Counterfactual Conditionals in Quantified Modal Logic

We present a novel formalization of counterfactual conditionals in a quantified modal logic. Counterfactual conditionals play a vital role in ethical and moral reasoning. Prior work has shown that moral reasoning systems (and more generally, theory-of-mind reasoning systems) should be at least as expressive as first-order (quantified) modal logic (QML) to be well-behaved. While existing work on moral reasoning has focused on counterfactual-free QML moral reasoning, we present a fully specified and implemented formal system that includes counterfactual conditionals. We validate our model with two projects. In the first project, we demonstrate that our system can be used to model a complex moral principle, the doctrine of double effect. In the second project, we use the system to build a data-set with true and false counterfactuals as licensed by our theory, which we believe can be useful for other researchers. This project also shows that our model can be computationally feasible. Introduction Natural-language counterfactual conditionals (or simply counterfactuals) are statements that have two parts (semantically, and sometimes syntactically): an antecedent and a consequent. Counterfactual conditionals differ from standard material conditionals in that the mere falsity of the antecedent does not lead to the conditional being true. For example, the sentence “If John had gone to the doctor, John would not be sick now” is considered a counterfactual as it is usually uttered when “John has not gone to the doctor”. Note that the surface syntactic form of such conditionals might not be an explicit conditional such as “If X then Y”; for example: “John going to the doctor would have prevented John being sick now”. Material conditionals in classical logic fail when used to model such sentences. Counterfactuals occur in certain ethical principles and are often associated with moral reasoning. We present a formal computational model for such conditionals. The plan for the paper is as follows. We give a brief overview of how counterfactuals are used, focusing onmoral reasoning. Then we briefly discuss prior art in modeling counterfactuals, including computational studies of counterfactuals. We then present our formal system, used as a founCopyright c © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. dation for building our model of counterfactual conditionals. After this, we present the model itself and prove some general properties of the system. We end by discussing two projects to demonstrate how the formal model can be used. Use of Counterfactual Conditionals Counterfactual reasoning plays a vital role in human moral reasoning. For instance, the doctrine of double effect (DDE) requires counterfactual statements in its full formalization. DDE is an attractive target for building ethical machines, as numerous empirical studies have shown that human behavior in moral dilemmas is in accordance with what the doctrine (or modified versions of it) predict. Another reason for considering DDE is that many legal systems use the doctrine for defining criminality. We briefly state the doctrine below. Assume that we have an ethical hierarchy of actions as in the deontological case (e.g. forbidden, morally neutral, obligatory); see (McNamara 2010). We also assume that we have a utility or goodness function for states of the world or effects, as in the consequentialist case. Given an agent a, an action α in a situation σ at time t is said to be DDE-compliant iff (the clauses are verbatim from (Govindarajulu and Bringsjord 2017a)): DDE Informal C1 the action is not forbidden C2 the net utility or goodness of the action is greater than some positive amount γ; C3a the agent performing the action intends only the good effects; C3b the agent does not intend any of the bad effects; C4 the bad effects are not used as a means to obtain the good effects; and C5 if there are bad effects, the agent would rather the situation be different and the agent not have to perform the action. That is, the action is unavoidable. Note that while (Govindarajulu and Bringsjord 2017a) present a formalization and corresponding implementation Moral dilemmas are situations in which all available actions have large positive and negative effects. There are also exist in the literature more fine-grained hierarchies(Bringsjord 2015). and “stopwatch” test of the first four clauses above, there is no formalization of C5. Work presented here will enable such a formalization. The last clause has been rewritten below to make explicit its counterfactual nature. C5 Broken Up C5a The agent desires that the current situation be different. C5b The agent believes that if the agent itself were in a different situation, the agent would not perform the action α. Separately, (Migliore