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 (pϕq) depends on {ϕ}
– Tr (pTr (pϕq)q) depends on {Tr (pϕq)} . . .
– ∀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 Φlf of grounded sentences in terms of direct or indirect
dependency on the non-semantic base language L .
Define truth for the extended language LTr in a way, such that all
T-biconditionals for members of Φlf 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,
LTr = L + Tr be our object language. LTr extended by the language of set
theory (and fragments of English) is our metalanguage.
For ϕ ∈ LTr , let ValΦ (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,
LTr = L + Tr be our object language. LTr extended by the language of set
theory (and fragments of English) is our metalanguage.
For ϕ ∈ LTr , let ValΦ (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 ∈ LTr , Φ ⊆ LTr ):
Definition
ϕ depends on Φ iff
for all Ψ1 , Ψ2 ⊆ LTr : if ValΨ1 (ϕ) , ValΨ2 (ϕ) then Ψ1 ∩ Φ , Ψ2 ∩ 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,
LTr = L + Tr be our object language. LTr extended by the language of set
theory (and fragments of English) is our metalanguage.
For ϕ ∈ LTr , let ValΦ (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 ∈ LTr , Φ ⊆ LTr ):
Definition
ϕ depends on Φ iff
for all Ψ1 , Ψ2 ⊆ LTr : if ValΨ1 (ϕ) , ValΨ2 (ϕ) then Ψ1 ∩ Φ , Ψ2 ∩ 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 ⊆ LTr : ValΨ (ϕ) = ValΨ∩Φ (ϕ)
Hannes Leitgeb (LMU Munich)
Semantic Groundedness
August 2011
10 / 20
Example
1
For ϕ ∈ L : ϕ depends on ∅.
2
Tr (pϕq), ¬Tr (pϕq) depend on {ϕ}.
3
The liar sentence λ depends on {λ}
(but λ ∨ ¬λ, λ ∧ ¬λ depend on ∅).
4
(Tr (pαq) → Tr (pβq)) ∧ (¬Tr (pαq) → Tr (pγq)) 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 LTr .
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 ∈ LTr , for all Φ, Ψ ⊆ LTr :
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 LTr .
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 ∈ LTr | 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 ∈ LTr | ϕ depends on Φ } is closed under sentential operations,
substitutional quantification, logical equivalence.
Furthermore, one can iterate D −1 :
Lemma
For all Φ, Ψ ⊆ LTr : if Φ ⊆ Ψ, then D −1 (Φ) ⊆ D −1 (Ψ).
There is a least fixed point Φlf of D −1 .
For all ϕ ∈ LTr :
ϕ ∈ Φlf iff ϕ depends on Φlf
ϕ ∈ Φlf iff Tr (pϕq) ∈ Φlf .
L & Φlf & LTr , and Φlf satisfies the closure conditions from above.
Hannes Leitgeb (LMU Munich)
Semantic Groundedness
August 2011
13 / 20
Definition
For ϕ ∈ LTr :
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 ∈ Φlf .
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 ∈ LTr :
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 ∈ Φlf .
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 ⊆ LTr : 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 (pϕq) 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 (p2 + 2 = 4q) depends directly on non-semantic soa’s.
Tr (pTr (p2 + 2 = 4q)q) 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 Γlf , such that for all ϕ ∈ Φlf :
ϕ ∈ Γlf iff ValΓlf (ϕ) = 1.
Hannes Leitgeb (LMU Munich)
Semantic Groundedness
August 2011
16 / 20
Finally, truth:
Theorem
There is a set Γlf , such that for all ϕ ∈ Φlf :
ϕ ∈ Γlf iff ValΓlf (ϕ) = 1.
So if we finally define for ϕ ∈ LTr ,
Definition
ϕ is true (in-LTr ) iff ϕ ∈ Γlf .
then this definition is formally correct and entails all T-biconditionals for Φlf .
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 Γlf , such that for all ϕ ∈ Φlf :
ϕ ∈ Γlf iff ValΓlf (ϕ) = 1.
