Comprehensive Study Guide on Propositions and Logical Assertions

Historical and Conceptual Foundations of Propositions

• Around the turn of the twentieth century, logicians engaged in an extensive exchange across academic journals regarding the formal definition of the term proposition.
• A foundational definition of a proposition is a statement expressible in a complete sentence containing a subject and a predicate, which is either true or false.
• A proposition fundamentally makes a claim about the state of the world.

Defining Characteristics of a Proposition

• A proposition possesses four primary defining characteristics:
  • It makes a claim about the world.
  • It is expressed in a complete sentence.
  • It consists of a subject and a predicate.
  • It is either true or false.
• Except for the requirement of being expressed in a complete sentence, these characteristics are essentially four different ways of stating the same underlying concept.
• The criterion of being either true or false serves as the standard test for identifying a proposition. A statement qualifies as a proposition if it is meaningful to ask, "True or false?":
  • Example: Asking "True or false: Today is Tuesday?" is meaningful and can be verified by checking a calendar.
  • Example: Asking "True or false: It's raining right now?" is meaningful and can be verified by observing the ground outside.
• Unstructured words or single names fail this test because asking "True or false: John?" is meaningless and asserts nothing.
• Expressing a complete claim, such as "Jack is a Baptist," provides a meaningful statement that can be evaluated as true or false.

Propositions versus Statements and Sentences

• A proposition is distinct from its linguistic expression in a statement or sentence.
• A single proposition can be expressed in various ways across different languages or communication mediums:
  • Expressing that it is sunny outside uses different word sets in English, Spanish, or American Sign Language, but all three expressions convey the exact same proposition.
  • In technical philosophy, the state of affairs or fact in the world (e.g., the sunniness outside) constitutes the proposition, whereas the vocalized or written words constitute the statement.
• Strict logical definition: A proposition is a fact in the actual world or in some possible world.
  • The statement "It's raining outside right now" represents a proposition because, even if false in the current actual world, it describes a fact in a possible world where it is raining.
• Sentences and propositions are not interchangeable for two primary reasons:
  • Non-propositional sentences: Certain complete sentences, such as questions and commands, do not express propositions because they make no claims about the world.
  • Multi-propositional sentences: A single sentence can express multiple propositions. While concise authors like Ernest Hemingway frequently construct sentences with a single subject and predicate, complex authors write long, multi-clause sentences that contain multiple embedded propositions.

Compound Sentences and Conjunctions

• Independent propositions can be combined into a single sentence using punctuation and conjunctions:
  • Combining "The sky is blue" and "The grass is green" with a comma and the word "and" creates one sentence containing two distinct component propositions.
• Functional Synonymy of "And" and "But":
  • While traditional grammar highlights stylistic differences, formal logic treats "and" and "but" as functional synonyms.
  • In natural language, "but" conveys rhetorical nuances or connotations (e.g., stating someone is a great student and/but scored well on a test), whereas "and" is neutral.
  • In formal logical notation, all rhetorical nuances are stripped away, and both "and" and "but" are represented using the exact same logical symbol.
  • Other functional synonyms for "and" include "although" (e.g., "Although the sky is blue, the grass is green"), semicolons, and commas paired with conjunctions.

Connotations, Metaphors, and Textual Exceptions

• Textual Section 1.7a emphasizes connotations: the distinction between a word's literal definition and its understood meaning in common language usage, which must be carefully separated when evaluating truth values.
• Textual Section 1.7b addresses non-literal figures of speech, such as metaphors:
  • Statements like "Life is just a bowl of cherries" are metaphorical rather than literally true.
  • The phrase "Life is just a bowl of cherries" comes from a Tin Pan Alley song that is over a hundred years old.

Compact Subjects and Compact Predicates

• Sentences can compress multiple propositions into concise forms using compound subjects or compound predicates.
• Compact Subjects:
  • The sentence "Jack went up the hill, and Jill went up the hill" can be expressed more efficiently as "Jack and Jill went up the hill."
  • Although condensed into a single predicate ("went up the hill"), the sentence still asserts two distinct component propositions because two distinct subjects independently perform the action.
• Compact Predicates:
  • The sentence "Jack fell down, and Jack broke his crown" can be expressed more efficiently using the nursery rhyme phrasing: "Jack fell down and broke his crown."
  • This structure contains one subject and two predicates, expressing two distinct underlying propositions.

Non-Asserted Component Propositions: Conditionals and Disjunctions

• Certain compound sentences contain embedded component propositions without asserting that those individual propositions are true.
• Conditional Statements (If-Then Structure):
  • Consider the component propositions "I study" and "I get A's".
  • Combining them with "and" ("I study, and I get A's") asserts that both propositions are true.
  • Combining them into a conditional statement ("If I study, then I get A's") alters their logical function.
  • The clause "If I study" does not assert that the speaker actually studies.
  • The clause "then I get A's" does not assert that the speaker actually receives A's, as it is strictly dependent on studying.
  • A conditional sentence asserts neither of its embedded component propositions independently.
  • Instead, the entire conditional sentence asserts exactly one claim about the world: that a conditional relationship exists where studying leads to getting A's. This overall claim is testable.
• Disjunctive Statements (Either-Or Structure):
  • Consider the component propositions "We're having fish for supper" and "We're having chicken for supper."
  • Standard conjunctions like "and", "but", and "or" are introduced in educational media such as Saturday morning Schoolhouse Rock ("Conjunction Junction, what's your function?").
  • In logic, "or" strips assertiveness from both constituent propositions.
  • The sentence "Either we're having fish for supper or we're having chicken for supper" does not assert that fish is definitely being served, nor does it assert that chicken is definitely being served.
  • Neither component proposition is asserted as true.
  • The disjunctive sentence asserts exactly one single claim about the world: that tonight's menu will include at least one of the two options, which is a testable claim.
• Practical Context:
  • A real-world instance involves an invitation to the Luna Club in New York with a jazz musician friend, illustrating menu options and conditional expectations.