Questions: Model-Theoretic Semantics and Truth Conditions

Question 1

In a model M with domain D = {Fido, Rex, Luna}, the predicate 'barks' is assigned the set {Fido, Rex}. The sentence 'Luna barks' is:

• A. True, because Luna is in the domain D
• B. False, because Luna is not in the extension of 'barks' in M
• C. Neither true nor false — model-theoretic semantics only evaluates quantified sentences
• D. True or false depending on the possible world — a single model cannot determine truth value

Answer: (B)

Model-theoretic truth is a relation between expressions and models. 'Luna barks' is true in M if and only if the entity assigned to 'Luna' is a member of the set assigned to 'barks.' Since Luna ∉ {Fido, Rex}, the sentence is false in M. Being in the domain D is not sufficient — D contains all entities that exist in the model, but a predicate's extension is the specific subset that has the property. This is the core mechanism: truth is determined by checking membership in the model's interpretation, not by any external facts.

Question 2

'The morning star' and 'the evening star' both pick out the planet Venus. In model-theoretic terms, these expressions have the same ___ but different ___.

• A. Intension; extension — they mean the same concept but refer differently in different worlds
• B. Extension; intension — they refer to the same object now but pick it out by different rules across possible worlds
• C. Semantic value; truth condition — they denote the same entity but describe it differently
• D. Reference; sense — this distinction belongs to Frege's framework, not model-theoretic semantics

Answer: (B)

Extension is the actual referent in the current model — both expressions pick out Venus, so their extensions are identical. Intension is the function from possible worlds to extensions — the concept or mode of presentation that determines what each expression refers to in any given world. 'The morning star' picks out the brightest object in the morning sky; 'the evening star' picks out the brightest object in the evening sky. In a world where these were different objects, the expressions would have different extensions. Same extension now; different functions across all possible worlds. This is why 'The morning star is the evening star' is informative — it tells you something non-trivial about the world.

Question 3

Compositionality in model-theoretic semantics means that the meaning of 'every dog barked loudly' can be systematically computed from the meanings of its parts without separately stipulating what the whole sentence means.

• T. True
• F. False

Answer: True

Compositionality is the principle that complex expressions derive their semantic values from their parts and the rules for combining them. This is what makes model-theoretic semantics a genuine theory rather than a lookup table. 'Every dog barked loudly' is computed: 'loudly' modifies 'barked' via function application, the resulting predicate combines with the universal quantifier 'every,' which takes 'dog' as its domain restrictor and applies to the whole. Each syntactic operation corresponds to a semantic operation. This compositional structure is what allows finite vocabulary and rules to generate infinitely many interpretable sentences — mirroring natural language productivity.

Question 4

Two expressions with the same extension should have the same intension, because they refer to the same object in the world.

• T. True
• F. False

Answer: False

The morning star/evening star case directly refutes this. Both expressions have the same extension (Venus) — but different intensions, because they are defined by different criteria that could diverge across possible worlds. Intension is a function from possible worlds to extensions; if two expressions pick out the same object now but through different descriptions, they have identical extensions but distinct intensions. This distinction is essential for analyzing intensional contexts: 'Astronomers believed the morning star was the morning star' (tautology) vs. 'Astronomers believed the morning star was the evening star' (a substantive discovery) cannot be distinguished if we collapse extension and intension.

Question 5

Explain the extension/intension distinction using 'the morning star' and 'the evening star' as examples, and say why it matters for analyzing natural language meaning.

Model answer: Extension is the actual referent of an expression in a given model — what it picks out in the actual world. Both 'the morning star' and 'the evening star' have the same extension: the planet Venus. Intension is the function from possible worlds to extensions — the concept or rule that determines what an expression refers to in any possible situation. 'The morning star' picks out the brightest object visible in the morning sky; 'the evening star' picks out the brightest object in the evening sky. These are different rules, which could pick out different objects in other possible worlds. The distinction matters because natural language has intensional contexts — attitude reports, modal claims, conditionals — where substituting co-extensional expressions changes truth value. 'Mary believes Venus is Venus' is trivially true; 'Mary believes the morning star is the evening star' reports a substantive empirical belief.

Without the extension/intension distinction, we cannot explain why coreferential terms are not always intersubstitutable, why identity statements can be informative, or how modal reasoning works. Frege noticed the problem (Sinn vs. Bedeutung); possible-worlds semantics and model-theoretic intensions provide the formal framework to handle it systematically.