et al. 2014) have an empirical study in which they elicit subjects to produce counterfactual answers to questions in a mix of situations with and without moral content. Their answers have the form ofC5a andC5b. Their study shows with statistical significance that humans spent more time responding to situations that had moral content. This suggests the presence of non-trivial counterfactual reasoning in morally-charged situations. Counterfactual reasoning also plays an important role in the intelligence community in counterfactual forecasting (Lehner 2017). In counterfacutal forecasting, analysts try to forecast what would have happened if the situation in the past was different than what we know, and as Lehner states there is a need for formal/automated tools for counterfactual reasoning. Prior Art Most formal modeling of counterfactuals has been in work on subjunctive conditionals. While there are varying definitions of what a subjunctive conditional is, the hallmark of such conditionals is that the antecedent, while not necessarily contrary to established facts (as is the case in counterfactuals), does speak of what could hold even if it presently doesn’t; and then the consequent expresses that which would (at least supposedly) hold were this antecedent to obtain. Hence, to ease exposition herein, we simply note that (i) subjunctives are assuredly non-truth-functional conditionals, and (ii) we can take subjunctive conditionals to be a superclass of counterfactual conditionals. A lively overview of formal systems for modeling subjunctive conditionals can be found in (Nute 1984). Roughly, prior work can be divided into cotenability theories versus possible-worlds theories. In cotenability theories, a subjunctive φ > ψ holds iff (C + φ) → ψ holds. Here C is taken to be a set of laws (logical/physical/legal) cotenable with φ. One major issue with many theories of cotenability is that they at least have the appearance of circularly defining cotenability in terms of cotenability. In possible-worlds theories, semantics of subjunctive conditionals are defined in terms of possible worlds. While conceptually attractive to a degree, such approaches are problematic. For example, many E.g., “If you were to practice every day, your serve would be reliable” is a subjunctive conditional. It might not be the the case that you’re not already practicing hard. However, “If you had practiced hard, your serve would have been reliable” is a counterfactual (because, as it’s said in the literature, the antecedent is “contrary to fact”). E.g., (Lewis 1973) famously aligns each possible world with an order of relative similarity among worlds, and is thereby able possible-worlds accounts are vulnerable to proofs that certain conceptions of possible worlds are provably inconsistent (e.g. see (Bringsjord 1985)). For detailed argumentation against possible-world semantics for counterfactual conditionals, see (Ellis, Jackson, and Pargetter 1977). Relevance logics strive to fix issues such as explosion and non-relevance of antecedents and consequents in material conditionals; see (Mares 2014) for a wide-ranging overview. The main concern in relevance logics, as the name implies, is to ensure that there is some amount of relevance between an antecedent and a consequent of a conditional, and between the assumptions in a proof and its conclusions. Our model does not reflect this concern, as common notions of relevance such as variable/expression sharing becomemuddled when the system includes equality, and become even more muddled when intensionality is added. Most systems of relevance logic are primarily possibleworlds-based and share some of the same concerns we have discussed above. For example, (Mares and Fuhrmann 1995; Mares 2004) discuss relevance logics that can handle counterfactual conditionals but are all based on possible-worlds semantics, and the formulae are only extensional in nature. Work in (Pereira and Saptawijaya 2016) falls under extensional systems, and as we explain below, is not sufficient for our modeling requirements. Differently, recent work in natural language processing by (Son et al. 2017) focuses on detecting counterfactuals in social-media updates. Due to the low base rate of counterfactual statements, they use a combined rule-based and statistical method for detecting counterfactuals. Their work is on detecting (and not evaluating, analyzing further, or reasoning over) counterfactual conditionals and other counterfactual statements. Needed Expressivity Our modeling goals require a formal system F of adequate expressivity to be used in moral and other theoryof-mind reasoning tasks. F should be free of any consistency or soundness issues. In particular, F needs to avoid inconsistencies such as the one demonstrated below, modified from (Govindarajulu and