4.2 A Neutral Framework
4.2.2 Tolerant Semantic Structures
Definition 4.2.3. We will take a sequence of wffs of L0 to be a function whose domain is some suitable initial segment of the ordinals and whose range is a subset of W F FL0.5 As usual:6
• Γ,∆ := ΓS
{hδ1, ϕi : for some δ0 ∈ dom(∆), δ1 = dom(Γ) +δ0 and
ϕ = ∆(δ0)};
• Γ, ϕ := Γ,hϕi,
where ‘:=’ is a metalinguistic symbol expressing the definition relation and dom(R) and ran(R) are, respectively, the domain and the range of R.
Definition 4.2.4. A logic L for L0 is any subset of {hΓ
0,Γ1i :
ran(Γ0),ran(Γ1)⊆W F FL0}.7
As anticipated, we start our semantic construction by developing a minimal logical basis, the basic tolerant logic T0, which already displays the core
idea of any tolerant logic—the failure of the transitivity property for the consequence relation—but which is otherwise completely neutral with regard to the issues concerning the philosophy of the logic of a vague language. By adding further and further constraints on the semantics, we will then be able to specify stronger and stronger tolerant logics. We will make use of standard lattice-theoretical semantics, introducing the modifications appropriate for obtaining failures of the transitivity property for the consequence relation.
Definition 4.2.5. AT0-structure SforL0 is a 5plehV
S, DS,S,tolS, OSi,
where:
5Every function is assumed to be total unless otherwise specified.
6Throughout, ‘Γ’, ‘∆’, ‘Θ’, ‘Λ’, ‘Ξ’ (possibly with numerical subscripts) are used as met-
alinguistic variables ranging over the set of sequences whose range is included inW F FL0.
7We will thus be working in a multiple-conclusion framework, this being required in
• VS is a non-empty set of objects (the “values”);
• DS is a non-empty subset of VS (the “designated values”) such that:
(D0) For every v0, v1 ∈ VS, if v0 ∈ DS and v0 S v1, then v1 ∈ DS
(see the next item for a definition ofS)
(DS is an upper set);
• S is a partial ordering (reflexive, anti-symmetric, transitive relation)
onVS such that:
(glb/lub20) For every v0, v1 ∈ VS, {v0, v1} has a greatest lower bound
(glb) and a least upper bound (lub)
(S thus corresponds to a lattice);
• tolS is a “tolerance” function from VS into pow(VS) (the powerset of
VS) such that:
(tol0) For every v ∈VS, v ∈tolS(v);
(tol1) For every v ∈VS, tolS(v) is an upper set.
Note in particular that tolS(v) is allowed to contain values which are
not contained in the upper set whose minimum element is v. As we will see, this “tolerating” feature of tol is crucial in generating failures of the transitivity property for the consequence relation;
• OS is a non-empty set of operations on VS. In particular,
{negS,implS} ⊆OS, where:
(neg⇒0 ) For everyv0, v1 ∈VS, ifv0 S v1, then negS(v1)S negS(v0);8
8Note that (neg⇒
0 ) and (neg⇒1 ) can be neatly packaged into the so-called “law of in- tuitionist contraposition” that, for every v0, v1 ∈ VS, if v0 S negS(v1), then v1 S
(neg⇒1 ) For everyv ∈VS,v S negS(negS(v));
(impl⇒0 ) For every v0, v1 ∈ VS, if v1 ∈ tolS(v0), then implS(v0, v1) ∈
DS;
(impl⇐0 ) For every v0, v1 ∈ VS, if implS(v0, v1) ∈ DS, then v1 ∈
tolS(v0).
Again, note in particular how impl relates to tol, and especially how (impl⇐0 ) allows implS(v0, v1) to belong toDS even ifv1 does not belong
to the upper set whose minimum element is v0. Henceforth, we will
focus on the case where {negS,implS} = OS, but it is clear how,
given the rich structure generated by VS, DS, S and tolS, many
other interesting operations may be defined and added to OS (and be
expressed by corresponding operators in some extension of L0).9 In
order to exploit the full power of (neg⇒0 ), we place another constraint on tolS:
(tol2) For every v0, v1 ∈ VS, if v1 ∈ tolS(v0), then negS(v0) ∈
tolS(negS(v1)).
A T0-structure can then be used to interpretL0 once it is equipped with an
interpretation function for W F FL0 and once suitable recursive clauses for the sentential operators are given.
