Semantic Groundedness Hannes Leitgeb LMU Munich August 2011 Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 1 / 20
Luca told you about groundedness in set theory. Now we turn to groundedness in semantics. Plan of the talk: 1 What is Semantic Groundedness? 2 Groundedness and Dependence 3 Beyond Semantic Groundedness 4 An Afterthought: Semantic vs. Set-Theoretic Groundedness 5 [Bibliography] Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 2 / 20
What is Semantic Groundedness? Herzberger (1970): grounding as a link between set theory and semantics (p. 146) Every sentence is assumed to be about a set of entities, its domain. The general notion of a domain is more readily indicated than explicated, but the analysis to follow depends on no problematic cases, and ultimately proves independent of any particular explication of domain (p. 148) A sentence is groundless iff it is the first member of some infinite sequence of sentences each of which belongs to the domain of its predecessor. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 3 / 20
Groundless sentences, like groundless classes, are pathological; to adapt Mirimanoff s term, they are extraordinary. Some of them give rise to paradox, which is to say they collide with cherished conceptual principles like the abstraction principle of naive set theory (that every condition determines a set) or its counterpart in naive semantics (that every sentence determines a statement). In both set theory and semantics there is a temptation to banish everything extraordinary by some grounding axiom that denies groundless classes the status of sets or denies groundless sentences the status of statement. In set theory, grounding requirements have wide-currency; in semantics they have been widely honored though seldom acknowledged, and hardly brought to the level of explicit formulation. A first effort in this direction might read: Semantic Grounding Condition: Any given sentence determines a statement only if it is grounded or is nonsemantic... (pp. 148f) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 4 / 20
Groundless sentences, like groundless classes, are pathological; to adapt Mirimanoff s term, they are extraordinary. Some of them give rise to paradox, which is to say they collide with cherished conceptual principles like the abstraction principle of naive set theory (that every condition determines a set) or its counterpart in naive semantics (that every sentence determines a statement). In both set theory and semantics there is a temptation to banish everything extraordinary by some grounding axiom that denies groundless classes the status of sets or denies groundless sentences the status of statement. In set theory, grounding requirements have wide-currency; in semantics they have been widely honored though seldom acknowledged, and hardly brought to the level of explicit formulation. A first effort in this direction might read: Semantic Grounding Condition: Any given sentence determines a statement only if it is grounded or is nonsemantic... (pp. 148f) The concept grounded term for any elementary conceptual system is inexpressible within that system (p. 157) no language is universal in the sense of Tarski (p. 164) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 4 / 20
Kripke (1975): The downwards view of groundedness (cf. Kremer 1988) if a sentence... asserts that (all, some, most, etc.) of the sentences of a certain class C are true, its truth value can be ascertained if the truth values of the sentences in the class are ascertained. If some of these sentences themselves involve the notion of truth, their truth value in turn must be ascertained by looking at other sentences, and so on. If ultimately this process terminates in sentences not mentioning the concept of truth, so that the truth value of the original statement can be ascertained, we call the original sentence grounded; otherwise, ungrounded (pp. 693f) whether a sentence is grounded is not [...] an intrinsic [...] property of a sentence, but usually depends on the empirical facts (p. 694) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 5 / 20
