Dialetheism/Categories of truth

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What is truth?

Categories of truth

We can establish that a sentence, in logic or maths, is true in a number of different ways. Each gives us a different level of confidence in the result. In order, they are:

  1. Proved directly from the axioms and rules of derivation - these proofs we can be confident are true if there are no errors in the proofs (Tp)
  2. Proved by negation. The opposite has been proved. These are less certain because the weaker condition of both the proof and its opposite could be true (Tn)
  3. Proved by virtue of being Gödel sentences. This relies upon Gödel's theorems, which may be unsafe because of relying on contradiiction (Tg)
  4. Proved by contradiction. These are the weakest proofs, open to reasonable doubt, because they rely on the unnecessary axiom of contradiction (Tc)

How can we establish that something is true

The conventional view is that if something is true, then that's it, it's as true, in the same way, as anything else is true.

That isn't quite the case though. Gödel has shown that there are sentences that are true that cannot be proved in an axiom system.

So we end up with a few different ways that a proposition x can be said to end up in the set T:

  1. If a ∈ T, b ∈ T, and @ is valid then a @ b => x => ( x ∈ Tp) [p proved true]
  2. or

  3. If ~x ∈ F => ( x ∈ Tn) [~p proved false]
  4. or

  5. x ∈ { Gödel sentences } => x ∈ Tg [p true Gödel sentence]
  6. or

  7. If ~x = a ∈ T, b ∈ T, and @ is valid then a @ b ∈ F => by contradiction x => ( x ∈ Tc) { a special case of Tp } [p true by contradiction]

There may be other ways of establishing x ∈ T, let's think of them belonging to the set To. Then:

T = ∪ ( Tp Tn Tg Tc To )

The set of all true things is made up of things that are true by virtue of being in one of the sub-sets of T. These subsets might intersect, so some things might be in T for more than one reason - which is important.

Where are we going to find dialethias?

We're pretty safe with x ∈ Tp - if we've proved that x is true from valid premises and valid rules, we can be pretty safe in saying that it's unlikely that x will also be false.

If x ∈ Tg we're probably on pretty good ground as well.

Tc relies on the 'law' of non-contradiction', so, if x ∈ T only because x ∈ Tc, then it is quite possible that x ∈ F.

Similarly if we simply have x ∈ T because x ∈ Tn, it's a thin argument and quite possible that x ∈ F.

How can we establish that something is false?

How can we know if x ∈ F?

Much the same arguments as above apply - T and F are reasonably symmetric:


If a ∈ F or b ∈ F, and @ is valid then a @ b => x => ( x ∈ Fp)

or

If ~x ∈ T => ( x ∈ Fn)

or

x ∉ { Gödel sentences } => x ∉ Tg => x ∈ Fg (a lot of things are members of Fg, many of them true statements)

or

If x = a ∈ T, b ∈ T, and @ is valid then a @ b ∈ F => by contradiction ~x => ( ~x ∈ Fc)

As before there might be other reasons, so we have the identity:

F = ∪ ( Fp Fn Fg Fo )

Where will we find dialethia?

We know that they are in the intersection ( T ∩ F ) but we can now be much more explicit about where they are, and are not:

If d is a dialethia, we can make these, tentative, claims:

d ∈ Fg ⋁ d ∈ Tn ⋁ d ∈ Tc ⋁ d ∈ To
d ∉ Fp
d ∉ Tp

so:

d ∈ ( ∪ ( Fn Fc Fg Fo ) ∩ ∪ ( Tn Tc Tg To ) )