The Uses of Slippery Slope Argument

PoT seems to be a reasonable principle to hold. In fact it is very similar to the principle of mathematical induction¹⁸:

However, the principle of mathematical induction is not applicable to every predicate. For instance, even if it is true that the number 0 is a small number, it is not true that every successor of a small number will always be a small number. Otherwise the number one billion will be a small number. We can understand the falsity of PoT in this case by reflecting on this sentence¹⁹:

We cannot say what is the first number, in the series of natural numbers, which is not small, but we know that one billion is not small.

One account that has surprising consequences is the epistemic theory of vagueness, according to which there is always a precise point that draws the line between the right application of a vague predicate and its complement, even though we are not able to draw that line.²⁰ PoT is always false, even though we are not able to assign truth-values in the penumbra of the predicate.

We have also accounts that reject the excluded middle: many-valued logics, from three-valued logics to infinite valued fuzzy logics.²¹ In this case, either some instances of PoT have an indefinite truth-value or the reiterated application of the modus ponens diminishes the truth-value of the conditionals until 0, the falsity.

Although this is not the place to discuss these accounts, I would like to suggest two considerations: (a) I never understood what an epistemicist thinks that we should know in order to be able to draw the boundaries of our vague concepts for epistemic accounts and (b) it seems to me that sacrificing the excluded middle, as the many-valued logics would have it, would be too high a cost to bear: for instance, it seems to me that even if Jim was a case of penumbra of the application of rich, the sentence ‘Either Jim is rich or Jim is not rich’ would be necessarily true, because it is a logical truth. However, in many-valued logics this sentence can be not true (indefinite or with a value as ½).

I prefer an approach that favours the idea that there are ‘many permissible boundaries or cut-offs. This runs counter to a familiar tradition, according to which vagueness is characterized as absence of cut-offs’.²² This is the supervaluation approach and its relatives.

The analysis of vagueness carried out by supervaluationism can throw light on the analysis of the Sorites Paradox.²³ A vague predicate fails to divide things precisely into two sets, its positive and its negative extensions. When this predicate is applied to a borderline case, we will obtain propositions which are neither true nor false. This gap reveals a deficiency in the meaning of a vague predicate. We can remove this deficiency and replace vagueness by precision by stipulating a certain arbitrary boundary between the positive and negative extensions, a boundary within the penumbra of the concept. Thus, we get a sharpening or completion of this predicate. However, there are not only one, but many possible sharpenings or completions. In accordance with supervaluationism, we should take all of them into account. For supervaluationism, a proposition p -containing a vague concept- is true if and only if it is true for all its completions; it is false if and only if it is false for all its completions; otherwise it has no truth-value -it is indeterminate. A completion is a way of converting a vague concept into a precise one. So now we should distinguish two senses of ‘true’: ‘true’ according to a particular completion, and ‘true’ according to all completions, or supertrue. If a number x of grains of sand is in the penumbra of the concept of a heap, then it will be true for some completions and false for others that x is a heap and, therefore, it will neither be supertrue nor superfalse.

Completions should meet some constraints. In particular, propositions that are unproblematically true (false) before completion should be true (false) after completion is performed. In this way, supervaluationism retains a great part of classical logic. Thus, for instance, all tautologies of classical logic are valid in a theory of supervaluations, ‘x is a heap or x is not a heap’ -a token of the law of excluded middle- is valid, because it is true in all completions independently of the truth-value of its disjuncts.

What about Slippery Slope Arguments? Well, PoT is, in fact, superfalse: in each sharpening there is a precise boundary and, therefore, an x that is F and an x′ that is not-F. But, no vague predicate has only one boundary; all of them have a plurality of boundaries. Many boundaries of the same concept produce the impression that vague concepts are concepts without boundaries, but imprecise boundaries are still boundaries.²⁴ All sorites arguments, and with it all the Slippery Slope Arguments which have a first premise with a vague predicate, are unsound arguments in virtue of the falsity of the second premise.

Nonetheless, supervaluational accounts are not without objections. Firstly, it has been argued that these accounts cannot provide a classical notion of logical consequence because they fail to preserve certain rules of inference.²⁵ Secondly, the definition of supertrue cannot be Tarskian (‘p’ is true if and only if p) because given that we cannot retain Excluded Middle and Bivalence, we must reject Bivalence.²⁶ Finally, supervaluationism has problems with the so-called higher order vagueness: (a) given that the same notion of admissible sharpening is vague, there will be not only borderline cases, but also borderline cases of borderline cases and so on,²⁷ and (b) the introduction of the determinacy operator (‘D’: ‘True in all admissible sharpenings’) leads to a contradiction if we assume the normal semantic behaviour of vague predicates.²⁸

Even though here I do not intend to reply to these powerful objections, it is worthwhile to note that there are several strategies in the literature to overcome these objections.²⁹ I would like to emphasize only two aspects: (a) Even if we cannot retain the Tarskian notion of truth, maybe a weaker version is sufficient (from p we can infer ‘p’ is true, and vice versa),³⁰ and (b) the clear understanding afforded by the supervaluation theory provides us with an approach to vagueness as a modal phenomenon and we need, as usual in modal accounts, to distinguish among several notions of true, not only supertrue.³¹

Despite my preferences for the supervaluationist account, nothing in my argument depends on that. For epistemic accounts and for many-valued logics PoT is not universally valid, though this feature of that principle is explained in a variety of ways.³² And this is, in my view, the point that logic and argumentation theory can contribute to prevent the abuse of Slippery Slope argumentation. The fact that there is no justification to draw the line in a precise place or step does not imply lack of justification to draw the line. We are often justified to draw the line and, in this way, to manage our concepts so as to be able to stop the slippery slope somewhere. In this sense, sometimes the ideal of treating like cases alike is not attainable. It is preferable to sacrifice in some cases such ideal, in order to be able to draw the line. The impossibility to draw the line is worse, in these cases, than not treating like cases alike.³³

In brief, the Slippery Slope Arguments are suspicious arguments, not in virtue of their logical validity –they are valid from a logical point of view-, but because they are often unsound, since if the first premise of the argument includes a vague predicate, as it is always the case with the Sorites cases, then the second premise incorporating PoT is not universally true, in fact either it is false or it has a limited application.