No. 111 Puddle reflecting night lights
The world pushes one way
the mind pushes in another
‘push’ is not even the right word
‘grind’ is better.
push
push
push
Brief life stories (2)
Once upon a time there was a snake. The snake lived inside a guitar. The snake felt sad and lonely and wanted to get out of the guitar and frolic with other snakes.
Every time the snake tried to crawl out through the guitar’s sound hole, it ran into the strings. The snake banged its head against the strings over and over again, but couldn’t get past them.
As it did so, the guitar strings vibrated and created pleasing music for a couple of nearby children. This made the children very happy.
One day the snake banged its head against the guitar strings so hard that they broke. The snake could now crawl out of the guitar and go away.
Which it did.
The snake left behind an empty guitar with broken strings.
This made the nearby children very sad. I’m not sure they ever fully recovered.
Contra deflationism
Anyway, what I really wanted to talk about was this argument that’s been offered against deflationism.
Now, of course, there’s no need to think that any axiomatic theory of truth is by default deflationist. So, people like Stewart Shapiro and Jeffrey Ketland say: look, suppose you’ve got this base theory A; and you want to add a truth-predicate T(x) to it; and you do; and you add axioms for T(x). Call the new theory AT. Then it’s only natural, they say, that AT be called a deflationist theory of truth only if AT doesn’t decide any new sentences of A – where by ‘new’ we mean sentences that A itself could decide, but doesn’t.
In other words, we say that T(x) is a deflationist truth-predicate in AT only if
for any sentence φ in the language of A, if AT ⊢ φ then A ⊢ φ
When this happens we say that AT is conservative over A.
So, their point is, AT is a deflationist theory of truth only if it’s conservative over the base theory A.
(see Shapiro [1998] and Ketland [1999])
The idea, put simply, is that when you add axioms for truth, the new theory needs to remain neutral with respect to the original theory.
Deflationism (2/3)
By the way, formulating a theory of truth for L in the form of axioms for some (primitive) predicate ‘T(x)’ has certain advantages in its own right.
For one thing, you don’t need to invoke some super-duper metalanguage like set-theory, or second-order quantification, or stuff like that. You just take L and add to it a predicate T(x), and tinker with axioms for T(x) until things feel right.
It’s true, if L is sufficiently expressive, you can’t have the unrestricted T-biconditionals:
T(φ) ↔ φ
as axioms, because that generates paradox (unrestricted means that instead of ‘φ’ you can put any sentence of the new language L + what you get when you add T(x) to L). The Diagonal Lemma gives you a sentence λ such that:
λ ↔ ~T(λ)
But from the unrestricted axiom schema:
T(λ) ↔ λ
which leads to a contradiction.
You can put as axioms, though, all of
T(φ) ↔ φ
when ‘φ’ is to be replaced only by sentences of the original language L.
Now, as I said, the advantage of this is that you don’t have to invoke a powerful metalanguge in order to carry this out – you can just use L augmented with T(x). The disadvantage, I guess, is that theories like this will have an infinite number of axioms, which usually can’t be reduced to a single formula. But maybe we can live with that.
Another thing about nudging truth in via axioms is that T(x) will be a primitive term – a black box. The only thing we know about it are the types of inferences it licenses.
Again, that can be thought of as an advantage. For example, a predicate introduced like this works the same way throughout all models – because the way it’s introduced, it’s not relative to a model. Model thinking can help us, though, because we can use it to show that some set of axioms we might wish to introduce is consistent.
Deflationism (1/2)
Now, to be perfectly honest, I think the main motivation behind deflationism is the desire to approach a theory of truth from a perspective different from Tarski’s – that is, instead of trying to define what truth means, you take ‘true’ as a primitive and give rules for its use.
Here’s how I see the difference in more detail.
Tarski wanted to define the truth-term for a certain language L, so he did two things in this regard.
First of all, he thought about the criteria which a definition of a predicate, say T(x), would have to satisfy so that T(x) counts as a truth-predicate. He required that any definition of a truth-term for the sentences of a language L be materially adequate – the defined predicate T(x) for L has to imply the corresponding T-biconditionals in L’ (the language in which the definition is formulated):
T(ψ) ↔ Ψ,
where ‘ψ’ is a canonical name for a sentence of L, and ‘Ψ’ is a translation of that sentence in L’.
Tarski, as mentioned, wanted to formally define the truth-term for a language L outright – this means building up a formula φ(x) in a formal metalanguage L’, such that φ(x) will characterize the set of true sentences of L. So the second thing Tarski did was, he actually provided such a formula for a range of formal languages.
Deflationism (0)
Ok, so here’s a quicker version of what it is to be a deflationist (you can scratch previous posts).
What we want, basically, is to understand the role of the word ‘true’ when it’s used by people in assertions. Alternatively, we want to understand what ‘true’ means. Alternatively, we want to understand what possession of the concept TRUTH consists in (we don’t want to limit our options about what we want (so as to make it easier that we get something)).
To put it differently. You hear someone saying that this or that is true – what exactly is that person saying? Or: you learn that this or that piece of information is true – what exactly have you learned? Or: you want to prove that some hypothesis is true – what exactly are you trying to prove? Or: you hear that there are sentences which (the set of axioms) PA, if consistent, can neither prove nor disprove, but which can be recognized to be true in the standard model of arithmetic – what does ‘true’ mean in this case?
Here’s what I take to be the basic claims deflationists want to cover:
(A) learning/asserting that some sentence p (in a language you already understand) is true is just learning / asserting that p.
(B) ‘true’, in our language, has an expressive role – it gives us the (logical) means to form assertions we couldn’t form without it.
Brief life stories (1)
Once upon a time there were two crows living on a boat in the middle of the ocean.
The weather in the middle of the ocean was cold and rainy. The crows were cold and afraid.
All of a sudden Francoise Hardy appeared and started to sing to the little crows. Francoise Hardy was made of ice-cream.
Because Francoise Hardy sang so nice the rain stopped and the clouds cleared and the sun came out. This made the crows feel very happy.
And the sun shone brightly and the weather became warm and Francoise Hardy, who was made of ice cream, melted. And the crows ate up what was left of Francoise Hardy.
This made the crows feel very happy.
No. 133 Walls
No. 24 Pictures from the Chinawoman concert in Bucharest, Thursday night (it was awesome, thanks to SoM for invitations)
