Strength Factors: An Uncertainty System for a Quantified Modal Logic
Abstract
We present a new system for handling uncertainty in a quantified modal logic (firstorder modal logic). The system is based on both probability theory and proof theory and is derived from Chisholm’s epistemology. We concretize Chisholm’s system by grounding his undefined and primitive (i.e. foundational) concept of reasonableness in probability and proof theory. We discuss applications of the system. The system described below is a work in progress; hence we end by presenting a list of future challenges.
1 Introduction
We introduce a new system for talking about uncertainty of iterated beliefs in a quantified modal logic with belief operators. The quantified modal logic we use is based on the deontic cognitive event calculus (), which belongs to the family of cognitive calculi that have been used in modeling complex cognition. Here, we use a subset of that we term micro cognitive calculus (). Specifically, we add a system of uncertainty derived from Chisholm’s epistemology [\citeauthoryearChisholm1987].^{1}^{1}1See the SEP entry on Chisholm for a quick overview of Chisholm’s epistemology: https://plato.stanford.edu/entries/chisholm/#EpiIEpiTerPriFou. The system is a work in progress and hence the presentation here will be abstract in nature.
One of our primary motivations is to design a system of uncertainty that is easy to use in enduser facing systems. There have been many studies that show that laypeople have difficulty understanding raw probability values (e.g. see [\citeauthoryearKaye and Koehler1991]); and we believe that our approach borrowed from philosophy can pave the way for systems that can present uncertain statements in a more understandable format to lay users.
can be useful in systems that have to interact with humans and provide justifications for their uncertainty. As a demonstration of the system, we apply the system to provide a solution to the lottery paradox. Another advantage of the system is that it can be used to provide uncertainty values for counterfactual statements. Counterfactuals are statements that an agent knows for sure are false. Among other cases, counterfactuals are useful when systems have to explain their actions to users (If I had not done , then would have happened). Uncertainties for counterfactuals fall out naturally from our system. Before we discuss the calculus and present , we go through relevant prior work.
2 Prior Work
Members in the family of cognitive calculi have been used to formalize and automate highly intensional reasoning processes.^{2}^{2}2By “intensional processes”, we roughly mean processes that take into account knowledge, beliefs, desires, intentions, etc. of agents. Compare with extensional systems such as firstorder logic that do not take into account states of minds of other agents. This is not be confused with “intentional” systems which would be modeled with intensional systems. See [\citeauthoryearZalta1988] for a detailed treatment of intensionality. More recently, using we have presented an automation of the doctrine of double effect in [\citeauthoryearGovindarajulu and Bringsjord2017].^{3}^{3}3This work will be presented at IJCAI 2017. We quickly give an overview of the doctrine to illustrate the scope and expressivity of cognitive calculi such as . The doctrine of double effect is an ethical principle that has been shown to be used by both untrained laypeople and experts when faced with moral dilemmas; and it plays a central role in many legal systems. Moral dilemmas are situations in which all available options have both good and bad consequences. The doctrine states that an action in such a situation is permissible iff — (1) it is morally neutral; (2) the net good consequences outweigh the bad consequences by some large amount ; and (3) at least one or more of the good consequences are intended, and none of the bad consequences are intended. The conditions require both intensional operators and a calculus (e.g. the event calculus) for modeling commonsense reasoning and the physical world. Other tasks automated by cognitive calculi include the falsebelief task [\citeauthoryearArkoudas and Bringsjord2008] and akrasia (succumbing to temptation to violate moral principles) [\citeauthoryearBringsjord et al.2014].^{4}^{4}4Arkoudas and Bringsjord \shortciteArkoudasAndBringsjord2008Pricai introduced the general family of cognitive event calculi to which belongs. Each cognitive calculus is a sorted (i.e. typed) quantified modal logic (also known as sorted firstorder modal logic). Each calculus has a welldefined syntax and proof calculus. The proof calculus is based on natural deduction [\citeauthoryearGentzen1935], and includes all the introduction and elimination rules for firstorder logic, as well as inference schemata for the modal operators and related structures.
On the uncertainty and probability front, there have been many logics of probability, see [\citeauthoryearDemey et al.2016] for an overview. Since our system builds upon probabilities, our approach could use a variety of such systems. There has been very little work in uncertainty systems for firstorder modal logics. Among firstorder systems, the seminal work in [\citeauthoryearHalpern1990] presents a firstorder logic with modified semantics to handle probabilistic statements. We can use such a system as the foundation for our work, and use it to define the base probability function used below. (Note that we leave unspecified for now.)
3 The Formal System
The formal system is a modal extension of the the event calculus. The event calculus is a multisorted firstorder logic with a family of axiom sets. The exact axiom set is not important. The primary sorts in the system are shown below.