So if we finally define for ϕ ∈ LTr ,
Definition
ϕ is true (in-LTr ) iff ϕ ∈ Γlf .
then this definition is formally correct and entails all T-biconditionals for Φlf .
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 Φlf , and
more generally there is no sentence in LTr for which Φlf 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
Bibliography:
Barwise, J. and J. Etchemendy (1987): The Liar, Oxford: Oxford University Press.
Field, H. (2008): Saving Truth from Paradox, Oxford: Oxford University Press.
Fine, K. (2010): “Some puzzles of ground,” Notre Dame Journal of Formal Logic 51, 97–118.
Fine, K. (forthcoming): “The pure logic of ground,” Review of Symbolic Logic.
Gaifman, H. (1992): “Pointers to truth,” The Journal of Philosophy 89, 223–261.
Herzberger, H. (1970): “Paradoxes of grounding in semantics,” Journal of Philosophy 67, 145–167.
Horsten, L. (2010): “Impredicative identity criteria,” Philosophy and Phenomenological Research 80, 411–439.
Horsten, L. and H. Leitgeb (2009): “How abstraction works,” in: A. Hieke and H. Leitgeb (eds.), Reduction, Abstraction, Analysis,
Frankfurt: Ontos Press, 217–238.
Incurvati, L. (2011): “How to be a minimalist about sets,” forthcoming in Philosophical Studies.
Kremer, M. (1988): “Kripke and the logic of truth,” Journal of Philosophical Logic 17, 225–278.
Kripke, S.A. (1975): “Outline of a theory of truth,” Journal of Philosophy 72, 690–716.
Leitgeb, H. (2002): “What is a self-referential sentence? Critical remarks on the alleged (non-)circularity of Yablo’s paradox,” Logique et
Analyse 177–178, 3–14.
Leitgeb, H. (2005): “What truth depends on,” Journal of Philosophical Logic 34, 155–192.
Leitgeb, H., (2008): “Towards a logic of type-free modality and truth,” in: C. Dimitracopoulos et al. (eds.), Logic Colloquium 05, Lecture
Notes in Logic, Cambridge: Cambridge University Press, 68–84.
Leitgeb, H. (2011): “Abstraction grounded. A note on abstraction and truth,” to appear in: P. Ebert and M. Rossberg (eds.),
Abstractionism in Mathematics: Status Belli.
Liggins, D. (2008): “Truthmakers and the groundedness of truth,” Proceedings of the Aristotelian Society 108, 177–196.
Linnebo, Ø. (2008): “Structuralism and the notion of dependence,” The Philosophical Quarterly 58, 59–79.
Linnebo, Ø. (2009): “Bad company tamed,” Synthese 170, 371–391.
Meadows, T. (under review): “Truth, dependence and supervaluation: living with the ghost”.
Rosen, G. (forthcoming): “Metaphysical dependence: Grounding and reduction”.
Russel, B. (1908): “Mathematical Logic as Based on the Theory of Types,” Amer. J. of Mathematics 30, 222–262.
Scharp, K. (under review): “Truthmakers for truths about truth: A problem”.
Schlenker, P. (2007): “The elimination of self-reference (generalized Yablo-eries and the theory of truth,” Journal of Philosophical Logic
36, 251–307.
Terzian, G. (2008): “Structure of the Paradoxes, Structure of the Theories: A Logical Comparison of Set Theory and Semantics”, in: A.
Hieke and H. Leitgeb (eds.), Proceedings of the 31st International Wittgenstein Symposium, Austrian Ludwig Wittgenstein Society,
346–348.
Van Vugt, F.T. and D. Bonnay (2009): “What makes a sentence be about the world? Towards a unified account of groundedness”,
unpublished manuscript.
Welch, P. (2009): “Games for truth,” The Bulletin of Symbolic Logic 15, 410–427.
Yablo, S. (1982): “Grounding, dependence, and paradox,” Journal of Philosophical Logic 11, 117–137.
Yablo, S. (1993): “Paradox without self-reference,” Analysis 53, 251–252.
Hannes Leitgeb (LMU Munich)
Semantic Groundedness
August 2011
20 / 20