Friday, March 25, 2011

Truth Be Told

Truth Be Told is the name of a conference on truth in Amsteram, which has just finished. The idea was to make philosophers and logicians interact about truth. Old Jaakko Hintikka, Hartry Field and Paul Horwich crossed swords. Hannes Leitgeb, Volker Halbach, Leon Horsten, Pascal Engel, Wolfgang Hinzen, Albert Visser, John Collins and Michael Sheard were the other invited speakers. Some interesting contributed papers topped it off. Wednesday-evening was the annual E.W. Beth Lecture, provided by Hartry Field.

So how did it go, this interaction? Mwah. Albert Visser noted that both philosophers of truth and logicians who work on truth in their respective discourse follow their own inner melodies. Nonetheless Visser made serious attempts to teach philosophers lessons based on a high-brow stuff concerning non-standard satisfaction and a fearful world without induction --- The Great Equaliser. One take home message was the distinction between syntactic and semantic conservativeness, two notions that do not coincide but are related. Conservative extensions can be very strong. So when Deflationists claim that the T-theory should be conservative in order to express precisely what the 'non-substantiveness' of the T-pred consists in, they bet on the wrong horse. Moreover, it is well-known that the T-pred is not conservative (work of Jeff Ketland comes to my mind). Volker Halbach commented that a more limited conservativeness result should be aimed for, proof-theoretical in nature. Horwich at one point said that Deflationism consists of only the T-schema and some restrictions to avoid liar-troubles. What kind of predicate the T-pred is, must follow from the T-schema. No additional principles ought to be added such as to which kinds of predicate the T-pred belongs.

Visser also had bad news for Davidsonians. Tarski's commutation conditions do not yield alpha-conversion in non-inductive contexts. Take that! Truth behaves like satisfaction in inductive contexts only. Sensitivity on syntax comes to the fore only in non-inductive contexts. Induction hides the essence ... How worried Davidsonians need to be was not entirely clear to me.

Suppose a philosopher of truth says: I am interested in truths about the world, in how the world makes propositions true. The world is not 'the standard model' of my theories, scientific and common-sense, it is the only model that matters. So who cares about non-standard models?
Here the Lewisian may say: other possible worlds are other 'models', you need them to. What if you reject possible worlds?

Volker Halbach discussed various proposals how to restrict the T-schema to avoid paradox. Typing? Grounding? T-positive sentences? The last-mentioned was Halbach's way. Not conservative! Halback bited to dust and gave up conservativity. Commentator Bruni recalled that classical logic plus expressibility of elementary facts about T-pred blows up! We can't have it all. Something's got to give ...

Leitgeb did set-theory all over again but now as a theory of propositional functions. Typefree semantics to avoid all paradoxes was the reward. Aboutness entered the stage. The commentator remarked there is little difference between Leitgeb's theory and ZFC + T-pred. There you go. We also have a firm intuition about collections and its iterative conception as codified (to some extent) in ZFC, whereas we have nothing corresponding in Leitgeb's theory. Aboutness is the converse of membership, Leitgeb responded.

Field's Beth Lecture on 'Property Theory and the Foundations of Mathematics' was slightly disappointing. A modest layer of property theory on set-theory was his aim. What for? Realms for non-classical logic opened up, then. Fine. So we can do constructive math in the new layer? Fields wanted a serious conditional, which is one that is reflexive and yields modus ponendo ponens. Ties with his theory of truth presumably were in the background, but remained there.

Hintikka asked Who is afraid of Alfred Tarski? His ghost was spotted outside ... Hintikka's familiar song of IF-logic was sung, with rasping voice I must add, because a 1st-order language behaves such that a definition of the T-pred is impossible, as Tarski has taught us. No set-theory for Hintikka. But what is the range of his function-quantifiers in his game-theoretical semantics of IF-logic, then? So asked commentator Sean Welsh. 'For some' means 'Find one', was Hintikka's constructivist-like answer, and further he could reduce a Sigma 1-1 fragment of 2nd-order logic to IF-logic, in order to have his cherished quantifiers. Ho do you like them apples! The semantics of the quantifiers of his IF-logic remains however an open if not unsolvable problem. Few like these apples, I'm afraid. On other occaisons Hintikka promulgated a new Hilbert Programme for the foundations of mathematics, based on IF-logic. Not this time.

Talk about trustworthy and untrustworthy informers by Micheal Sheard was VERY accessible and rather unsatisfactory. He defined a trustworthy sourse as someone who gives you sentences that are consistent with your own beliefs. But they need not be true? Against another the informer may become untustworthy. When starting from a false belief and hearing a true one, one can revise. Beautiful co-organiser Theodora Achourioti slapped Shear around a bit in this vein. Belief-revision strangely seemed beyond Sheard's horizon. Revise, please.

A contributed paper by a Gang of Four defined tolerant and strict truth. New possibilities opened up. Their system ST+ is a conservative extension of classical logic. Without abandoning classical logic, paradoxes could be avoided due to failure of transtitivity of the strong-tolerant satisfaction relation.

Field's talk about Naive Truth Theories was in my judgment much better than his Beth Lecture.
Acceptance Logic, Paraconsistent Dialethism, Paracompleness, the near miss of Lukasiewicz system L-aleph-0 and the search for something weaker, but stronger than Strong Kleene. Commentator Speck came up with Silence Logic. The equivalence of L-aleph-0\U (where U is one axiom) and BCK was reported (I have forgotten what 'BCK' stands for, sorry).
A paracomplete truth theory with a BCK condition seemed the latest thing for Field.
Acceptence Logic yielded also a novel argument against LEM (Law Excl. Middle).

Leon Horsten talked about truth-hierarchies in Field's theory and the Revision Theory of Truth.
They are close. The Revenge Liar remains a problem for both theories. Horsten judged Kripke-Fefermann simpler than Rev. Theory. Ineffable liars. Stable truth, nearly stable truth and ultimate truth. Give it to me, Leon ...

Wolfgang Hinzen presented a theory about grammar at the speed of sound. Aristotle had it right when he said in De Interpretatione that ''falsity and truth have to do with combination and separation''. Only context-dependence has to be added. The T-predicate is a grammatical predicate, Hinzen argued. Our inclination to adher to the T-schema is a consequence of the grammar of language (broadly construed, so as to include the T-predicate) and can be analysed; therefor it is an unalysable starting point, as Deflationists hold. Lexical differences are unimportant. Because the T-pred is grammatical, it is not substantive, as Deflationists claim.

An equivocation seems to occur: Deflationists maintain that the T-schema is the basis for all philosophical explanations of 'truth-phenomena'. This is not in contradiction to analysing the T-schema grammatically. I asked Hinzen what then the grammar of intuitionists, constructivists and all other humans is who
reject LEM --- for LEM is a consequence of the T-schema, plus Tr(not-p) implies Fa(p). Hinzen promised me he was going to think about this.

Pascal Engel argued against Alethic Pluralism, the thesis there are many kinds of truths, over and above kinds of propositions and accompanying kinds of truth-conditions; he focussed on the norm of belief, debating with absent Lycan. Truisms about truth (Swiss army knife: Engel is stationed in Geneva nowadays) are substantial, so truth must be substantial too, right?

The finale was Paul Horwich's undermining diagnosis of truth-relativism: we don't need it, we only need Deflationism. Horwich argued that truth-relativism originates in inflationary intuitions. Relativist theses are relational statements and they are true or false, in line with the T-schema. No independent relative-truth concept is needed. An otiose product of misunderstandings about truth. WHAM!

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