Sort  Description 

Agent  Human and nonhuman actors. 
Moment or Time  Time points and intervals. E.g. simple, such as , or complex, such as . 
Event  Used for events in the domain. 
ActionType  Action types are abstract actions. They are instantiated at particular times by actors. E.g.: “eating” vs. “jack eats.” 
Action  A subtype of Event for events that occur as actions by agents. 
Fluent  Used for representing states of the world in the event calculus. 
Full has a suite of modal operators and inference schemata. Here we focus on just two: an operator for belief and an operator for perception . The syntax of and inference schemata of the system are shown below. is the set of all sorts, are the core function symbols, shows the set of terms, and is the syntax for the formulae.
[linecolor=white, frametitle=Syntax, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
The above calculus lets us formalize statements of the form “John believes now that Mary perceived that it was raining.” One formalization could be:
The figure below shows the inference schemata for . captures that perceptions get turned into beliefs. is an inference schema that lets us model idealized agents that have their beliefs closed under the proof theory. While normal humans are not deductively closed, this lets us model more closely how deliberate agents such as organizations and more strategic actors reason. Assume that there is a background set of axioms we are working with.
[linecolor=white, frametitle=Inference Schemata, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
4 The Uncertainty System
In the uncertainty system, we augment the belief modal operators with a discrete set of uncertainty factors termed as strength factors. The factors are not arbitrary and are based on how derivable a proposition is for a given agent.
Chisholm’s epistemology has a primitive undefined binary relation that he terms reasonableness with which he defines a scale of strengths for beliefs one might have in a proposition. Note that Chisholm’s system is agent free while ours is agentbased. Let denote that is more reasonable than to an agent at time . We require that be asymmetric: i.e., irreflexive and antisymmetric. That is, for all , ; and for all and ,
We also require that be transitive. In addition to these conditions, we have the following five requirements governing how interacts with the logical connectives and (the first three conditions can be derived from the defintion of sketched out later):
We also add a belief consistency condition which requires that:
For convenience, we define a new operator, the withholding operator (this is simply syntactic sugar):
We now reproduce Chisholm’s system below. Note the formula used in the definitions below are metaformula and not strictly in . {mdframed}[linecolor=white, frametitle=Strength Factor Definitions, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
 Acceptable

An agent at time finds acceptable iff withholding is not more reasonable than believing in .
 Some Presumption in Favor

An agent at time has some presumption in favor of iff believing at is more reasonable than believing at time :
 Beyond Reasonable Doubt

An agent at time has beyond reasonable doubt in iff believing at is more reasonable than withholding at time :
 Evident

A formula is evident to an agent at time iff is beyond reasonable doubt and if there is a such that believing is more reasonable for at time than believing , then is certain about at time .
 Certain

An agent at time is certain about iff is beyond reasonable doubt and there is no such that believing is more reasonable for at time than believing .
The above definitions are from Chisholm but more rigorously formalized in . The definitions and the conditions give us the following theorem.
[linecolor=white, frametitle= Theorem: Higher Strength subsumes Lower Strength, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt] For any and , if , we have:
Proof: and by definition. by the second clause in the definitions of and . by the asymmetry property of .
For , we have a proof by contradiction. Assume that (in shorthand):
Using on the former and on the latter, we get
Using transitivity, we get . This violates irreflexivity, therefore .
For , if the condition for does not hold, by we have:
Using the condition for and transitivity, we get
giving us , and we started with . This violates the belief consistency condition.
The definitions almost give us except for the fact that is undefined. While Chisholm gives a careful and informal analysis of the relation, he does not provide a more precise definition. Such a definition is needed for automation. We provide a three clause defintion that is based on both probabilities and proof theory.
There are many probability logics that allow us to define probabilities over formulae. They are well studied and understood for propositional and firstorder logics. Let be the set of all formulae in . Let be a pure firstorder subset of . Assume that we have the following partial probability function defined over ^{5}^{5}5Something similar to the system in [\citeauthoryearHalpern1990] that accounts for probabilities as statistical information or degrees of belief can work.:
Then we have the first clause of our definition for . {mdframed}[linecolor=white, frametitle=Clause I. Defining , frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt] If and are defined then:
We might not always have meaningful probabilities for all propositions. For example, consider propositions of the form “I believe that Jack believes that .” It is hard to get precise numbers for such statements. In such situations, we might look at the ease of derivation of such statements given a knowledge base . ^{6}^{6}6Another possible mechanism can leverage DempsterShafer models of uncertainty for firstorder logic [\citeauthoryearNunez et al.2013]. Given two competing statements and , we can say one is more reasonable than the other if we can easily derive one more than the other from . This assumes that we can derive and from . We assume we have a cost function that lets us compute costs of proofs. There are many ways of specifying such functions. Possible candidates are length of the proof, time for computing the proof, depth vs breadth of the proof, unique symbols used in the proof etc. We leave this choice unspecified but any such function could work here. Let denote provability w.r.t. to agent at time .
