The Propagation of Suspension of Judgment.

Abstract

It is not uncommon in the history of science and philosophy to encounter crucial experiments or crucial objections the truth-value of which we are ignorant, that is, about which we suspend judgment. Should we ignore such objections? Contrary to widespread practice, I show that in and only in some circumstances they should not be ignored, for the epistemically rational doxastic attitude is to suspend judgment also about the hypothesis that the objection targets. In other words, suspension of judgment “propagates” from the crucial objection to the hypothesis. In this paper I study under which conditions this phenomenon occurs, and discuss its significance for the topics of skepticism, scientific realism, and peer disagreement.

Appendices

Appendix A: Proof in Epistemic Logic

The following tableaux shows by reductio ad absurdum that the argument is valid.¹⁵

$$\begin{aligned} \begin{array}{lll} 1. K(p \rightarrow \lnot q), 0 \\ 2. Ip, 0 \\ 3. K(\lnot p \rightarrow q), 0 \\ 4. \lnot Iq, 0 \\ 5. (Kq \vee K \lnot q), 0; [4, \textrm{def. \, of \; I} ] \\ 6. \diamondsuit p, 0; [2] \\ 7. \diamondsuit \lnot p, 0; [2]\\ 8. 0r1; [6]\\ 9. p, 1; [6]\\ 10. 0r2; [7]\\ 11. \lnot p, 2; [7]\\ 12. (p \rightarrow \lnot q), 1; [1]\\ 13. (p \rightarrow \lnot q), 2; [1]\\ 14. (\lnot p \rightarrow q), 1; [3]\\ 15. (\lnot p \rightarrow q), 2; [3]\\ \end{array} \end{aligned}$$

Therefore, Iq.

Appendix B: Proof in 3-Valued Logic

We can also model our argument in 3-valued logic, where the truth-values T, F, and i are understood epistemically:

‘$v(p)=T$’ is interpreted as ‘the agent believes p’;

‘$v(p)=F$’ is interpreted as ‘the agent disbelieves p’;

‘$v(p)=i$’ means that the agent is ignorant as to whether p is the case; in other words, the agent suspends judgment about p.

Moreover, in 3-valued logic we can add symbols for expressing that a proposition can be ‘definitely true’ (T), with the symbol $+$ and ‘not definitely true’, i.e. either false (F) or indeterminate (i), with the symbol −. Here we do not capture the degrees or strengths of beliefs, just full beliefs or disbeliefs: we only need the objection to be crucial, so that it implies disbelief in the hypothesis (which can be captured with a material conditional, as in premise 1 above). (Cf. our remarks in Sect. 3 p. 13.)

1.1 Preliminary Justification of the Formalism

There are different 3-valued logics; the most appropriate to model our epistemic scenario is arguably Łuckasiewickz’s version. The crucial difference with strong and weak Kleene logic is the conditional, which according to Łuckasiewickz is:

$$\begin{aligned} \begin{array}{cr|lll} &{} &{} &{} q &{} \\ &{} p \supset _{L} q &{} T &{} i &{} F \\ \hline &{} T &{} T &{} i &{} F \\ &{} p \; \quad i &{} T &{} T &{} i \\ &{} F &{} T &{} T &{} T \\ \end{array} \end{aligned}$$

The center cell, with value T, distinguishes Łuckasiewickz’s from the Kleene’s conditionals. The latter give an i in this place. We must justify that this is the correct conditional to represent the logic between crucial objections and hypotheses.¹⁶ A sufficient reason seems to be that which leads Łuckasiewickz to justify his conditional. He introduced the value T in the center cell in order to preserve the logical truth in classical logic that $( p \rightarrow p )$. We are in fact interested in preserving it; this is especially clear given that we are interpreting the truth-values epistemically: strange epistemological reasons would be needed to deny that a conditional with identical antecedent and consequent is known to be true whatever the epistemic status of p (T, F, or i). The controversial case in $( p \rightarrow p )$ is when $v(p)=i$ (the center cell in the truth-table above). In this case I think that the conditional should be true, for it states that: if we are ignorant as to whether p, then we are ignorant as to whether p. This is trivially known to be true. (This, however, does not show that Łuckasiewickz’s conditional captures the semantic values of an objection p which would refute a hypothesis q. This could still be disputed, as I have not provided a justification of this.)

