**A Partially Truth Functional Approach to Indicative Conditionals ***
Richard Grandy
Philosophy
Rice University
rgrandy@rice.edu
Daniel Osherson
Psychology
Princeton University
osherson@princeton.edu
*This paper is largely drawn from our on-line work *Logic: A Primer for Psychologists. *It is available at http://www.ruf.rice.edu/~rgrandy/primer.pdf and http://www.princeton.edu/~osherson/primer.pdf
**A Partially Truth Functional Approach to Indicative Conditionals**
**1. ***Theories of Conditionals*
Debates about the adequacy of the material conditional as a formal rendering of the indicative use of “if then” (or its ancient Greek equivalent) date back at least 23 centuries. Exactly how to specify the class of English indicative conditionals is still unclear (Lycan, chap. 1), but paradigm cases have the form “If A then B” where A and B are statements that can “stand alone” (unlike “I were rich” in “If I were rich then I would own a yacht”). Such constructions will here be denoted “A > B.” Interpreting A > B as the material conditional implies that it is either true or false depending on just the respective truth values of A and B. Grice (1989, 158-185) and D. Lewis (1976), *inter alia*, argue for the adequacy of this account. Others have proposed that the truth value of A > B depends on something more (Lycan 2001, Gauker 2005).
Starting with Adams (1965,1975), and continuing through Edgington (1991) and Bennett (2003), many authors have argued that indicative conditionals lack truth values, and that the proper way to understand them is in terms of assertibility. A slightly different proposal is that A > B has no truth value, but is rather a conditional assertion. On this view, someone who states A > B has not yet made an assertion but is committed to B should A prove true (Quine 1982, 21; DeRose & Grandy 1999).
**2. ***A New Theory*
We propose to explore a relatively unknown alternative theory which holds that indicative conditionals sometimes have truth values, sometimes not. Specifically, we propose that A > B has the usual value when A is true. Hence, T T yields T, and T F yields F. But A > B has no truth value when A is false. We use U to denote the latter case, understood as the absence of a truth value rather than some new one.
Although we arrived at this approach in ignorance of its history, the idea was sketched in DeFinetti (1935, translated by Angell, 1995) and developed in more detail by McDermott (1996). However, there is a difference between our theory and theirs: for Definetti and McDermott, U is infectious — if a sentence contains a subsentence which lacks a truth value (on some interpretation), then the entire sentence also lacks truth value.
We use an approach closely related to supervaluationism, and consider the ways in which U might have been defined. This is perhaps made clearest by considering our truth table for conjunction.
A B A and B
T T T
T U U
T F F
U T U
U U U
U F F
F T F
F U F
F F F
Since a conjunction with false conjunct is false no matter how the other one is evaluated, we assign F to conjunctions whose component values are F U and U F. Definetti and McDermott assign U. (Kearns, 2004, has a similar truth table with different motivation.) In the same spirit, our table for A > B is:
A B A > B
T T T
T U U
T F F
U T T
U U U
U F F
F T U
F U U
F F U
Most of the lines are evident, given our approach, but the entry for U > F calls for comment. Since we are concerned with the ways U could be replaced by T or F, we consider that if the antecedent were T the conditional would be false, and if the antecedent were F the conditional would be undefined, so the only definite outcome starting from U F is F.
To accommodate U, the definition of “tautology” needs attention. We don't wish to require that tautologies always evaluate to T since that renders A > A non-tautologous (counter intuitively); for it has value U when A is F. The alternative definition requiring that the sentence is never F is too lax since both (A and not A) > B and not-[(A and not A) > B] would then be labeled tautologies. We believe that requiring of tautologies that they are true in some interpretations and false on none is just right (verification is left to the reader).
**3.*** Entailment *
Entailment is defined in terms of *invalidating partial truth assignments. *A partial truth assignment is invalidating for an argument in case
a. it renders all premises T or U
b. it renders the conclusion F or U
c. it has at least one defined (T or F) statement among the premises and conclusion.
We call an argument form *valid* iff it has no invalidating partial truth assignment. For example, A > B, A / B is valid (provided A, B are atomic) whereas A > C / (A and B) > C is not. To verify the last claim, let A and C be true and B false. Then A > C is true but (A and B) > C is undefined. For another example, transitivity A > B, B > C / A > C is not valid on our definition (let A be assigned false, B and C true). Note that our definitions of tautology and entailment are not united via a deduction theorem.
Call our theory “STV” (sometimes has a truth value), to be contrasted with the familiar NTV theory (never has a truth value).
**4. *** Evaluating the STV Theory: Syntax*
Our theory (like NTV) interprets conditionals differently from statements based on other natural language connectives, such as disjunction and conjunction (the latter don't produce U from T,F). This may seem peculiar, but there are syntactic respects in which the constructions differ. For example, there seem to be no counterfactual conjunctions or disjunctions:
*I were rich and/or I have a yacht.
Furthermore, there are grammatical transformations that work smoothly on conjunctions and disjunctions, but produce ungrammatical conditionals. Consider the following sentences:
1a Chipmunks live on Venus and chipmunks don’t mind heat..
2a. Chipmunks live on Venus or chipmunks don’t mind heat.