Additionally, Kripke presents an upwards view of groundedness: Suppose we are explaining the word true to someone who does not yet understand it. We may say that we are entitled to assert (or deny) of any sentence that it is true precisely under the circumstances when we can assert (or deny) the sentence itself. [...] In this manner, the subject will eventually be able to attribute truth to more and more statements involving the notion of truth itself. There is no reason to suppose that all statements involving true will become decided in this way, but most will. Indeed, our suggestion is that the grounded sentences can be characterized as those which eventually get a truth value in this process (p. 701) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 6 / 20
Additionally, Kripke presents an upwards view of groundedness: Suppose we are explaining the word true to someone who does not yet understand it. We may say that we are entitled to assert (or deny) of any sentence that it is true precisely under the circumstances when we can assert (or deny) the sentence itself. [...] In this manner, the subject will eventually be able to attribute truth to more and more statements involving the notion of truth itself. There is no reason to suppose that all statements involving true will become decided in this way, but most will. Indeed, our suggestion is that the grounded sentences can be characterized as those which eventually get a truth value in this process (p. 701) Either understanding motivates a jump operator, such that ϕ can be defined as grounded if it has a truth value in the least fixed point of that operator. Such semantical notions as grounded, paradoxical, etc. belong to the metalanguage. [...] The ghost of the Tarski hierarchy is still with us (p. 714) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 6 / 20
We find two justifications for ascribing truth or falsity (non-trivially) only to grounded sentences: all paradoxical sentences seem to be ungrounded, and ascribing truth or falsity to ungrounded sentences means to go beyond the facts. But: Ultimately this commits one to give up on universality. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 7 / 20
We find two justifications for ascribing truth or falsity (non-trivially) only to grounded sentences: all paradoxical sentences seem to be ungrounded, and ascribing truth or falsity to ungrounded sentences means to go beyond the facts. But: Ultimately this commits one to give up on universality. As Kremer (1988) summarizes: Thus a grounded sentence is one whose truth-value can be ascertained on the basis of facts not involving the concept of truth. We can say that the original sentence depends on non-semantic facts (p. 227) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 7 / 20
Groundedness and Dependence What is conceptually prior: groundedness or dependency? The latter. (Groundedness is well-foundedness of... what?) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 8 / 20
Groundedness and Dependence What is conceptually prior: groundedness or dependency? The latter. (Groundedness is well-foundedness of... what?) But it is not so clear what this dependency relation is meant to be in fact, there are different proposals: Yablo (1982): compositional, Strong-Kleene dependence E.g., what ϕ ψ depends on is determined by what ϕ depends on and what ψ depends on. Leitgeb (2005): non-compositional, classical dependence E.g., ϕ ϕ depends on nothing (the empty set). (There are further approaches, e.g., Gaifman 1992.) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 8 / 20
The way to proceed in the classical case: Define a semantic notion of dependence, such that, e.g., Tr( ϕ ) depends on {ϕ} Tr( Tr( ϕ ) ) depends on {Tr( ϕ )}... x(p(x) Tr(x)), x(p(x) Tr(x)) depend on the extension of P the Liar sentence λ depends on {λ} Define the set Φ lf of grounded sentences in terms of direct or indirect dependency on the non-semantic base language L. Define truth for the extended language L Tr in a way, such that all T-biconditionals for members of Φ lf are derivable. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 9 / 20
We consider an example: Let L be the language of first-order arithmetic, L Tr = L + Tr be our object language. L Tr extended by the language of set theory (and fragments of English) is our metalanguage. For ϕ L Tr, let Val Φ (ϕ) be the truth value of ϕ as being given by (i) the standard model of arithmetic, (ii) with Φ as the extension of Tr (mod coding). Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 10 / 20
