PHI-DSC-202 (Logic II): Unit 4: Quantification

Contact Hours: 60 | Full Marks: 100 (ESE=70/CCA=30)

Table of Contents

1. Quantification: Symbolization
2. Rules of Quantification
3. Proof Construction in Predicate Logic

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Quantification: Symbolization

**Quantification** is the process of translating categorical and singular propositions into the language of Predicate Logic (or First-Order Logic), which can express internal structure not possible with Propositional Logic. It uses **quantifiers** and **predicates**.

Key Symbols

• Predicates: Capital letters (A, B, C...) representing properties or relations (e.g., Ax = "x is an Animal").
• Individual Constants: Lowercase letters (a, b, c...) representing specific individuals (e.g., s = Socrates).
• Individual Variables: Lowercase letters (x, y, z...) representing unspecified members of the universe of discourse.
• Universal Quantifier (x): Means "For all x" or "Every x".
• Existential Quantifier (∃x): Means "There is at least one x" or "Some x".

Symbolizing Standard Form Propositions

Symbolization Tip: A/E use the Arrow (→); I/O use the Dot (⋅)

Rules of Quantification

In addition to the Nineteen Rules, four new rules are required for proofs involving quantifiers. These rules allow moving between quantified statements and instantiated (singular) statements.

1. UI (Universal Instantiation):

   From a universally quantified statement, you can deduce any statement that results from replacing the variable with an **individual constant (a, b, c...)**.

   Formula: (x) (Fx) / ∴ Fa

2. UG (Universal Generalization):

   From an instantiated statement about an **arbitrarily chosen** individual (a specific variable *y* that is not tied to any premise), you can deduce the corresponding universally quantified statement.

   Formula: Fy / ∴ (x) (Fx) (Requires strict constraints on *y*)

3. EI (Existential Instantiation):

   From an existentially quantified statement, you can deduce an instantiated statement using an **individual constant (a, b, c...) that has not occurred previously** in the proof.

   Formula: (∃x) (Fx) / ∴ Fg (where 'g' is new)

4. EG (Existential Generalization):

   From an instantiated statement about a **specific** individual, you can deduce the corresponding existentially quantified statement.

   Formula: Fa / ∴ (∃x) (Fx)

Proof Construction in Predicate Logic

Proofs in Predicate Logic combine the Nineteen Rules (for propositional steps) with the four Rules of Quantification (for handling quantifiers).

Steps in a Quantified Proof:

1. **Symbolize:** Translate the premises and conclusion into symbolic quantified form.
2. **Instantiate:** Use **UI** and **EI** to remove the quantifiers from the premises, yielding instantiated statements (propositional logic statements). **EI must be used first**, and always with a new constant.
3. **Deduce:** Use the Nineteen Rules (MP, MT, Simp, etc.) to manipulate the instantiated lines and derive the required intermediate instantiated conclusion.
4. **Generalize:** Use **UG** or **EG** to re-introduce the required quantifier to match the final conclusion.

Common Mistake: The Instantiation Rule Constraint

The biggest pitfall is violating the **EI** constraint: **The constant used in EI must be NEW.** Also, violate the **UG** constraint: You cannot generalize from a constant introduced by **EI** if that constant appears in a premise line or in the line you are trying to generalize from (if the line depends on a premise containing that constant).

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