PHIL222 L16 GPLI

What GPL Can Do
Formalization of Arguments: General Predicate Logic (GPL) allows for the formalization of arguments that involve relationships among entities in a structured manner. This capability is essential for logical reasoning, demonstrating how information can be expressed mathematically to draw conclusions.
Example Argument:

• Premise 1 (P1): Brutus was a friend of Caesar. This states a relational fact about two individuals.

• Premise 2 (P2): Brutus betrayed Caesar. This indicates an action that has implications about the relationship established in P1.

• Conclusion (C): Therefore, Brutus betrayed one of his friends. This summarizes the logical deduction drawn from premises P1 and P2.
  Formalized Representation:

• P1 can be expressed as F(b, c), where F represents a friendship relation between Brutus (b) and Caesar (c).

• P2 can be formalized as B(b, c), where B indicates the act of betrayal.

• The conclusion can be formalized as ∃x(F(b, x) ∧ B(b, x)), indicating there exists some entity x such that Brutus has a friendship with x and also has betrayed x.
  What GPL Cannot Do
  Limitations in Formalization: Despite its utility, GPL has inherent limitations. Certain types of propositions cannot be effectively formalized, which constrains its applicability in specific contexts.
  Examples of Non-formulizable Statements:

• Example 1: "Alice is the tallest person." This subjective statement lacks a relational structure usable in traditional GPL.

• Example 2: "Mark Twain is Samuel Longhorne Clemens." Here, the identity of the same person is presented in two forms, which requires additional predicates for representation.

• Example 3: "Mark Twain is not Mary Ann Evans." This statement involves negation of identity, complicating formalization.

• Example 4: "There are two dogs." Such existential statements about quantity cannot be simply expressed in GPL as it lacks quantification capability.

• Example 5: "There are between ten and twenty dogs." This complexity further exemplifies the limitations of GPL in handling statements about number ranges.
  Need for Expansion: To address these limitations, an expansion of the predicate logic language is necessary. This includes new predicates and constructs to encapsulate more complex ideas and relationships.
  Introducing GPLI
  New Predicate: The introduction of a new two-place identity predicate, noted as I2, enhances GPL’s functionality. It accounts for identity relationships crucial for logical discussions about equivalent entities.
  Syntactic Aspect: This new predicate behaves like other two-place predicates in GPL, integrating seamlessly into logical expressions.
  Semantic Aspect: However, it requires special treatment due to its unique conceptual implications regarding identity and existence.
  The Identity Relation
  Concept of Identity: In the realm of logic, identity refers to the quality of being the same object or entity. Clear definitions help avoid confusion in logical relations.
  Identity Defined: An object a is said to be identical to an object b if and only if they refer to the same entity. This allows for the use of different names (a and b) to refer to one underlying object.
  Logical Identity: Each object possesses the property of being identical to itself, which can be denoted as a = a, b = b, emphasizing that identity is reflexive.
  Identity and Lack of Identity
  Visual Representation: Identity and the absence of identity can be graphically represented through diagrams that illustrate the relationships among objects, enhancing understanding of logical structures.
  Example of Two-place Relations
  Relations among Objects: To illustrate relations, a diagram can be employed to show ordered pairs which signify various relational properties among items.
  Relation: "x is taller than y":

• Set of pairs: {⟨b, a⟩, ⟨c, a⟩, ⟨d, a⟩, ⟨d, b⟩, ⟨d, c⟩}, where the first element of each pair is taller than the second.
  Relation: "x is the same height as y":

• Set of pairs: {⟨a, a⟩, ⟨b, b⟩, ⟨b, c⟩, ⟨c, b⟩, ⟨c, c⟩, ⟨d, d⟩}, indicating equality in height.
  Relation: "x is identical to y":

• Set of pairs: {⟨a, a⟩, ⟨b, b⟩, ⟨c, c⟩, ⟨d, d⟩}, reinforcing the concept of identity among objects.
  Translations of Propositions
  Different Functions of "is": The verb “is” can serve various predicational roles within sentences.

1. "Mark Twain is a novelist" - represents a predication indicating membership in a category.

2. "Mark Twain is Samuel Longhorne Clemens" - this emphasizes identity between two designations of the same person.

3. "Mark Twain is not Mary Ann Evans" - this negates identity, asserting that the two are distinct.
   Glossary for Translations:

• m: shorthand for Mary Ann Evans

• s: shorthand for Samuel Longhorne Clemens

• t: shorthand for Mark Twain

• N_x: indicates that x is classified as a novelist
  Translations of Propositions:

1. N_t, indicating that Mark Twain is a novelist.

2. I2(t, s), signifying the identity between Mark Twain and Samuel Longhorne Clemens.

3. ¬I2(t, m), indicating the distinctness of Mark Twain from Mary Ann Evans.
   Additional Translations

4. "Alice is taller than Bob":

• T(a, b), representing the relation of height between Alice and Bob.

1. "Alice is not taller than herself":

• ¬T(a, a), stating the non-tallness comparison to oneself.

1. "Alice is the tallest person":

• ∀x((P(x) ∧ ¬I2(x, a)) → T(a, x)), stating that for all x, if x is a person and not identical to Alice, then Alice is taller than x.

1. "Mark Twain is taller than some other novelist":

• ∃x((N(x) ∧ ¬I2(x, t)) ∧ T(t, x)), signifying the existence of some novelist taller than Mark Twain.

1. "Samuel Longhorne Clemens is taller than someone other than Mary Ann Evans":

• ∃x((P(x) ∧ ¬I2(x, m)) ∧ T(s, x)), indicating an existence of a person taller than Mary Ann Evans, distinct from Samuel Longhorne Clemens.

1. "Mark Twain is taller than everyone except Samuel Longhorne Clemens":

• ∀x((P(x) ∧ ¬I2(x, s)) → T(t, x)), asserting Mark Twain's superiority in height relative to all except himself.
  "Some novelist other than Mark Twain is taller than Mary Ann Evans":

• ∃x((N(x) ∧ ¬I2(x, t)) ∧ T(x, m)), stating that there exists a novelist taller than Mary Ann Evans who is not Mark Twain.
  "Mary Ann Evans is taller than every novelist apart from herself":

• ∀x((N(x) ∧ ¬I2(x, m)) → T(m, x)), suggesting Mary Ann Evans's height is greater than any novelist excluding herself.
  Abbreviations
  Formal Abbreviation of Identity: This allows one to write I2(t1, t2) simply as t1 = t2, making expressions more concise.
  Negation Abbreviation: Writes ¬t1 = t2 in the form t1 ̸= t2, which is equivalent to ¬I2(t1, t2), thus simplifying negation of identity.
  Convenience: These abbreviation symbols serve as informal tools that facilitate easier logical discourse and comprehension without losing clarity.