ULTIMATE TRUTHVIS-À-VISSTABLE TRUTH

Welch, Philip D.

doi:10.1017/s1755020308080118

Cited by 26 publications

(38 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For the limit case, in [22] Lemma 2.2, this stronger reduction in (ii) of T 2 ø ∏ ∑ 1 F ∏ was shown only uniformly for those ∏ with L ø ∏ | = ß 1 -Separation: this was sufficient for our arguments at that time. However we had missed the uniformity over all ∏ < ß that can be obtained from the F -Limit Lemma 3.3 below.…”

Section: Lemma 31 (I) There Is An Effective Procedures For Testing F mentioning

confidence: 89%

“…The same methods can be used to show that for Field's construction in [4] which we showed in [22] essentially constructed a recursively isomorphic copy of the stability set H ≥ of the Herzberger sequence, that we can say the same for his sets.…”

Section: Truth Hierarchiesmentioning

confidence: 95%

“…Field's first 'acceptable point' ¢ 0 of his sequence was shown in [22] to coincide with ≥, and the second with ß. (It is a feature of these kinds of inductive sequence, that the limit stages are determined by the liminf rule, which is in effect some form of infinitary rule; and this wipes out differences in what one does at successor stages; one could even have much stronger (or weaker) successor stage operations than Field considers, but if we stick with the liminf rule at limits one again ends up with the same pair of 'stability' ordinals (≥, ß) reappearing.…”

Section: Truth Hierarchiesmentioning

confidence: 99%

“…(ii) Moreover the sequence hF ∏+k | 0 ∑ k < ni is uniformly recursive in F ∏+n for any such ∏ and n 2 N. 2 The reader may notice that in [22] we used the slightly different sets C AE rather than F AE ; there C AE contained only pairs of the form h>°! Aq,1i, hpA°!…”

Section: Lemma 35 (I) Therementioning

confidence: 99%

See 3 more Smart Citations

Some Observations on Truth Hierarchies

Welch

2014

Rev. symb. logic

Self Cite

View full text Add to dashboard Cite

We show how in the hierarchies F AE of Fieldian truth sets, and Herzberger's H AE revision sequence starting from any hypothesis for F 0 (or H 0 ) that essentially each H AE (or F AE ) carries within it a history of the whole prior revision process.As applications (1) we provide a precise representation for, and a calculation of the length of, possible path independent determinateness hierarchies of Field's construction in [4] with a binary conditional operator.(2) We demonstrate the existence of generalised liar sentences, that can be considered as diagonalising past the determinateness hierarchies definable in Field's recent models. The 'defectiveness' of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are 'ineffable liars'. We may consider them a response to the claim of [4] that 'the conditional can be used to show that the theory is not subject to "revenge problems".' The ScopeThe purpose of this note is to investigate more closely the hierarchies of truth sets produced by the revision sequence process. The first hierarchy, the one produced by Herzberger, [12], [11], was invented to test how various self-referential sentences in a language containing names for elements of a ground model M , and sufficient to define such diagonalising sentences, would behave under repeated applications of the Tarskian definability scheme which produced repeatedly truth sets. Herzberger allowed this process to proceed into the transfinite by using a liminf rule (all of which we specify in more detail below). This revision process has been the subject of various investigations and extensions, notably by Gupta and Belnap in a series of papers, but also in the book [9]. 1More recently Field in e.g. [4], has used such a liminf revision process, to analyse the consequences of adding a binary operator°! to a language similar to the above, with Tr a truth predicate. Field takes at each successive stage not just a new level of definability in the Tarskian sense, but a strong Kleenean fixed point (à la Kripke [15]).The two sequences of sets we shall dub here hH AE | AE 2 Oni (the "H -sets") and hF AE | AE 2 Oni (the "F -sets") (where On denotes the class of ordinals). When defined over the same model, such as M = they are, mathematically at least, surprisingly similar. Indeed we showed in [22] the stability sets consisting of those sentences that are in all the H -sets from some point on, and Field's ultimate truth sets are recursively isomorphic -that is there is a pencil and paper algorithm for converting members of one set into the other, and conversely. Of course this is not to say that the members of the final sets are the same or have the same intended meaning. The phenomenon we are seeing here is that the liminf rule is acting as some kind of very powerful infinitary logical rule. One can show that whatever one does (within some considerably wide bounds) at successor steps will be swamped in effect by the limit rule. This is why the two ultimate sets a...

show abstract

Section: Lemma 31 (I) There Is An Effective Procedures For Testing F mentioning

confidence: 89%

Section: Truth Hierarchiesmentioning

confidence: 95%

Section: Truth Hierarchiesmentioning

confidence: 99%

Section: Lemma 35 (I) Therementioning

confidence: 99%

See 2 more Smart Citations

Some Observations on Truth Hierarchies

Welch

2014

Rev. symb. logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…However Kreisel's squeezing argument that shows that the intuitive notion of validity coincides with the technical one of logical validity using models, would be problematic in this context. Indeed for Kreisel we need a completeness theorem, which as Field remarks, [13] shows is not possible here.…”

mentioning

confidence: 96%

TRUTH, LOGICAL VALIDITY AND DETERMINATENESS: A COMMENTARY ON FIELD’SSAVING TRUTH FROM PARADOX

Welch

2011

The Review of Symbolic Logic

Self Cite

View full text Add to dashboard Cite

show abstract

Classicality Lost: K3 and LP after the Fall

Jenny

2017

Thought: A Journal of Philosophy

View full text Add to dashboard Cite

It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this paper, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality o ered by proponents of LP are far less promising than those available to proponents of K3. This undermines the ability of proponents LP to engage in public reasoning in classical domains. IntroductionGottlob Frege famously held that "nothing is added to [a] thought by . . . ascribing to it the property of truth" ( , ). This idea is commonly expressed with the slogan that truth is transparent: φ and T r ( φ ) -the sentence that says that φ is true-are fully intersubstitutable in extensional contexts. Unfortunately, in classical logic, the law of excluded middle, i.e.φ ∨ ¬φ, and the rule of explosion, i.e. φ, ¬φ ∴ ψ, allow us to derive any sentence from the liar sentence if we have transparency. It's tempting to put the blame on transparency here.However, it isn't entirely obvious what to replace transparency with. That is why a number of authors have instead blamed classical logic. Saul Kripke ( ), Robert Martin and Peter ' φ ' is a term for φ in the object language. Note that the corner quotes here are Gödel quotes, not Quine quotes. To be fully precise, I would need to use both kinds of quotes; however, since corner quotes are commonly used for both, I allow myself the usual use-mention sloppiness here and throughout.See McGee ( ) and Halbach ( ) for surveys of some of the options.

show abstract

ULTIMATE TRUTHVIS-À-VISSTABLE TRUTH

Cited by 26 publications

References 13 publications

Some Observations on Truth Hierarchies

Some Observations on Truth Hierarchies

TRUTH, LOGICAL VALIDITY AND DETERMINATENESS: A COMMENTARY ON FIELD’SSAVING TRUTH FROM PARADOX

Classicality Lost: K3 and LP after the Fall

Contact Info

Product

Resources

About