[linecolor=white, frametitle=Clause II. Defining , frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
If one of and is not defined, but if and :
Clauses I and II might not always be applicable as the premises in the definitions might not always hold. A more common case could be when we cannot derive the propositions of interest from our background set of axioms . For example, if we are interested in the uncertainty values for statements that we know are false, then it should be the case that they be not derivable from our background set of axioms. In this situation, we look at and see what minimal changes we can make to it to let us derive the proposition of interest. Trivially, if we cannot derive from , we can add it to to derive it, as . This is not desirable for two reasons.
First, simply adding to might result in a contradiction. In such cases we would be looking at removing a minimal set of statements from . Second, we might prefer to add a more simpler set of propositions to rather than itself to derive . Recapping, we go from to below:
When we go from to we would like to modify the background axioms as minimally as possible. Assume that we have a similarity function for sets of formulae. We then choose and as given below ( denotes that is consistent):
Consider a statement such as “It rained last week” when it did not actually rain last week, and another statement such as “The moon is made of cheese.” Both statements denote things that did not happen, but intuitively it seems that former should be more easier to accept from what we know than the latter. There are many similarity measures which can help convey this. Analogical reasoning is one such possible measure of similarity. If the new formulae are structurally similar to existing formulae, then we might be more justified in accepting such formulae. For example, one such measure could be the analogical measure used by us in [\citeauthoryearLicato et al.2013].
Now we have the formal mechanism in place for defining the final clause in our definition for our reasonableness. Let be the distance between and closest consistent set under that lets us prove :
[linecolor=white, frametitle= Clause III. Defining , frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
If one of and is not defined, and one of and does not hold, then
The final piece of is inference rules for belief propagation with uncertainty values. This is quite straightforward. Inferences propagate uncertainty values from the premises with the lowest strength factor; and inferences happen only with beliefs that are close in their uncertainty values, with maximum difference being parametrized by , with default .
[linecolor=white, frametitle=Inference Schemata for , frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt]
5 Usage
In this section, we illustrate by applying it solve problems of foundational interest such as the lottery paradox [\citeauthoryearKyburg Jr1961, p. 197] and a toy version of a more real life example, a murder mystery example (following in the traditions of logic pedagogy). Finally, we very briefly sketch abstract scenarios in which can be used to generate uncertainty values for counterfactual statements and to generate explanations for actions.
5.1 Paradoxes: Lottery Paradox
In the lottery paradox, we have a situation in which an agent comes to believe and from a seemingly consistent set of premises describing a lottery. Our solution to the paradox is that the agent simply has different strengths of beliefs in the proposition and its negation. We first go over the paradox formalized in and then present the solution.
Let be a meticulous and perfectly accurate description of a 1,000,000,000,000ticket lottery, of which rational agent is fully apprised. Assume that from it can be proved that either ticket 1 will win or ticket 2 will win or or ticket 1,000,000,000,000 will win. Letâs write this (exclusive) disjunction as follows (here is an exclusive disjunction):
The paradox has two strands of reasoning. The first strand yields and the second strand yields with .
Strand 1: Since believes all propositions in , can then deduce from this the belief that there is at least one ticket that will win, a proposition represented as:
Strand 2: From it can be proved that the probability of a particular ticket winning is .
For the next step, note that the probability of ticket winning is lower than, say, the probability that if you walk outside a minute from now, you will be flattened on the spot by a door from a 747 that falls from a jet of that type cruising at 35,000 feet. Since you, the reader, have the rational belief that death won’t ensue if you go outside (and have this belief precisely because you believe that the odds of your sudden demise in this manner are vanishingly small), the inference to the rational belief on the part of that won’t win sails through — and this of course works for each ticket. Hence we have as a valid belief (though not derivable in from ):
From and above, we get:
Applying to the above and , we get:
The two strands are complete, and we have derived contradictory beliefs labeled S and S. Our solution consists of two new uncertainty infused strands that result in beliefs of sufficiently varying strengths that block inferences that could combine them.
Strand 3: Assume that is certain of all propositions in , then using , we have:
Strand 4: Since , using Clause I and the strength factor definitions, we have now that for all
Using the reasoning similar to that in Strand 2, we get:
Strands 3 and 4 resolve the paradox. Note that cannot be applied to S and S and churn out arbitrary propositions, as the default value of the parameter in requires beliefs to be no more than 2 levels apart.
