Commonsense Reasoning

A Logicistic Perspective

First, let’s recall from this article what logicism was about:

L.1 An axiomatic theory of the given topic,
L.2 deduction rules given by logic.

Secondly, we briefly discussed here how logicism can be applied to language to create the theory of formal semantics called Montague Grammar. There we learned how to translate the sentence ‘every woman sees a man’ to the logical form ∀x(W OM AN (x) → ∃y(M AN (y) ∧ SEES(x, y)).

With all these preparations, here, I’d to show how this applies to current trends in commonsense reasoning. Additionally, we will even see how it relates to the different philosophical definitions of commonsense.

Trends

As we could already discuss, the methods of logicism correspond roughly to the perspectives of critical faculties and common beliefs.

To recall, the first school defines commonsense as a universal phenomenon. More concretely, it is assumed that everyone regardless of region and time period the person lives in possesses the same capabilities of judgement as given by birth.

The second school defines commonsense as universal beliefs that rather complement the natural capabilities we receive at birth. So it does not represent knowledge everyone achieves through critical judgement but principles that we take for granted in the common life without the capability to reason for it.

In commonsense reasoning, the goal is to study the structure of this implicit knowledge and beliefs of our everyday life while at the same time investigating the associated judgment. These two philosophical approaches can be also seen in the current trends of commonsense reasoning, which essentially consists of knowledge representation together with rules of inference.

John McCarthy first remarked that in order to encode explicitly these two sides of commonsense reasoning one should use formal logic. To rephrase McCarthy, the arguments for such a declarative knowledge representation are its explicitness, its modularity and its support for easy modifications in contrast to the implicit and procedural representation. Of course, at this point, a formal language similar to Montague Grammar is needed to encode and manipulate knowledge.

Can Penguins fly?

Let’s make this more concrete with the following example.

Consider the set of facts:
1. Every bird flies.
2. Tux is a penguin.
3. Tux is a bird.
4. Tux does not fly.

Of course, in classical logic, this would be a contradiction as (1) every bird flies but at the same time(3) Tux being a bird (4) does not fly. Nevertheless, the assumption is that we are still able to make sense of this set of facts by revising our knowledge/beliefs.

For example, a possible revision would be to infer:
5. Every bird but penguins fly.

To be able to infer knowledge from a consistent set of facts, one has therefore to assume that truth values are not necessarily inherited by subsets, i.e. even if for all elements in the set bird the predicate fly holds, it does not have to be the case that it holds for all elements in a subset of bird. The logics in which this phenomenon occurs are called \textit{non-monotonic}.

Logicistic Interdependency

We could see again the interdependency of the two parts L.1 and L.2 of logicism. We are confronted with a set of base facts that we consider true in our theory and need a new set of rules to correctly infer knowledge from these facts.

Elaborating further on how the two sides materialize in the challenges of modern approaches to commonsense reasoning, one can introduce the theory of Yehoshua Bar-Hillel: first, the side of L.2 that studies valid inference rules consists mostly of solving foundational issues.

In the example above, we already saw one special instance of the investigated logics. In most cases, a modified version of first-order logic is used. Further logics besides such non-monotonic logic also include temporal, spatial, deontic, epistemic, and doxastic logics. As mentioned in the example, additional reasoning patterns like belief revision were investigated.

In order to solve such foundational issues about reasoning, one has to ensure that the given logical axioms fit the observed facts and study how the system is able to make precise inferences.

Then for L.1, we need to include a set of base facts in the model or theory. The attempts to solve this issue are done by creating a database for commonsense facts.

The Cyc research project focused on creating an encyclopedia of facts that are needed for commonsense reasoning. While they created a large-scale database with millions of facts, it was only their intention to get as many correct statements as possible. Similarly to commonsense, some facts might be wrong and thus, the representational language of Cyc is imperfect.

This approach is opposing the foundational side of finding logics that create correct inferences. Nevertheless, one could argue that the two approaches combined could results in a similar interdependence as L.1 and L.2: on one hand, we have an axiomatic theory given by a large-scale database while on the other hand, we find the correct deduction rule. Therefore, it might be a good approximation to the philosophical aspects of the critical faculties and common beliefs.

At the same time, the Cyc research project also simulates partially the collective representations, the third philosophical aspect. By simply collecting commonsense facts as uttered inside a linguistic community, the knowledge we obtain corresponds to the meaning given by the social realization since we only observe communication without the intrinsic meaning of utterances.

Sources

Carlos E. Alchourrón, Peter Gärdenfors, and David Clement Makinson. 1985. On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2).

Ernest Davis and Leora Morgenstern. 2004. Introduction: Progress in formal commonsense reasoning. Artificial Intelligence, 153(1–2):1–12.

Antony Galton. 1995. Time and change for ai. Handbook of logic in artificial intelligence and logic programming, 4.

Joseph Halpern, Yoram Moses, Ronald Fagin, and Moshe Y. Vardi. 1995. Reasoning About Knowledge. MIT Press, Cambridge, MA.

Douglas B. Lenat. 1995. Cyc: a large-scale investment in knowledge infrastructure. Communications of the ACM.

John L. McCarthy. 1959. Programs with common sense. In Proceedings of the Teddington Conference on the Mechanization of Thought Processes, pages 75–91.

Jack Minker. 1993. An overview of nonmonotonic reasoning and logic programming. The Journal of Logic Programming, 17(2–4):95–126.

Henry Prakken. 1996. Two approaches to the formalisation of defeasible deontic logic. Studia Logica, 57(1).