Questions: Formal Language and Natural Language Semantics

Question 1

A logician translates 'Mary believes the president is corrupt' into logic and then substitutes 'the CEO of Axiom Corp' for 'the president,' since they are the same person. The resulting sentence attributes to Mary the belief that the CEO of Axiom Corp is corrupt. What goes wrong?

• A. Nothing — if the president and the CEO are identical, substitution preserves truth in all contexts
• B. The substitution fails because 'the president' is an indexical whose reference shifts with context
• C. Attitude reports create opaque contexts where substituting co-referring terms can change truth value — Mary may not know the two descriptions refer to the same individual
• D. The problem is grammatical, not semantic — the substitution is syntactically ill-formed

Answer: (C)

Attitude reports like 'Mary believes that...' create what are called intensional (opaque) contexts. Standard first-order logic assumes that co-referring terms are intersubstitutable everywhere (the principle of extensionality). But within the scope of 'believes,' substitution can change truth value: Mary may sincerely believe the president is corrupt without knowing the president is the CEO. This failure of substitutivity is one of the central challenges that formal semantics must address when extending logical tools to natural language.

Question 2

The sentence 'It's raining' cannot be assigned a definite truth value by a standard model-theoretic semantics alone. What additional machinery is required?

• A. A possible-worlds framework to evaluate the sentence across all possible states of affairs
• B. A context parameter specifying at minimum a location (and possibly a time) relative to which truth is evaluated
• C. A probability distribution over rain events, since the sentence is inherently probabilistic
• D. No additional machinery — standard models assign truth values to all sentences

Answer: (B)

'It's raining' is an indexical sentence: its truth conditions depend on where and when it is uttered. Without a context specifying location (and time), there is no determinate answer. Formal semantics handles this by adding a context parameter — a tuple including speaker, location, time, etc. — that supplements the model. Option A (possible worlds) is used for modality and counterfactuals but does not by itself resolve context-dependence. Option D is wrong because standard Tarskian model theory was designed for formal languages with no indexicals.

Question 3

Indexical expressions like 'I,' 'here,' and 'now' cannot be handled adequately by standard model theory and require a separate context parameter that varies with each utterance situation.

• T. True
• F. False

Answer: True

Standard model theory assigns fixed interpretations to constants and predicates relative to a model. Indexicals violate this: 'I' refers to the speaker (who changes with each utterance), 'here' refers to the location of utterance, 'now' to the time. To handle this, Kaplan and others extended the semantic framework with a context — a tuple of speaker, location, time, world — so that the interpretation of an indexical is a function from contexts to referents. This extension is necessary and not available within standard first-order model theory alone.

Question 4

Applying formal logic to natural language is primarily a matter of translation — once you identify the correct logical form of an English sentence, standard first-order semantics handles the rest.

• T. True
• F. False

Answer: False

This is the naive view that the topic is designed to refute. Natural language features — ambiguity (the same sentence has multiple logical forms), context-dependence (truth conditions shift with speaker/location/time), opacity in attitude reports, non-truth-functional conditionals, generics, tense, aspect, modality, and questions — cannot be handled by a simple translation into first-order logic. Formal semantics for natural language is an active, ongoing empirical and theoretical enterprise that requires substantial extensions: possible-worlds semantics, type theory, dynamic logic, context parameters, and more.

Question 5

Why does the context-dependence of gradable adjectives like 'tall' pose a challenge for compositional semantics, and how does formal semantics attempt to address it?

Model answer: Gradable adjectives like 'tall' are implicitly relative to a comparison class: 'tall for a jockey' and 'tall for a basketball player' can be simultaneously true and false of the same person. Compositional semantics builds sentence meanings from parts, but if the meaning of 'tall' shifts with context rather than being a fixed predicate, the composition machinery must be extended. Formal semantics addresses this by treating gradable adjectives as relations to a standard or as functions from contexts (including a comparison class parameter) to extensions, rather than as context-independent predicates. The key insight is that compositionality is preserved, but the semantic values of context-sensitive expressions are themselves context-dependent functions rather than fixed sets.

This example generalizes to many natural language expressions: evaluative terms ('expensive,' 'large'), relational expressions ('local,' 'nearby'), and pronouns all require contextual parameters. Formal semantics must track these parameters systematically throughout the compositional derivation — a significant complication over standard first-order semantics for formal languages.