In Search of the Right Logic

Rami Koskinen

The later development of logic in the 20th century has emphasized the importance and centrality of first-order logic (FOL) at the expense of second-order logic (SOL) and other “nonstandard” logics. There are, of course, many perfectly valid mathematical and pragmatic reasons for the popularity of FOL. However, it is sometimes argued in philosophical circles that there are also deeper reasons to regard FOL as the one right Logic that grounds, or is at least behind, all normatively correct reasoning (or even the fundamental furniture of the world, whatever this means). Here, I will criticize some strands of this line of argument.

In contrast to what is sometimes said about the timeless and absolute character of logical notions, the road to modern formal logic was colored by numerous historical contingencies. Higher-order logic is actually an older theoretical invention than FOL and the extraction of the latter as its own formal system was not as obvious as one might at first think (Moore 1988; Ferreirós 2001). Nevertheless, it is clear that FOL has many properties for which there are good reasons to consider it a central logical theory. Its design seems to enclose the essential core required of almost any theory of logic. At the same time, its assumptions are innocent and its syntax and semantics simple, precise, and easy to grasp. In many ways, it seems to correspond to our intuitive ideal of a complete system of elementary logic. As a guarantee of good behavior, a number of favorable metatheoretical results have also been proven for FOL, the most important being Gödel’s completeness theorem, establishing the correspondence between syntactic deducibility and semantic entailment. This is not the case with SOL.

W. V. O. Quine (1970: 79) has argued that the completeness theorem is one of the most significant reasons for adhering to first-order languages. A somewhat similar “metatheoretical equilibrium” argument has also been provided by Leslie Tharp in his article “Which Logic Is the Right Logic?” (1975). For Tharp, FOL appears to have all the desired or expected metalogical properties that result from limiting quantifier ranges to their familiar places. The inclusion of stronger quantifiers may increase expressive power, but at the same time weakens the control of metalogical properties. Thus, FOL is not a mere compromise dictated by historical chance (Tharp 1975). However, there seems to be a weakness in this line of argumentation. One can always ask why the meta-properties favored by the logician in question are the most relevant or natural ones. Corresponding pieces of circumstantial evidence can easily be provided for SOL as well. For what it’s worth, the mathematical fact that all higher-order logic is in a sense reducible to SOL (Hintikka 1955; Shapiro 1991: 141) could be used to argue for setting the natural boundaries of logic at the second-order level.

On the other hand, the metalogical side of FOL is not entirely what one would hope from an ideal “universal language of science” either. It is a well-known fact that despite its different completeness and compactness properties, full first-order predicate logic is not decidable. For example, a computer cannot, in a manageable finite time, check whether an arbitrary first-order formula is logically valid or not. This certainly would not have pleased Leibniz, who, as early as the 17th century dreamed of a universally valid calculus that could solve all human problems. Leibniz’s dream later led to the creation of computer science. It is true that FOL contains sub-fragments that are decidable (for example, when limited to monadic predicates only), but this is in fact true for SOL, too (Boolos 1975)!

If we are looking for a system that is metalogically even more well-behaved, we could return all the way down to the traditional logic of propositions, for which the familiar truth-table method guarantees complete decidability. This could hardly be considered a totally unnatural move: for example, contra Aristotle, the ancient Stoics identified logic solely as the study of the truth-functional connectives (Ferreirós 2001: 452). However, limiting oneself to mere sentential connectives (like “and”, “or”, “if… then…”) would result in a steep decline in the expressive power of logic, and many feel that the ability of propositional calculus to adequately formalize linguistic or mathematical phenomena is radically deficient.