5.2 Application: Solving a Murder
We look at a toy example in which an agent has to solve a murder that happened at time . believes that either or is the murderer. The agent knows that there is a gun gun involved in the murder and that the owner of the gun at committed the murder. also knows that is the owner of the gun initially at time .
[linecolor=white, frametitle= Presumption in Favor of Being the Murderer, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt] From just these facts, the agent has some presumption for believing that is the murderer.
Proof Sketch: All the above statements can be taken as certain beliefs of . For convenience, we consider the formulae directly without the belief operators.
In order to prove the above, we need to prove that it is easier for the agent to derive that is the murderer than to derive that is not the murderer. First, to prove the former, the agent just has to assume that ’s ownership of the gun did not change from to . Second, in order for the agent to believe that did not commit the murder but committed it, the agent must be willing to admit that something happened to change ’s ownership of the gun from time to that results in owning the gun. One possibility is that simply sold the gun to . Both the scenarios are shown as proofs in the Slate theorem proving workspace [\citeauthoryearBringsjord et al.2008] in the Appendix. Figure 1 shows a proof modulo belief operators of from and Figure 2 shows a proof of from .
If we assume that and exhaust the space of allowed additions, then it easy to see how syntactic measures of complexity will yield that as is more complex than . This lets us derive that has some presumption in favor of .
What happens if the agent knows or has a belief with certainty that ’s ownership of the gun did not change from to ?
[linecolor=white, frametitle= Beyond Reasonable Doubt that is the Murderer, frametitlebackgroundcolor=gray!25, backgroundcolor=gray!10, roundcorner=8pt] If the agent is certain that ’s ownership of the gun did not change from till , the agent has beyond reasonable doubt that she is the murderer.
Proof Sketch: In this case we directly have that:
In order to flip the last two statements above, we need to modify , but we can derive that is the murderer without any modifications, and since , it easier to believe is the murderer than to withhold that is the murderer.
5.3 Counterfactuals
At time , assume that an agent believes in a set of propositions and is interested in propositions and with and:
We may need nontrivial uncertainty values, but in this case, will assign a trivial value of to both the propositions. We can then look at closest consistent sets to under :
Clause III from the definition for reasonableness gives us:
5.4 Explanations
The definitions of the strength factors and reasonableness above can be used to generate highlevel schemas for explanations. These schemas can be used instead of simply presenting raw probability values to endusers. While we have not fleshed out such explanation schemas, we illustrate one possible schema. Say an agent performs an action on the basis of . In this case, the agent could generate an explanation that at the highest level simply says that it is more reasonable for the agent to believe than for the agent to believe in . The agent could then further explain why it was reasonable for it by using one of the three clauses in the reasonableness definition.
6 Inference Algorithm Sketch
Describing the inference algorithm in detail is beyond the scope of this paper, but we provide a highlevel sketch here.^{7}^{7}7More details can be found here: https://goo.gl/2Vz2nJ Our proof calculus is simply an extension of standard firstorder proof calculus under different modal contexts. For example, if believes that believes in a set of propositions and , then believes that believes . We convert into the pure firstorder formula and use a firstorder prover. The conversion process is a bit more nuanced as we have to handle negations, properly handle substitutions of equalities, uncertainties and transform compound formulae within iterated beliefs.
7 Conclusion and Future Work
We have presented initial steps in building a system of uncertainty that is both probability and proof theory based that could lend itself to (1) solving foundational problems; (2) being useful in applications; (3) generating uncertainty values for counterfactuals; and (4) building understandable explanations.
Shortcomings of can be cast as challenges, and many challenges exist, some relatively easy and some quite hard. Among the easy challenges are defining and experimenting with different candidates for , , and . On the more difficult side, we have to come up with tractable computational mechanisms for computing the in the definition for . Also on the difficult side, is the challenge of coming up efficient reasoning schemes. While we have an exact inference algorithm, we believe that an approximate algorithm that selectively discards beliefs in a large knowledge base during reasoning will be more useful. Future work also includes comparison with other uncertainty systems and exploration of conditions under which uncertainty values of are similar/dissimilar with other systems (thresholded appropriately).
Acknowledgements
We are grateful to the Office of Naval Research for their funding of projects titled “Advanced Logicist Machine Learning” and “Making Morally Competent Robots” and to the Air Force Office of Scientific Research for funding the project titled “Great Computation Intelligence: Mature and Further Applied” that enabled the research presented in this paper. We are also thankful for the insightful reviews provided by the three anonymous referees.
Appendix A Appendix: Slate Proofs
The figures below are vector graphics and can be zoomed to more easily read the contents.
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