: “… is valuable”.
The English translation is “Everything that is actually valuable is valuable.”
⊨2D σ
Trivial to prove: we just evaluate at diagonal pairs ⟨w, w⟩, but then the actual world is the same as the evaluation world and we’re simply evaluating Vx → Vx for every x at every world.
This supports the idea that σ is knowable a priori.
⊭2D □ σ
□σ at a diagonal pair ⟨w, w⟩ requires σ to be true at ⟨w, w’⟩ for every w’. So you can easily make a countermodel: just have two worlds where some object is V at one and not V at the other.
This supports the idea that σ is (superficially) contingent: there are models where σ is false at some worlds.
⊨2D F@σ
To prove, note that F@σ just requires truth of σ all along the diagonal. And we’ve already showed by 2D-validity of σ that this holds in all models. So F@σ is true everywhere, and in particular on the diagonal (generally 2D valid and 2D valid).
This supports the idea that σ is deeply necessary – no matter which world is actual, σ will be true at that world.
Complications
There’s a strong intuitive argument that a priori entails deep necessity, but it’s hard to make this rigorous.
Intuition: if you can know φ a priori, then φ’s truth doesn’t depend on any contingent feature of reality, so φ is true no matter what (deeply necessary).
But a priori justification is empirically defeasible. Even when you have a priori justification to believe φ, there could be some background empirical conditions whose obtaining you simply take for granted.
You’re entitled to ignore those conditions, but they’re contingent – and if evidence of them failing arose, your justification would be undermined.
For example, maybe the sentence s = “If anyone uniquely invented the zip, then Julius [stipulated to pick out the zip inventor] did” has some sort of defect (e.g. that “Julius” is somehow incoherent). In worlds where this background condition fails, the sentence isn’t even truth-evaluable! So you can’t say s is true in every world; at best it’s true in all worlds where the presupposed condition holds.
Davies raises the suggestion that maybe there could even be a priori justified beliefs where we’re entitled to ignore the possibility of empirical evidence that would render the belief false, not merely contentless.
The claim that 2D-validity is the formalisation of a priori rests on the epistemic equivalence of s and @s – i.e., that understanding the @ operator suffices to know a priori that @s ↔ s.
This would imply that any competent speaker of a language could recognise the truth of a 2D-valid sentence without any empirical evidence.
Note that @s and s are true in the same worlds, but differ in truth with respect to worlds: they agree on-diagonal, but disagree elsewhere.
@s is superficially necessary, and that’s just an artefact of how @ interacts with □; it doesn’t reflect genuine necessity.
Identity statements with ordinary proper names (e.g. Cicero = Tully) are both superficially and deeply necessary, yet they’re a posteriori.
This suggests that the relationship between a priori and deep necessity is asymmetric: a priori ⇒ deep necessity, but deep necessity ≠> a priori.
Intuitively, this is because two names can pick out the same object (so the identity holds necessarily at every level) while presenting it via different routes that make the identity non-obvious and discoverable only empirically.
More formally, Davies draws on the Fregean distinction between sense and reference. The state of affairs of identity (reference) has deep necessity necessarily, whereas the mode of presentation (sense) is where the epistemic status of a priori vs a posteriori comes in.
Since sense determines reference but not vice versa (multiple senses can present a single reference), deep necessity ≠> a priori. A state of affairs could be deeply necessary (like Marcus is self-identical), but presented in a way that requires empirical a posteriori knowledge.
You could close this gap by treating all names as descriptive names, but there Kripke raises objections to that.
Interactions between F, □, @ and X:
□ ranges over evaluation worlds i.e. checks truth along a horizontal, starting in whichever row we were already in. This gives us superficial necessity.
F ranges over reference worlds, i.e. checks truth along a vertical.
But on its own, that’s not very interesting: if φ contains no occurrences of @, then Fφ is just equivalent to φ (since varying the reference world doesn’t affect a formula that never needs it).
X takes the current evaluation world w and makes it also the reference world – i.e., it lets you proceed “considering w as actual”.
F@ quantifies over both coordinates locked together, i.e. checks truth on the diagonal. This gives us deep necessity.
To interpret: F@φ holds at ⟨v, w⟩ iff for all v’, V^2^(@φ, v’, w) = 1. And that holds iff for every v’, V^2^(φ, v’, v’) = 1, since @ tells us to evaluate at the reference world.
X□ does exactly the same job as F@, checking for truth along the diagonal.
□Xφ holds at ⟨v, w⟩ iff for all evaluation worlds w’, V^2^(Xφ, v, w’) = 1. But applying X, that’s just whenever V^2^(φ, w’, w’) = 1 for all w'.
F□ and □F both quantify over everything. This gives us general validity.
□@φ collapses to @φ.
The outer □ ranges over each evaluation world w’ to check @φ, but this just tells us to check φ at the initial reference world v.
So, if φ is actually true, then it’s necessarily actual. This motivates why we need to introduce F: it’s contingent which world is actual, but □ is unable to capture that.
@□φ collapses to □φ.
□ doesn’t care about your current evaluation world (given than for 2-dimensionalism we use a universal accessibility relation), so @ changing it to the reference world v is immaterial.
@Xφ collapses to @φ.
@ sends you to the evaluate the inner formula at the reference world v. But then the inner X has no effect, since you’re setting the now-v evaluation world to the reference world, which remained as v.
X@φ collapses to Xφ.
X stores the evaluation world w as the reference world. But then the inner @ has no effect, since it just tells you to evaluate at reference world w, which started off as your evaluation world anyway.
dSQML vs SQML validity
The reason we change the definition of validity for dSQML (only evaluating truth at the designated world of each model, not all worlds) is so that we get the expected validities for sentences containing @.
@φ → φ is dSQML-valid, and not SQML-valid, but intuitively we want it to hold.
We also want ◇@φ → φ to hold, since ◇@φ is semantically identical to @φ (actuality has a constant value across all worlds). But it would not be valid according to the SQML definition.
Similarly with φ → @φ.
Together, this gives us φ ↔ @φ, which is the simplest example of a contingent a priori sentence: it’s 2D valid, but its necessitation fails.
@ creates a privileged world in the model, and logical truth should be assessed from the standpoint of that privileged world.