Then there’s the problem of ontology. For Quine (1970: 68), famously, “second-order logic is set theory in sheep’s clothing” and thus far too ontologically committed to count as a proper logic. Tharp similarly also draws attention to ontology, but from another direction: “Evidently our conceptual scheme is such that we think of the world in terms of objects and relations. Sentential Logic deals with whole sentences and, unlike [FOL], suppresses this prior analysis and prior commitment.” (1975: 18; emphasis in original). In Tharp’s view, therefore, adequate logic would appear to be forced to bend to a certain kind of ontological commitment. Despite its good metalogical properties, sentential logic is simply an inadequate tool to adequately describe various conceptual phenomena. It is on this basis that Tharp considers that FOL has a legitimate place as our unquestionable right logic. However, it is interesting that at the same time Tharp rejects SOL mainly on the basis of its metalogical properties. I do not think it is self-evident that such an approach is entirely justified. It can be argued that, in many ways, our pre-theoretical ontological commitments go beyond what is required for accepting first-order semantic tools alone. The same goes for many mathematical practices that utilize SOL (Shapiro 1991). The question arises, why should our basic logic be blind in one part of our conceptual worldview when it is already firmly tied to our conceptual practices elsewhere?

What I suggest is that the choice of a logic as a formal system of representation is always contextual and goal-bound to some extent. More specifically, there are no universal philosophical reasons that forbid the use of logics beyond FOL. For example, because of its stronger expressive power, SOL has many uses, where FOL is simply inadequate as a formal tool. Examples of this “nonfirstorderizability” include certain known sentences of natural language (“Some critics admire only one another”), numerous mathematical properties (like strong Peano induction or the least upper bound of a set of reals) and even what are traditionally deemed logical concepts like identity. SOL can deal with all of these (see Boolos 1975, 1984; Shapiro 1991).

It is probably not far off the mark to suggest that the reasons behind the success of FOL are mainly pragmatic in nature and do not stem from the rationalist arguments that try to establish why it deserves its reigning status – why it must be the one correct logic (cf. Wagner 1987). As a formal representation of logical inference, it captures patterns of reasoning shared by both ordinary and mathematical arguments. It has also enough semantic power to account for an important class of conceptual phenomena, but not all. However, this relative lack of expressive power results in ease-of-use metalogical properties elsewhere, making it, for example, a good tool for teaching.

Admitting that different goals and practices can affect the choice of a logic, does this lead to logical anarchy, where anyone can choose their own logic? Or that we have to abandon the idea that logic is normative?

Simply put, no. This is because if we want to communicate with others in the pursuit of mutual goals, we sooner or later have to converge on our conceptual tools and language schemes. Also, logical pluralism does not automatically mean that logic doesn’t have any normative power. To give a simple example, both propositional and predicate logic can be normative in how they treat the basic features of logical consequence (failing to grasp material implication in either system is usually bad news). However, due to the differences in their semantic power, they differ in how they can be used to represent various ordinary and scientific phenomena. While some problems can be nicely dealt with, say, propositional truth-tables, others might require more fine-grained expressive machinery. In other words, to quote Ilkka Pättiniemi, it is a question of picking the right tool for the right job. Or, if this sounds too mundane in the case of logic (which, after all, is the science of the general and the abstract), it is recommended to give at least some consideration for the purpose in question before deciding whether a particular logic is the right one.

References

Boolos, G. (1975). “On Second-Order Logic.” The Journal of Philosophy 72: 509–527.

Boolos, G. (1984). “To Be Is to Be a Value of a Variable (Or to Be Some Values of Some Variables).” The Journal of Philosophy 81: 430–449.

Ferreirós, J. (2001). “The Road to Modern Logic – An Interpretation.” The Bulletin of Symbolic Logic 7: 441–484.

Hintikka, J. (1955). “Reductions in the Theory of Types.” Acta Philosophica Fennica 8: 57–115.

Moore, G. (1988). “The Emergence of First-Order Logic.” In: M. Aspray & P. Kitcher (Eds.), History and Philosophy of Modern Mathematics. Minneapolis, MN: University of Minnesota Press, 95–135.

Quine, W.V.O. (1970). Philosophy of Logic. Englewood Cliffs, NJ: Prentice-Hall, Inc.

Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-order Logic. New York: Oxford University Press.

Tharp, L. (1975). “Which Logic Is the Right Logic?” Synthese 31: 1–21.

Wagner, S. (1987). “The Rationalist Conception of Logic.” Notre Dame Journal of Formal Logic 28: 3–35.

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