Bringsjord 2017a). In the inference chain below, we have an agent a who knows that the murderer is the person who owns the gun. Agent a does not know that agentm is the murderer, but it’s true thatm is the owner of the gun. If the knowledge operator K is a simple first-order logic predicate, we get the proof shown below, which produces a contradiction from sound premises. to capture in clever fashion the idea that a counterfactual φ > ψ holds just in case the possible world satisfying φ that is the most similar to the actual world also satisfies ψ. While as is plain we are not fans of possible-worlds semantics, those attracted to such an approach to counterfactuals would do well in our opinion to survey (Fritz and Goodman 2017). By extensional logics, we refer broadly to non-modal logics such as propositional logic, first-order logic, second-order logic etc. By intensional logics, we refer to modal logics. Note that this is different from intentions which can be represented by intensional operators, just as knowledge, belief, desires etc can be represented by intensional operators. See (Zalta 1988) for an overview. Modeling Knowlege (or any Intension) in First-order Logic 1 K (a, Murderer (owner (gun))) ; given 2 ¬K (a,Murderer (m)) ; given 3 m = owner (gun) ; given 4 K (a,Murderer (m)) ; first-order inference from 3 and 1 5 ⊥ ; first-order inference from 4 and 2 Even more robust representation schemes can still result in such inconsistencies, or at least unsoundness, if the scheme is extensional in nature (Bringsjord and Govindarajulu 2012). Issues such as this arise due to uniform handling of terms that refer to the same object in all contexts. This is prevented if the formal system F is a quantified modal logic (and other sufficiently expressive intensional systems). We present one such quantified modal logic below. Background: Formal System In this section, we present the formal system in which we model counterfactual conditionals. The formal system we use is deontic cognitive event calculus (DCEC). Arkoudas and Bringsjord (2008) introduced, for their modeling of the false-belief task, the general family of cognitive event calculi to which DCEC belongs. DCEC has been used to formalize and automate highly intensionalmoral reasoning processes and principles, such as akrasia (giving in to temptation that violates moral principles) (Bringsjord et al. 2014). and the doctrine of double effect described above. Briefly, DCEC is a sorted (i.e. typed) quantified modal logic (also known as sorted first-order modal logic). The calculus has a well-defined syntax and proof calculus; outlined below. The proof calculus is based on natural deduction (Gentzen 1935), commonly used by practicing mathematicians and logicians, as well as to teach logic; the proof calculus includes all the standard introduction and elimination rules for first-order logic, as well as inference schemata for the modal operators and related structures. Syntax DCEC is a sorted calculus. A sorted system can be regarded analogous to a typed single-inheritance programming language. We show below some of the important sorts used in DCEC. Among these, the Agent, Action, and ActionType sorts are not native to the event calculus. The syntax can be thought of as having two components: a first-order core and a modal system that builds upon this first-order core. The figures below show the syntax and inference schemata of DCEC. The syntax is quantified modal logic. The first-order core of DCEC is the event calculus (Mueller 2006). Commonly used function and relation symbols of the event calculus are included. Other calculi (e.g. the situation calculus) for modeling commonsense and physical 6 The description of DCEC here is mostly a subset of the discussion in (Govindarajulu and Bringsjord 2017a) relevant for us. reasoning can be easly switched out in-place of the event calculus. The modal operators present in the calculus include the standard operators for knowledgeK, beliefB, desireD, intention I, etc. The general format of an intensional operator is K (a, t, φ), which says that agent a knows at time t the proposition φ. Here φ can in turn be any arbitrary formula. Also, note the followingmodal operators:P for perceiving a state, C for common knowledge, S for agent-to-agent communication and public announcements, B for belief, D for desire, I for intention, and finally and crucially, a dyadic deontic operator O that states when an action is obligatory or forbidden for agents. It should be noted that DCEC is one specimen in a family of easily extensible cognitive calculi. The calculus also includes a dyadic (arity = 2) deontic operatorO. It is well known that the unary ought in standard deontic logic lead to contradictions. Our dyadic version of the operator blocks the standard list of such contradictions, and beyond.