3a. If Chipmunks live on Venus then chipmunks don’t mind heat.
Now consider their transformations:
1b Chipmunks live on Venus and don’t mind heat.
2b Chipmunks live on Venus or don’t mind heat.
3b *If chipmunks live on Venus then don’t mind heat.
1c What lives on Venus and doesn’t mind heat?
2b What lives on Venus or doesn’t mind heat?
3b *What if lives on Venus then doesn’t mind heat?
Such syntactic dissimilarities suggest that the semantics of A > B is not parallel to the semantics of A-and-B and A-or-B, hence not truth functional in the same way. (See Lycan, 2001, for fuller discussion of the syntax of conditionals.)
**5. ***Evaluating the STV theory: Assertibility*
Given a finite set S of sentential variables, let a *distribution of probability *be any mapping of truth assignments --- themselves, maps from S to {T,F} --- to [0,1] with unit sum. We define the assertibility of a statement to be the probability of its truth given that it has a truth value. In other words, relative to a given distribution *Pr* of probability, the assertibility of statement A is:
Σ {*Pr*(α) | α makes A evaluate to T}
---------------------------------------------------------------------------------------
Σ {*Pr*(α) | α makes A evaluate to T} + Σ {*Pr*(α) | α makes A evaluate to F}
It then turns out that the assertibility of A > B equals Pr(B given A). We believe this fact explains many of the phenomena that led Adams, Edgington and others to embrace a NTV theory. Our approach has the advantage of making sense of the close connection between indicative conditionals and conditional probability, without entirely giving up truth values for the former.
It is well known that in the context of certain non-triviality conditions on a language, there is no connective ^ such that *Pr*(A ^ B) is the conditional probability Pr(B given A) (Lewis 1976 and many others). Our theory does not violate this theorem because it is the assertibility of A > B rather than its probability that is equated with conditional probability. The difference hinges on cases in which A > B is assigned no truth value.
**6. ***Some of the Things We Think STV Gets Right*
We are pleased that Strengthening the Antecedent and Transitivity fail. These were two goals of the theory because we regard the following arguments as invalid (the examples are not original with us).
If I put sugar in my coffee then it will be sweet
--------------------------------------------------
If I put sugar and kerosene in my coffee then it will be sweet.
If Queen Elizabeth dies then there will be a state funeral a fortnight later.
If a comet obliterates Earth tomorrow then Queen Elizabeth will die.
-----------------------------------------------------
If a comet obliterates Earth tomorrow then there will be a state funeral a fortnight later.
A subtler point is that while STV validates modus ponens for atoms, the following form is ruled invalid (via A,C true and B false).
A > (B > C), A / ( B > C)
McGee (1985) offers an instance of this form and argues for its invalidity. If one accepts his example (as we do sometimes), this is another plus for our approach. Other victories include STV's invalidation of the suspicious not(A > B) / A along with its validation of the inoffensive A > (B and C) / A > B. Also, the notorious [(A > B) or (B > A)] is excluded from the tautologies.
**7. *** A Disaster*
For atomic sentences A and B, STV validates A,B / A-and-B, but beyond atoms lies trouble. The inference A, B > C / (A-and-(B > C)) fails. For, let A and C be true, B false; then on the left we have one true statement and one undefined, but the conjunction is undefined.
A related problem arises for transitivity: We were pleased that STV does not validate A > B, B > C / A > C, but STV *does* validate (A > B)-and-(B > C ) / A > C. It is unacceptable that merely expressing conjunction affects validity in this way. So STV is not quite right.
**8. ***What Now?*
We have fiddled extensively with alternative truth tables for STV, and with its definition of entailment. An alternative system was discovered that is as good as the original STV but no better. It fixes the problems with conjunction above, but it produces dual problems with disjunctions. John MacFarlane suggested in discussion that we define entailment in terms of the conjunction of the premises. This would validate A, B > C / A-and-(B > C) but at the price of certifying transitivity as a valid inference form. (Recall that one of our motivations was to predict the invalidity of transitivity.) Similarly, defining validity in terms of the relative assertibility of premises and conclusion (across distributions) seems to lead nowhere.
A different approach is to restrict the formation rules to exclude unwanted embedded conditionals. For example, A > (B > C) could be barred whereas its apparent equivalent (A and B) > C could be allowed. Likewise, not-(A > B) might be excluded in favor of the equivalent A > not-B (assuming they are indeed equivalent). This idea was explored helpfully in McGee but seems to surrender too much of the recursive structure of language. In particular, to be helpful to STV, the conjunction A-and-(B > C) would need exclusion, which seems arbitrary. Conversely, instead of compromising syntactic compositionality, the truth functionality of the semantics of A > B could be abandoned, as in Lycan (2001).
Finally we could view indicative conditionals as bereft of truth values, thereby abandoning the core assumption of STV (and of many other theories). We might then adopt the Adams, Bennett, Edgington, model where assertibility replaces truth, or the Grandy and DeRose model (following Quine) where the consequent of a conditional is conditionally asserted, although the conditional is not. But our hope is rather that an identity-preserving modification of STV will ultimately come to mind.
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