For the sake of completeness, and to better understand the conditions that are needed for the propagation to occur, we can take the opportunity to express a slightly different version of our argument. In particular, we will get rid of the stronger premise 3, which described a scenario, optimistic for the opponent, in which if the objection was not the case, then the hypothesis was true. The expressive capabilities of 3-valued logic easily capture the argument we are looking for:

$$\begin{aligned} \begin{array}{l} 1. (p \rightarrow \lnot q), + \\ 2. (p \vee \lnot p), - \\ \hline C. q, - \\ \end{array} \end{aligned}$$

The untrue disjunction of premise 2 is a way to express that the value of p is indeterminate, that is, i; that is to say, that we suspend judgment about p. (This is a convoluted way of expressing that p is neither true nor false.)

We can see that the argument is valid by looking at the corresponding truth-table of a conditional with the consequent negated (where, still, the left-column values represent p’s values and the top-row values represent q’s values (not $\lnot q$)), which follows from the table for the conditional above:

$$\begin{aligned} \begin{array}{cr|lll} &{} &{} &{} q &{} \\ &{} p \rightarrow _{L} \lnot q &{} T &{} i &{} F \\ \hline &{} T &{} F &{} i &{} V \\ &{} p \; \quad i &{} i &{} T &{} T \\ &{} F &{} T &{} T &{} T \\ \end{array} \end{aligned}$$

Focusing on the row where the crucial objection p has an indeterminate epistemic truth-value (the row in the middle), we have two cases for the hypothesis q where the conditional is, as we are supposing, true: the values where q is false or indeterminate. That is, as C states, the hypothesis q is known to be not true.

Additionally, if we would agree that our meta-language follows a 3-valued logic—and we have been doing so from the beginning, where I introduced suspension of judgment as an acceptable doxastic state alongside belief and disbelief—we could interpret the ‘or’ above accordingly, concluding that the value of q is just indeterminate, that is, that we should suspend judgment about the hypothesis q.¹⁷

We can also see that the argument is valid with a proof by reductio in the following tableaux:¹⁸

$$\begin{aligned} \begin{array}{lll} 1. (p \rightarrow \lnot q), + \\ 2. (p \vee \lnot p), - \\ 3. q, + \\ \end{array} \end{aligned}$$

Therefore, $q, -$.

Appendix C: Propagation from a Premise to Its Conclusion

This phenomenon can also be rephrased so that it occurs from the premise of an argument to its conclusion.

Rosa (2019) denotes as the ‘logical principle of agnosticism’ (A5) the principle according to which one shouldn’t disbelieve a conclusion and suspend judgment about one of the premises. Formally (with irrelevant variations for the sake of clarity),

$$\begin{aligned} (A5): \; \; \; \qquad \mathrm {If} \; \; \Phi _1, \ldots , \Phi _n, \vDash \Psi \; \; \mathrm { then } \; \; \lnot ( K \Phi _1 \wedge \cdots \wedge K \Phi _{n-1} \wedge S \Phi _n \wedge K \lnot \Psi ). \end{aligned}$$

where, relative to an agent, Kp stands for ‘knows that p’ and Sp stands for ‘suspends judgment about whether p’.¹⁹ In our case, it follows from (A5) that, since one is agnostic about one of the premises, one should not believe that the conclusion is false. That is,

$$\begin{aligned} ( S \Phi _n \rightarrow \lnot K \lnot \Psi ) \end{aligned}$$

The conclusion $\Psi$ is the objection’s conclusion, as in the example of §2.5, that ‘the evidence from the unobservable universe may completely differ from the evidence of the observable region’. From (A5) we thus arrive at the result that one does not know that $\lnot \Psi$; that is to say, one should not believe that the evidence from the unobservable universe describes a universe similar to that from the observable region. This is to say, as I have argued above, that one should not believe the objection’s conclusion to be false.

An informal explanation of why (A5) holds is that, given that there are premises that lead to $\Psi$ and we are ignorant as to whether one of those premises is the case, $\Psi$ is an open possibility; hence, you are not justified in believing that $\Psi$ is not the case. (A5) is deduced in Rosa (2019, §3.2) from another plausible principle of agnosticism, (A2).²⁰$^{,}$²¹