SEM121 - Propositions

Main ideas & lessons

• Why sentence meaning isn’t just word meaning

  • Earlier discussion argued that meaning can’t reliably be built compositionally from individual word meanings.
  • Problems included:
    • Function words (small grammatical words)
    • Cross-linguistic differences
    • Sentences whose meaning is not captured by literal word meanings, e.g.:
      • Metaphors: “Time is money”
      • Idioms: “It’s raining cats and dogs”

• Truth-conditional / formal semantics as an alternative

  • Uses tools from propositional logic to give sentences formal meaning descriptions.
  • Focus: sentences are treated as having truth conditions—ways the world must be for the sentence to be true (or false).
  • Meaning relations between sentences can be expressed via:
    • Entailment
    • Presuppositions (mentioned as part of the broader approach)

• Central concept: propositions

  • A meta-language is used (a formal language for describing another language, e.g., to describe Japanese).
  • Truth-conditional semantics uses logical formulas.
  • The basic units in these formulas are propositions, described as:
    • The basic semantic content of a sentence
    • What a sentence “says about the world”
  • Key properties of propositions:
    • They have a truth value (true/false)
    • They can be known, believed, doubted, asserted, denied, queried
    • Propositions are constant across translations (the proposition content is the same; only the expression differs by language)
    • Truth depends on:
      • World knowledge
      • Additional parameters, such as:
        • Speaker (e.g., what the speaker knows)
        • Time and place context (e.g., “The sky is blue” might depend on when/where “daytime” is referenced, per the example setup)
  • Some propositions are:
    • Analytically true (true by meaning alone)
      • Tautologies (always true, e.g., “Either it’s raining or it’s not raining”)
      • Phrases like “Stupid people are stupid” as “informationally useless” tautologies
    • Analytically false (contradictions)
      • Contradictions can be:
        • Internal: “Today is not today”
        • Across propositions: “Brutus killed Caesar” but “Caesar is not dead”
    • Synthetic (not determined by meaning alone; must be checked against reality)
      • “My father is a rocket scientist”
      • “All linguists are unhappy” (speaker uncertainty is noted)

Propositional logic methodology (connectives + how complex propositions work)

1) Build complex propositions from simple propositions

• Use logical constants / connectives (functional operators).
• Five main connectives are taught:
  • Conjunction (AND)
  • Disjunction (OR)
  • Implication (IF…THEN)
  • Negation (NOT)
  • Equivalence (IF AND ONLY IF, abbreviated as “iff”)

2) Determine truth values via truth tables

• The truth value of a complex proposition is fully determined by:
  • The truth values of its component propositions
  • The specified meaning of each connective

3) Connective-by-connective behavior (as described)

• Conjunction (P ∧ Q) — “and”

  • True only if both P and Q are true
  • Becomes false if:
    • P is false, or
    • Q is false, or
    • either conjunct is false

• Disjunction (P ∨ Q) — “or”

  • The video distinguishes two interpretations:

  • Inclusive OR (described first)

    • True if at least one disjunct is true
    • False only if both are false

  • Exclusive OR (described later)

    • Triggered by phrasing like “either P or Q” (“a stricter type of disjunction”)
    • True only if exactly one disjunct is true
    • False if both are true
    • (The example uses the “either…or” style to motivate the need for a different truth table.)

• Implication (P → Q) — “if P then Q”

  • Only false when:
    • P is true, and
    • Q is false
  • In all other cases, the implication is treated as true, including when:
    • P is false (the example argues it can’t be “invalidated”)
    • Q is true

• Negation (¬P) — “not P”

  • Produces the opposite truth value of P:
    • If P is true, ¬P is false
    • If P is false, ¬P is true
  • Example mentioned: “Mary is not a woman” vs “Mary is a woman”

• Equivalence (P ↔ Q) — “P iff Q” (“if and only if”)

  • True when P and Q have the same truth value
  • False when P and Q differ
  • Example: “Mary is a woman iff she is female”

Limitations: what propositional logic leaves out

• Propositional logic loses ordering/relational constraints

  • Example:
    • “Lynda arrived late and missed her plane” vs
    • “Lynda missed her plane and arrived late”
  • They can share the same truth values under conjunction, but natural language acceptability depends on event ordering, so they don’t mean the same thing in context.

• Propositional logic ignores contrast/adversative and other discourse relations

  • Example pair:
    • “My parents were poor and honest” vs “My parents were poor but honest”
  • Same truth-value structure, but “but” introduces contrast/adversity that logic doesn’t capture just by “and.”

• Adverbial relations are not represented

  • Example:
    • “Linda studied a lot and passed her exams”
    • vs “Linda didn’t study at all and passed her exams”
  • Even with the same broad conjunction, propositional logic does not encode:
    • Causal/concessive/contrast meanings
  • The video explicitly notes that notions signaled by words like “yet,” “still,” “already” are ignored.

• Other meaning dimensions are outside propositional logic

  • Mentioned dimensions from semantics (e.g., attributed to Jeffrey/Leech in the subtitle):
    • Stylistic meaning
    • Affective meaning
    • Attitudes
    • Politeness degrees
  • These “additional dimensions” are not part of propositional logic.

What the course plans to do next

After presenting propositional logic’s machinery, the next steps are:

• Entailment: how to draw true inferences from true propositions
• Predicate logic: formalizing the internal semantic structure of propositions

These will be covered in separate lectures (as stated).

Speakers / sources featured

• Jeffrey Leech (mentioned as associated with semantic dimensions)