Model Theory and Frege’s Philosophy of Language September 13, 2018 CSCI 2952C: Computational Semantics Instructor: Ellie Pavlick HTA: Arun Drelich UTA: Jonathan Chang
“The basic aim of semantics is to characterize the notion of a true sentence (under a given interpretation) and of entailment.” Montague, 1970
Model Theory
Model Theory x > 17
Model Theory x > 17 ✔ ‘x’ ≔ 32
Model Theory x > 17 ✘ ‘x’ ≔ 14
Model Theory x > 17 ‘x’ ≔ 378 ‘x’ ≔ 18 ‘x’ ≔ 14 ‘x’ ≔ 17 ‘x’ ≔ 32
Model Theory x > 17 Language ‘x’ ≔ 378 ‘x’ ≔ 18 ‘x’ ≔ 14 ‘x’ ≔ 17 ‘x’ ≔ 32
Model Theory x > 17 Language ‘x’ ≔ 378 ‘x’ ≔ 18 ‘x’ ≔ 14 ‘x’ ≔ 17 The World ‘x’ ≔ 32
Model Theory Language x > 17 The World (TBD)
Model Theory Language x > y y > z x > z The World (TBD)
Model Theory Variables Language (to be grounded) x > y y > z x > z The World (TBD)
Model Theory Language x > y y > z Logical Symbols (defined) x > z The World (TBD)
Model Theory Entailment x > y z > w x > w The World (TBD)
Model Theory Entailment x > y z > w x > w x = 10 y = 5 z = 11 w = 8
Model Theory Entailment x > y z > w x > w x = 10 y = 5 z = 11 w = 8
Model Theory Entailment x > y z > w x > w x = 10 y = 5 z = 11 w = 8
Model Theory Entailment x > y z > w x > w x = 10 y = 5 z = 11 w = 8
Model Theory Entailment x > y ✔ z > w x > w x = 10 y = 5 z = 11 w = 8
Model Theory Entailment x > y ✘ z > w x > w x = 10 y = 5 z = 12 w = 11
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Frege’s Puzzles • Identity statements: • “The morning star is identical to the evening star.” • “a = a” is true by inspection, “a = b” requires knowledge about the world • Are “equivalent” by model-theoretic definitions (true in exactly the same interpretations) a b a a = =
Frege’s Puzzles • Propositional Attitude Reports • relationship between a person and a proposition: “x believes that p” • Principle of Identity Substitution: • John believes that Mark Twain wrote Huckleberry Finn. • Mark Twain=Samuel Clemens. • Therefore, John believes that Samuel Clemens wrote Huckleberry Finn. n h n o h t J a o n n h J t a h h t h o o s t t t J J e a a s v h h e e t t v i s e s l e e e i a b b a l a v = e v = e e b i i l l e e b b
Sense and Reference
Sense and Reference
Sense and Reference “the robot” “the autonomous agent” “that little guy”
Sense and Reference “the autonomous agent” 😓
Sense and Reference “that little guy” 😎
Compositionality John loves Mary.
Compositionality John loves Mary. John(x): entity -> {0,1} 1 if x is John else 0
Compositionality John loves Mary. John(x): entity -> {0,1} Mary(x): entity -> {0,1} 1 if x is John else 0 1 if x is Mary else 0
Compositionality f s.t. f(y) =1 if y loves x else 0 loves(x): entity -> {f: entity -> {0,1}} John loves Mary. John(x): entity -> {0,1} Mary(x): entity -> {0,1} 1 if x is John else 0 1 if x is Mary else 0
Compositionality the idea of love loves(x) John loves Mary. John(x) Mary(x) the idea of John the idea of Mary
Sense and Reference “Frege calls the sense of a sentence a thought, and whereas there are only two truth values , he supposes that there are an infinite number of thoughts .”
Discussion! What does Model Theory make of “context”? • Model theory suggests that when an interpretation I happens to make a sentence S true it is said to satisfy it . But I would like to know how does this vary with the person or entity in context ? For example, "He is killing all of them" might be true (when 'he' refers to "Alfonso" and 'them' refers to pigeons) for a person actually observing that but might be false for someone who has not observed the event/situation. In short, does model theory tackle this issue or does it assume something that I might have missed while reading the paper? • I find their example not clear cut: "if there is a father, therefore there is a child" . For instance, the child could have passed away or something. I think the issue stems from the ambiguity of the term father -- in logic one might only encode 'has a child', but semantically we consider one who has had a kid to always be a father. Would this information be encoded in context, or some sort of common knowledge? Perhaps this could also be resolved by using temporal logic?
Discussion! How do we know what needs to be resolved? What about in real, complicated sentences? • The document states that “If we go on to add [extra] information, so that [some expression ] S comes to express a true or false statement..” The rest of the document discusses what to do after we have identified the parts where we need information, but how is this even decided? What decides on whether there is enough information or not to decide whether the statement can be resolved or not? Isn’t it possible to forever debate the semantics of words and what implications they have on what they statement mean, especially when grounding to real world things ?? What symbols count as nonlogical? • We can use this framework to model basic human sentences like "The third grade boy called his mom" or "Every patient was seen by a doctor", but is it feasible to apply this theory to complicated sentences like "But he was unable to spell out the details, and there is some evidence that his contemporaries (and some more recent commentators) thought he was saying that the axioms may not determine the meanings of ‘point’ and ‘line’, but they do determine those of relational terms such as ‘between’ and ‘incident with’!" (from the passage) • The formal representation of thought and concepts and reasoning is restrictive in that now, thoughts are not some sort of mental state, but concrete entities (however abstract they may be) . We want sentences (does he also just restrict to declarative sentences?) to convey thought, however, how can this happen with more complex and intertwined concepts that reflect the context they are used in and the intent of the speaker who utters it?
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