We consider an example: Let L be the language of first-order arithmetic, L Tr = L + Tr be our object language. L Tr extended by the language of set theory (and fragments of English) is our metalanguage. For ϕ L Tr, let Val Φ (ϕ) be the truth value of ϕ as being given by (i) the standard model of arithmetic, (ii) with Φ as the extension of Tr (mod coding). Now we can define (for ϕ L Tr, Φ L Tr ): Definition ϕ depends on Φ iff for all Ψ 1, Ψ 2 L Tr : if Val Ψ1 (ϕ) Val Ψ2 (ϕ) then Ψ 1 Φ Ψ 2 Φ. This is semantic supervenience: no difference concerning the truth value of ϕ without a corresponding difference concerning the presence/absence of members of Φ in/from the extension of Tr. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 10 / 20
We consider an example: Let L be the language of first-order arithmetic, L Tr = L + Tr be our object language. L Tr extended by the language of set theory (and fragments of English) is our metalanguage. For ϕ L Tr, let Val Φ (ϕ) be the truth value of ϕ as being given by (i) the standard model of arithmetic, (ii) with Φ as the extension of Tr (mod coding). Now we can define (for ϕ L Tr, Φ L Tr ): Definition ϕ depends on Φ iff for all Ψ 1, Ψ 2 L Tr : if Val Ψ1 (ϕ) Val Ψ2 (ϕ) then Ψ 1 Φ Ψ 2 Φ. This is semantic supervenience: no difference concerning the truth value of ϕ without a corresponding difference concerning the presence/absence of members of Φ in/from the extension of Tr. Equivalently: for all Ψ L Tr : Val Ψ (ϕ) = Val Ψ Φ (ϕ) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 10 / 20
Example 1 For ϕ L: ϕ depends on. 2 Tr( ϕ ), Tr( ϕ ) depend on {ϕ}. 3 The liar sentence λ depends on {λ} (but λ λ, λ λ depend on ). 4 (Tr( α ) Tr( β )) ( Tr( α ) Tr( γ )) depends on {α,β,γ}. 5 For P in L: x(p(x) Tr(x)), x(p(x) Tr(x)) depend on the extension of P. 6 x(tr(x) Tr(. x)) depends on L Tr. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 11 / 20
Fix the left-hand side of dependency, and this will yield a filter: Lemma For all ϕ L Tr, for all Φ, Ψ L Tr : If ϕ depends on Φ, Φ Ψ, then ϕ depends on Ψ. If ϕ depends on Φ, ϕ depends on Ψ, then ϕ depends on Φ Ψ. ϕ depends on L Tr. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 12 / 20
Fix the right-hand side of dependency, and this will yield (something like) an algebra: Lemma D 1 (Φ) := {ϕ L Tr ϕ depends on Φ} is closed under sentential operations, substitutional quantification, logical equivalence. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 13 / 20
Fix the right-hand side of dependency, and this will yield (something like) an algebra: Lemma D 1 (Φ) := {ϕ L Tr ϕ depends on Φ} is closed under sentential operations, substitutional quantification, logical equivalence. Furthermore, one can iterate D 1 : Lemma For all Φ, Ψ L Tr : if Φ Ψ, then D 1 (Φ) D 1 (Ψ). There is a least fixed point Φ lf of D 1. For all ϕ L Tr : ϕ Φ lf iff ϕ depends on Φ lf ϕ Φ lf iff Tr( ϕ ) Φ lf. L Φ lf L Tr, and Φ lf satisfies the closure conditions from above. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 13 / 20
Definition For ϕ L Tr : ϕ depends directly on non-semantic soa s iff ϕ depends on L. [Groundedness:] ϕ depends (directly or indirectly) on non-semantic soa s iff ϕ Φ lf. ϕ is ungrounded iff ϕ does not depend on non-semantic soa s. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 14 / 20
Definition For ϕ L Tr : ϕ depends directly on non-semantic soa s iff ϕ depends on L. [Groundedness:] ϕ depends (directly or indirectly) on non-semantic soa s iff ϕ Φ lf. ϕ is ungrounded iff ϕ does not depend on non-semantic soa s. But dependency gives us more than just groundedness: Definition ϕ is selfreferential iff for all Φ L Tr : if ϕ depends on Φ, then ϕ Φ. (cf. Leitgeb 2002) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 14 / 20
Lemma ϕ depends on non-semantic soa s iff Tr( ϕ ) depends on non-semantic soa s. If ϕ is selfreferential, then ϕ is ungrounded. There are sentences ϕ for which there is no least set Φ on which they depend. (From this point of view, Herzberger s criterion of ungroundedness is sufficient, but not necessary.) Tr( 2 + 2 = 4 ) depends directly on non-semantic soa s. Tr( Tr( 2 + 2 = 4 ) ) depends on non-semantic soa s. λ and x(tr(x) Tr(. x)) are self-referential. The members of Yablo s sequence are ungrounded but not self-referential (cf. Yablo 1993, Schlenker 2007; search also m-phi.blogspot.com) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 15 / 20