9In particular, given the characteristic “lowering” behaviour of the conjunction opera-
tion (to be specified shortly), it may prove useful to define an equivalence operation equivS
such that:
(equiv⇒0 ) For everyv0, v1∈VS, ifv1∈tolS(v0) andv0∈tolS(v1), then equivS(v0, v1)∈
DS;
(equiv⇐0 ) For every v0, v1 ∈ VS, if equivS(v0, v1) ∈ DS, then v1 ∈ tolS(v0) and v0 ∈
tolS(v1),
and extend L0 with a new primitive, biconditional-like, 2ary operator ↔ expressing equivS. Save for a brief remark later (fn 18) substantiating this point, in this essay I
Definition 4.2.6. A T0-model M for L0 based on a T0-structure S is a
6ple hVM, DM,M,tolM, OM,intMi, where VM, DM, M, tolM and OM are
identical toVS,DS,S, tolS andOS respectively, and intM :V ARL0 7→VM is an interpretation function for V ARL0.
Definition 4.2.7. intM is extended to a full valuation function valM :
W F FL0 7→VM by the following recursion:
(valV ARL0) If ϕ ∈V ARL0, valM(ϕ) = intM(ϕ); (val¬) valM(¬ϕ) = negM(valM(ϕ));
(val∧) valM(ϕ∧ψ) = glbM({valM(ϕ),valM(ψ)});
(val∨) valM(ϕ∨ψ) = lubM({valM(ϕ),valM(ψ)});
(val→) valM(ϕ →ψ) = implM(valM(ϕ),valM(ψ)).
Determining which valuev ∈VM a wff has in M, valM a fortiori determines
whether v ∈DM or not—in other words, valM determines whether the value
of a wff is designated or not. Now, in standard many-valued semantics, the role played by designated values can be (informally) explained as follows. It is assumed that the actual semantics of an interpreted language J whose logical properties one is interested in studying exhibits at least the general featuresof the semantics used in the mathematical study of a formal language
K into whichJ can be adequately regimented. For example,J might be a fragment of English expressing first-order Peano-Dedekind arithmetic andK be a standard formal first-order language (with identity and functors): then just as, in every model of K , every sentence (closed wff) of K is assigned either 1 or 0 as value (but not both),10 so it is assumed that every sentence of J is either true or false (but not both). If the formal semantics of K
10Of course, even if usual, the particular choice of 1 and 0 to model truth and falsity
is completely conventional—any two other objects recognized by the background mathe- matical theory will do.
is many-valued, there will typically be, included in the set of all the values a sentence can be assigned in a model, a set of designated values. What do such values correspond to in the semantics of J?
The usual answer in philosophy of logic to this question is, roughly, that they correspond to the “good” values a sentence can have—that is, those which make a sentence good enough to be asserted, good enough to be believed, good enough to be acted upon etc. (see Priest [2001], p. 216).11 At least in this respect, then, what designated values correspond to plays the same role in a many-valued framework as truth plays in a two-valued framework, for, in a two-valued framework, it is truth that which warrants assertion, belief and action.
This is the place to enter a crucial clarification concerning the present use of a many-valued semantics.12 Such a semantics is here used with the
main purpose of inducing a certain (family of) logic(s). The different values are supposed to model the different levels of goodness a sentence can have in terms of its assertability, believability, enactability etc., where the notion of a level of goodness can be reduced for simple predications to the position of the relevant object in the ordering generated by the contextually relevant dimension of comparison. The values assigned to compound sentences by the semantics are supposed to model the way in which we understand the level of goodness of a compound sentence to be determined by the level of goodness of its components. Emphatically, the different intermediate levels of goodness are not different ways in which a sentence can be neither true nor false (and so neither do the extreme levels of goodness—if they exist—coincide with truth and falsity), nor do levels of goodness represent an ordering of truth among sentences (on this very last point I thus diverge from the interpreta-
11“Semantically” good enough. For it may well be that other non-semantic features
(e. g. epistemic ones) contribute to the determination of a sentence’s assertability, believ- ability, enactability etc. A similar qualification concerning truth should be understood below as implicit.
tion of the lattice-theoretical many-valued framework offered by Weatherson [2005a]). The truth about truth is that a sentence ‘P’ is true iff P, and false otherwise—and this is manifestly too simple a notion to use if one is trying to develop a non-classical logic in a classical metatheory. It is levels of goodness, understood in the minimalist way just explained, which allow us to draw the fine-grained distinctions required by non-classical reasoning (I thus agree with Michael Dummett’s strictures against those who “reduce the semantic notion of logical consequence to a purely algebraic tool” (Dum- mett [1975b], p. 293), but disagree with him when he claims that “[o]n the assumption that all our sentences possess determinate truth-values, there is simply nothing that one can think of that a truth-table would leave unex- plained concerning the meaning of the sentential operator for which it was correct” (Dummett [1975b], p. 294)).
