Advanced Logic and Fallacies – CUET PG Philosophy – Notes

Quantification Logic

From Propositional to Predicate Logic

Propositional logic, also known as sentential logic, is the branch of logic that studies the ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences. The fundamental units are whole statements, represented by letters like p, q, and r. For example, in the argument “All men are mortal; Socrates is a man; Therefore, Socrates is mortal,” propositional logic can represent this as p, q, therefore r. However, it cannot analyze why this argument is valid. It cannot look inside the sentences to see the relationship between “men,” “mortal,” and “Socrates.” The validity of this argument depends on the internal structure of these propositions.

This is the limitation that leads us to Quantification Logic, also known as Predicate Logic or First-Order Logic. It provides the tools to analyze the internal structure of propositions by breaking them down into subjects and predicates. It introduces new symbolic machineryvariables, predicates, and quantifiers-to handle statements about quantities, such as “all,” “some,” or “none.”

Core Components of Predicate Logic

To understand quantification, we must first understand its building blocks. These are the elements that allow us to translate natural language sentences into a precise, symbolic form.

• Individuals and Constants: Individuals are the specific objects or entities that our statements are about. These can be people, places, numbers, or any specific thing. We represent specific individuals with lowercase letters from the beginning of the alphabet, called individual constants (e.g., a, b, c). For instance, ‘s’ could stand for Socrates, and ‘a’ could stand for Aristotle.
• Variables: When we want to talk about individuals in a general way, without referring to a specific one, we use individual variables. These are represented by lowercase letters from the end of the alphabet (e.g., x, y, z). A variable is a placeholder that can be replaced by an individual constant.
• Predicates: A predicate is a property or relation that can be attributed to one or more individuals. We represent predicates with uppercase letters (e.g., F, G, H).
  • A monadic predicate expresses a property of a single individual. For example, “is human” can be symbolized as Hx, where H stands for the predicate “is human” and x is a variable representing any individual. If we say “Socrates is human,” we write Hs.
  • Apolyadic predicate (or relational predicate) expresses a relation between two or more individuals. For example, “loves” is a dyadic (two-place) predicate. “Romeo loves Juliet” can be symbolized as Lrj, where L stands for “loves,” r for “Romeo,” and j for “Juliet.” “Teaches” could be a triadic (three-place) predicate, as in Txyz for “x teaches y to z.”
• Quantifiers: Quantifiers are the symbols used to specify the quantity of individuals that have a certain property. They “bind” variables, meaning they specify the range of the variable. There are two primary quantifiers.

The Universal Quantifier (∀)

The universal quantifier, symbolized as (∀x), is used to assert that all individuals of a certain kind have a particular property. It corresponds to English phrases like “for all x,” “for every x,” or “for any x.”

Consider the statement “Everything is material.” If we let Mx mean “x is material,” we can symbolize this as:

(∀x)Mx

This formula states that for any individual x you choose from the domain of discourse, that x has the property of being material.

A more complex and common form is the universal affirmative ‘A’ proposition from categorical logic: “All humans are mortal.” To symbolize this, we need two predicates: Hx for “x is a human” and Mx for “x is mortal.” We are not saying that everything in the universe is a mortal human. We are saying that if something is a human, then it is mortal. This conditional relationship is key. The correct symbolization is:

(∀x)(Hx → Mx)

This reads: “For all x, if x is a human, then x is mortal.” The use of the conditional(→) is standard for universal affirmative statements. Using a conjunction (^) here would be a mistake. (∀x)(Hx^Mx) would mean “Everything is a human and everything is mortal,”which is a much stronger and false claim.