Finally, truth: Theorem There is a set Γ lf, such that for all ϕ Φ lf : ϕ Γ lf iff Val Γlf (ϕ) = 1. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 16 / 20
Finally, truth: Theorem There is a set Γ lf, such that for all ϕ Φ lf : ϕ Γ lf iff Val Γlf (ϕ) = 1. So if we finally define for ϕ L Tr, Definition ϕ is true (in-l Tr ) iff ϕ Γ lf. then this definition is formally correct and entails all T-biconditionals for Φ lf. Contra Field (2008): Do we really want more than just the grounded instances of the T-scheme? Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 16 / 20
Finally, truth: Theorem There is a set Γ lf, such that for all ϕ Φ lf : ϕ Γ lf iff Val Γlf (ϕ) = 1. So if we finally define for ϕ L Tr, Definition ϕ is true (in-l Tr ) iff ϕ Γ lf. then this definition is formally correct and entails all T-biconditionals for Φ lf. Contra Field (2008): Do we really want more than just the grounded instances of the T-scheme? Note that there is no arithmetical formula whose extension would be Φ lf, and more generally there is no sentence in L Tr for which Φ lf would be the least set on which it depends. (Herzberger s Paradoxes of Grounding!) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 16 / 20
Beyond Semantic Groundedness Groundedness and dependence are studied also in other contexts: Generalized dependency: Leitgeb (2005), Van Vugt and Bonnay (unpublished), Meadows (under review). Grounded truth and games: Welch (2009). Groundedness and abstraction / individuation: Linnebo (2008, 2009), Horsten & Leitgeb (2009), Horsten (2010), Leitgeb (forthcoming). Groundedness for modal predicates: Leitgeb (2008), Fischer (forthcoming project). Groundedness and truthmakers: Liggins (2008), Scharp (under review) Grounding and the in-virtue-of relation: Fine (2010, forthcoming), Rosen (forthcoming). Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 17 / 20
An Afterthought: Semantic vs. Set-Theoretic Groundedness Philip Welch and I are working on a paper in which we build an axiomatic theory of propositional functions ( sets), with a primitive relation of aboutness ( inverse ) that determines what a propositional function quantifies over ( members) such that two propositional functions are equal iff (i) they have the same conceptual structure and (ii) they are about the same objects ( Extensionality; cf. Barwise & Etchemendy 1987) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 18 / 20
An Afterthought: Semantic vs. Set-Theoretic Groundedness Philip Welch and I are working on a paper in which we build an axiomatic theory of propositional functions ( sets), with a primitive relation of aboutness ( inverse ) that determines what a propositional function quantifies over ( members) such that two propositional functions are equal iff (i) they have the same conceptual structure and (ii) they are about the same objects ( Extensionality; cf. Barwise & Etchemendy 1987) and finally we will inductively define satisfaction and truth for propositional functions in a quasi-tarskian manner. Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 18 / 20
An Afterthought: Semantic vs. Set-Theoretic Groundedness Philip Welch and I are working on a paper in which we build an axiomatic theory of propositional functions ( sets), with a primitive relation of aboutness ( inverse ) that determines what a propositional function quantifies over ( members) such that two propositional functions are equal iff (i) they have the same conceptual structure and (ii) they are about the same objects ( Extensionality; cf. Barwise & Etchemendy 1987) and finally we will inductively define satisfaction and truth for propositional functions in a quasi-tarskian manner. every propositional function has a certain range of significance, within which lie the arguments for which the function has values. Within this range of arguments, the function is true or false; outside this range, it is nonsense. (Russell 1908) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 18 / 20
!"# $"%# " "#$%&!! Our universe PF of propositional functions, including propositions, will thus be subject to an iterative conception (cf. Incurvati 2011). Groundedness in set theory and semantics coming together? (cf. Terzian 2008) Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 19 / 20
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