Let me note that understanding logical operations as operating on the fine structure of levels of goodness rather than simply on truth and falsity does not mean of course that they are not significantly constrained by truth and falsity—indeed, they should be such that the logic they generate is not too strong as to rule out intuitively possible assignments of truth and falsity and not too weak as to allow intuitively impossible assignments of truth and falsity. Let me also note that an additional layer of complexity is induced by the fact that talk of ‘good values’ should presumably be vague, so that our use of a classical metalanguage risks to misrepresent what good values are. This problem connects with some of the issues mentioned in fn 2—here I will only add that the risk is at least partially averted by the quantification over models in the definition of the consequence relation and by the rejection that one of the classically described models is the intended model of a vague language. Let me finally stress that I am painfully aware that a much more detailed discussion of these issues would be needed in order to make my claims persuasive, but that I hope that what I have just said gives enough indication as to how to understand the framework to be developed.
of the consequence relation: the informal characterization of consequence as necessary truth preservation from the premises to the conclusions gets formally translated as preservation of 1 from the premises to the conclusions in every model. In a many-valued framework, it is then very natural to define consequence as preservation of designated value from the premises to the conclusions (in the sense that, if every premise has a designated value in a model M, then some conclusion also has a designated value in M). It is thus guaranteed that, when one argues validly from good premises, one will reach some good conclusion.
This is the point of entry of the crucial modification I would like to propose in order to generate failures of the transitivity property for the con- sequence relation. Consider again the tolerance function tol:13 informally, it
implements the idea that, if v0 counts as very good a value for a sentence
to have, any v1 ∈ tol(v0) will also count as good enough a value for a sen-
tence to have (to be asserted, believed, acted upon etc.). Of course, since we are working in a classical (and thus transitive) metalanguage, we cannot require that, if v0 counts as good enough a value for a sentence to have, any
v1 ∈tol(v0) will likewise count asgood enough a value for a sentence to have,
since such a principle would breed paradox (given the fact that it need not be the case that, for every v0, v1, v2 ∈ V, if v2 ∈ tol(v1) and v1 ∈ tol(v0),
then v2 ∈tol(v0)).14 Still, it might reasonably be argued that our inferential
practices with a vague language lend support to the idea that all consequence in a vague language guarantees is that, when one argues validly from very good premises, one will reach some good enough conclusion. I will not at- tempt here to establish this claim in its generality—rather, I will briefly try to support its plausibility by illustrating how it is supposed to work in a particular case.
13Henceforth, I will drop subscripts for models and structures if no ambiguity threatens. 14Letting tol∗ = {hX, Yi : X ⊆ V and, for every v0, v0 ∈ Y iff, for some v1 ∈ X,
v0∈tol(v1)}, the point in the text can be put by saying that tol∗ is non-idempotent, in the sense that it need not be the case that, for every X⊆V, tol∗(tol∗(X)) = tol∗(X).
(4) A man with i hairs is bald
might have a very good value, because it has been accepted on the basis of perception, or intuition, or testimony etc.
(5) If a man withi hairs is bald, then a man with i+ 1 is bald
might also have a very good value, because it belongs to the naive theory of vagueness. Given this, it is reasonable to expect that:
(6) A man with i+ 1 hairs is bald,
which presumably follows from (5) and (4), will have a good enough value, and thus that it will also be assertable, believable, enactable etc. However, it is not equally reasonable to expect that, because:
(7) If a man withi+ 1 hairs is bald, then a man with i+ 2 is bald
has also a very good value (since it too belongs to the naive theory of vague- ness),
(8) A man with i+ 2 hairs is bald,
which presumably follows from (7) and (6), will also have a good enough value: for (6) only has a good enough value, and it is not equally reasonable to expect that a sentence having a merely good enough value can always be fed as a premise into an inferential process to yield a conclusion with a good enough value (as it is the case, on the contrary, when every premise has not just a good enough value, but a very good value).
It seems to me that the description of the case I have given agrees with those which would be given by many speakers if subjected to this short stretch of a soritical series. Typically, they would accept (4), (5) and (7) (and (6))
without feeling thereby compelled to accept (8), even though they would accept the validity of both modus-ponens arguments (note that, typically, they would not feel compelled to accept (8) even if there were no evidence that it is false). Typically, they would justify this complex pattern of accep- tance by claiming that, even though in itself valid, the argument in question (i.e.modus ponens) should not be “pushed too far”. In order to explain these reactions, it is plausible to conjecture that the underlying implicit concep- tion of validity is such that it is only thought to guarantee that very good premises yield a good enough conclusion, so that it need not be thought to guarantee that a double application ofmodus ponens starting from very good premises will ultimately issue in a good enough conclusion (as the case just described makes clear). On this conception—which is the key to the failure of transitivity—validity is not a matter of preservation of anything (neither of being very good from the premises to the conclusions nor of being good enough from the premises to the conclusions), but a matter ofconnection be- tween the premises’ being very good and the conclusions’ being good enough (see section 5.3.2 for more on this point).