Abstract
Analytic philosophy in the Frege-Russelian tradition often overlooks significant contributions made by American pragmatists to the field of logic. While the development of predicate logic is typically attributed to Frege, a return to Peirce’s writings from the late 19th century suggests that pragmatists were well-equipped to respond to the objections raised by analytic philosophers (the ‘great paradoxes’). Furthermore, Peirce’s reformulation of Boolean logic may have anticipated many of the solutions to these paradoxes. This essay illustrates 1) Peirce’s reactions to earlier symbolic or Boolean logic, 2) Peirce’s development of a proto-quantified predicate logic, and 3) his pragmatic motivations underlying these developments.

Peirce’s Critique of Boolean Logic

Boole’s logic allowed for the computation of validity for propositions using only three logical operators: “or,” “and,” and “not.” To make this clearer, think of “or” as a choice between options, “and” as a condition where both parts need to be true, and “not” as a way of denying something. For instance:

• “Either it will rain or it will snow.” (Here, “or” gives us two possibilities.)
• “It will rain and it will be cold.” (For this to be true, both rain and coldness must happen.)
• “It will not rain.” (This means rain is excluded as a possibility.)

In his 1870 Description of a Notation for the Logic of Relatives, Peirce critiques Boole’s theory by expanding the scope of formal logic. As he writes, “It is interesting to inquire whether [Boole’s logical algebra] cannot be extended over the whole realm of formal logic, instead of being restricted to that simplest and least useful part of the subject, the logic of absolute terms.” This critique highlights the limited capacity of Boolean logic to express a broader range of logical relationships.

Peirce’s objections to Boole’s system were primarily driven by two concerns:

1. Boolean logic does not necessarily imply the existence of the subjects of a particular proposition.
2. Boolean logic could not express a logic of relations or hypothetical propositions.

First Objection: Existential Import in Propositions

Boole’s logic symbolized particular propositions as equations of indefinite classes. For example:

• (a) “Some Y’s are X’s.” v⋅y=v⋅xv \cdot y = v \cdot xv⋅y=v⋅x
• (b) “Some Y’s are not X’s.” v⋅y=v⋅(1−x)v \cdot y = v \cdot (1 – x)v⋅y=v⋅(1−x)

Think of a “Y” as “dogs” and “X” as “animals.” The proposition (a) says “Some dogs are animals.” Boole’s system treats this as a relationship between groups, but Peirce argues that this ignores the existence of the dogs themselves. We don’t just want to say that “some dogs” are animals, but that “there are some dogs out there, and they are animals.”

To help explain, imagine you are looking for dogs in a park. Saying “some dogs are animals” is true but doesn’t actually acknowledge that the dogs exist in that park. Peirce’s point is that Boole’s logic misses that element of existence.

This issue isn’t just semantic; Peirce demonstrates that using Boole’s algebra leads to wrong conclusions. For example, if we take (b), “Some Y’s are not X’s,” Boole’s logic would incorrectly allow us to conclude that “some X’s are not Y’s,” which doesn’t follow logically in the real world. This shows how Boolean logic can misinterpret propositions that involve existence.

Second Objection: The Logic of Relations and Hypotheticals

Peirce’s second critique is about the limitations of Boolean logic when it comes to representing relations or hypothetical propositions. Let’s take a simple conditional statement, like:

• (d) “If it rains, then I will wear an umbrella.”
  In plain language, this means that rain is a condition for me wearing an umbrella. If it doesn’t rain, then the condition for wearing an umbrella is not met.

In Boolean logic, this would be represented as:
r=r∩ur = r \cap ur=r∩u, which says “instances of rain are instances where I am wearing an umbrella.” But this is an overly simplistic interpretation. It ignores the idea that “if it rains, then I will wear an umbrella” means rain is a condition for wearing the umbrella, not that both things always happen together. The logic here is more complex, as it deals with a potential scenario rather than a straightforward relationship between two things.

Peirce’s critique is that Boolean logic is not able to capture this nuance. He argues that relationships like “if…then” need a new type of logical operator, which leads him to introduce a copula for inclusion and implication.

Peirce’s Proto-Predicate Logic

To solve the problems he identified, Peirce developed a proto-predicate logic, which incorporated existential quantifiers and material implication. These modifications allowed logic to express the existence of subjects and relationships more effectively.

For example, instead of saying “Some dogs are animals” as an equation of indefinite classes, Peirce would express it as:

• (h) “Some Y’s are X’s.” ∃x((x=Y)∩(x⊃X))\exists x ((x = Y) \cap (x \supset X))∃x((x=Y)∩(x⊃X))
  This translates to: “There exists at least one Y (dog) such that it is X (an animal).” The inclusion of the existential quantifier “∃” acknowledges that we are talking about real, existing dogs.

Peirce also introduced material implication, which allows for statements like:

• (g) “X implies Y.” X⊃YX \supset YX⊃Y
  This can be understood as: “If X is true, then Y must follow.” It provides a more nuanced way to express conditional statements, like “If it rains, then I will wear an umbrella,” where the relationship between rain and wearing an umbrella is clearly defined.

Pragmatism and Logic

Peirce’s motivation behind these changes was grounded in his pragmatic philosophy. Unlike Frege, who sought a universal language of logic, Peirce was concerned with how logic could help solve practical problems. He was more interested in making logic a tool for reasoning about real-world scenarios. As he writes, the “third grade of clearness consists in such a representation of the idea that fruitful reasoning can be made to turn upon it, and that it can be applied to the resolution of difficult practical problems.”

For Peirce, the value of logic was not in its abstract truth but in its ability to guide action and understanding in everyday life. Logic, for him, had to be grounded in real, measurable effects and connected to the consequences of actions and decisions in the world.

Conclusion

Peirce’s critiques of Boolean logic and his development of proto-predicate logic were essential for advancing formal logic. By addressing the problems of existential import and relational logic, Peirce offered a more flexible system that anticipated key elements of modern predicate logic. His pragmatic approach—focusing on how logic could be used to understand and resolve practical problems—helped make his system more relevant to everyday reasoning and decision-making.

Peirce’s work offers an important bridge between the symbolic logic of Boole and the more formalized predicate logic developed by Frege. By emphasizing the need for logic to engage with real-world consequences, Peirce made crucial contributions to the future development of logical systems.

Sources:

Peirce, C. S. (1870). Description of a notation for the logic of relatives. In Collected Papers of Charles Sanders Peirce (Vol. 2, pp. 317-358). Harvard University Press.