You might think that Lukasiewicz is better because English sentences of the form “if φ, then φ” are always true.
But that seems false for a variety of sentences:
Presupposition failure, e.g. φ = “if the King of France is bald”.
Meaningless sentences e.g. φ = “if green ideas sleep furiously”
There are costs of Ł
Linearity is valid: ⊨ (P → Q) ∨ (Q → P). But that’s not sensible in natural language!
Conversely, contraction fails: ⊭(P → (P → Q)) → (P → Q). But that does seem like it’s valid in English.
It depends on the application.
Lukasiewicz makes sense for future contingents – e.g. “if there will be a sea-battle, there will be a sea-battle”.
For vagueness, it seems like supervaluationism itself is better – and it also validates the P → P sentence.
Priest’s logic of paradox
This is where we take # to represent the state of being both true and false, rather than neither true nor false.
i.e., a truth-value glut rather than gap
And then correspondingly 1 means true and only true; 0 means false and only false.
The motivation for this is dialetheism; the idea that some natural language sentences really are both true and false.
E.g., the liar sentence “this sentence is false”.
Or, at the instant Socrates dies, is he alive or dead? Maybe both.
Recall that semantic consequence in LP is very different to PL.
Modus ponens fails, as does ex falso quodlibet (the latter meaning LP is “paraconsistent”).
This is because Priest accepts that φ and ¬φ can both be true under a given interpretation – but then if you have ex falso quodlibet, every sentence ψ would follow as a semantic consequence.
Note that this isn’t necessitated by accepting that φ can be both true and false – see section below
The truth tables are the same as Kleene, but you can motivate them independently.
Specifically, a complex sentence φ is # iff there are grounds for it being both true and false; true if there are grounds for it being true; false if there are grounds for it being false.
Arguments against there being a truth-gap
Consider a sentence which is putatively neither true nor false – e.g., “Middling Mary is rich”. We can show that (given two natural axioms) this leads to a paraconsistent logic.
The two relevant principles:
(i) Transparency/disquotation: “φ” is true iff φ.
That is, for any meaningful sentence φ, you can interchange φ and “‘φ’ is true”.
This lets you move from meta-level claims about sentences to object-level claims about propositions.
(ii) Falsity is identical to truth of negation.
That is, “φ” is false iff ¬φ.
So now consider: suppose we claim that “Mary is rich” is neither true nor false.
Then, we’re saying that (a) “Mary is rich” is not true, and (b) “Mary is rich” is not false.
Applying (ii) to (b), “Mary is not rich” is not true.
Now applying transparency, we have (a*) Mary is not rich, and (b*) Not: Mary is not rich.
i.e., there’s an object-level contradiction ¬R ∧ ¬¬R.
So if you deny bivalence by asserting that we have truth-value gaps, that gets you to a dialetheic logic. But the gap theorist doesn’t want that!
They have a few escape routes:
Rejecting (i), and restricting transparency. But transparency looks very close to what constitutes our concept of truth.
Rejecting (ii). But this seems very unnatural and drives a wedge between object-language and the meta-linguistic predicate “false”.
Distinguishing between meta-language “not” and object-language ¬.
The idea here is that “not” is bivalent – when applied to a sentence that isn’t 1 (i.e. 0 or #), it outputs 1, and otherwise outputs 0 – while ¬ swaps 1 and 0 but leaves # alone.
But now natural-language “not” is different to ¬.
Priest argues that none of these strategies work: transparency and falsity-as-negation seem inviolable.
So if you want to deny bivalence, then you’ll need to accept contradictions anyway.
You might as well do this openly and use a paraconsistent logic.
Supervaluationism is the one view that might escape, though it gives up on truth-functionality.
They are only committed to the idea that each sentence is either super-true or not super-true.
So you can hold that every instance of φ ∨ ¬φ is super-true, without validating bivalence (since not every φ is either super-true or super-false)
This means that the step of the argument above applying transparency (to go from “φ” is not true ⇒ ¬φ) fails – i.e., super-truth failing doesn’t entail super-falsity, because it’s possible to have neither super-truth nor super-falsity.
Tim Williamson disputes this move and thinks it invalidates how we normally reason (see below).
Supervaluationism
Kit Fine’s argument in their support, from penumbral connections.
Suppose H symbolises “Henry is tall”, and Henry is a borderline case.
Then intuitively, the following sentences should all be #: H, H ∨ H, H ∧ H.
But H ∧ ¬H should be false, while H ∨ ¬H should be true.
A truth-functional valuation scheme cannot deliver this.
For example, when H is #, it must be the case that H ∧ ¬H takes the same value as H ∧ H, since ¬H and H are both # (on the Kleene definition of ¬).
This motivates supervaluationism.
But Tim Williamson objects that supervaluations “invalidate our natural mode of deductive thinking”.
Under SV, ordinary argumentative tools like the deduction theorem, reductio, argument by cases, and contraposition of assumptions no longer work.
E.g., we often make an assumption P, derive Q, and then discharge the assumption to conclude that P → Q.
Or, the inference that if φ ⊨ ψ then ¬ψ ⊨ ¬φ fails in SV too.
The reason this fails in SV is analogous to why it fails in MPL.
SV semantic consequence is concerned with global truth (on all precisifications), whereas classical inference only relies on local truth (on a particular precisification), which SV does not preserve.
In fact, you can formalise supervaluations using Kripke MPL models, where each world is a precisification and the modal operator Δ is “definitely” (2022Q3).
And we know that in MPL, DT doesn’t hold (because of Nec), etc.
On the other hand, the defender of SV can point out that SV-semantic consequence is co-extensive with PL-semantic consequence, so we have exactly the same tautologies as before.
Although we lose contraposition and conditional proof, these are just the costs of dealing with vagueness. And we want to be able to accommodate indeterminate truth-values.