Model Theory and Freges Philosophy of Language September 13, 2018 - - PowerPoint PPT Presentation

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

✔

‘x’ ≔ 32

Model Theory

x > 17

x > 17

✘

‘x’ ≔ 14

Model Theory

Model Theory

‘x’ ≔ 32 ‘x’ ≔ 18 ‘x’≔ 378 ‘x’ ≔ 17 ‘x’ ≔ 14

x > 17

Model Theory

x > 17

‘x’ ≔ 32 ‘x’ ≔ 18 ‘x’≔ 378 ‘x’ ≔ 17 ‘x’ ≔ 14

Language

Model Theory

x > 17

‘x’ ≔ 32 ‘x’ ≔ 18 ‘x’≔ 378 ‘x’ ≔ 17 ‘x’ ≔ 14

Language The World

Model Theory

x > 17

Language The World (TBD)

Model Theory

x > y y > z x > z

Language The World (TBD)

Model Theory

x > y y > z x > z

Language

Variables (to be grounded)

The World (TBD)

Model Theory

Language

Logical Symbols (defined)

x > y y > z x > z

The World (TBD)

Model Theory

Entailment

x > y z > w x > w

The World (TBD)

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 11 w = 8

Entailment

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 11 w = 8

Entailment

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 11 w = 8

Entailment

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 11 w = 8

Entailment

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 11 w = 8

✔

Entailment

Model Theory

x > y z > w x > w

x = 10 y = 5 z = 12 w = 11

✘

Entailment

Model Theory

∀I((I | = p) ⇒ (I | = h))

A premise (p) entails a hypothesis (h) iff, in every possible world in which p is true, h is also true.

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 a = a b =

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.

a b = a a =

J

• h

n b e l i e v e s t h a t J

• h

n b e l i e v e s t h a t J

• h

n b e l i e v e s t h a t J

• h

n b e l i e v e s t h a t

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”

😎

Sense and Reference

Compositionality

John loves Mary.

Compositionality

John loves Mary.

John(x): entity -> {0,1} 1 if x is John else 0

Compositionality

John loves Mary.

Mary(x): entity -> {0,1} 1 if x is Mary else 0 John(x): entity -> {0,1} 1 if x is John else 0

Compositionality

John loves Mary.

Mary(x): entity -> {0,1} 1 if x is Mary else 0 f s.t. f(y) =1 if y loves x else 0 loves(x): entity -> {f: entity -> {0,1}} John(x): entity -> {0,1} 1 if x is John else 0

Compositionality

John loves Mary.

Mary(x) the idea of Mary the idea of love loves(x) John(x) the idea of John

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?

• 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

• f 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?

Discussion! How do we know what needs to be resolved? What about in real, complicated sentences?

Discussion! Is there a difference between linguistic meaning and beliefs?

• The Principle of Identity Substitution seems to fail for propositional attitude reports due to

knowledge gaps and different ways of conceiving the denotation of the term. If we had all the information about the objects/terms being referred to, would this Principle still fail for propositional attitude reports?

• Why does Frege's Theory of Sense And Denotation not incorporate the general concept

what we've learned about sub languages in earlier study? The issue he presents with 'John believes Mark Twain wrote Huckleberry Finn' (a) and 'John believes Samuel Clemens wrote Huckleberry Finn' (b) do not present difficulties to the general theory at all, so long as a general domain changes when in the context of Johns mind….The issue here is a conflation of knowledge domains, not of denotation.

• …in the case “John believes that Mark Twain wrote Huckleberry Finn.” is true, the document

states that by substituting in the name, “John believes that Samuel Clemens wrote Huckleberry Finn.” is now false if John does not know who Samuel Clemens is. However, doesn’t the fact that Samuel Clemens is Mark Twain mean that if John knew that Samuel Clemens is Mark Twain, he would know that the last statement was true. Should a person’s knowledge about a subject affect how true it is? Should the “sense”

• f a name be purely based on how an individual interprets it, or something more grounded?

• When trying to understand Frege's distinction between denotation

and thought ( a distinction I am not completely convinced should exist), I am wondering how to distinguish (computationally) between 'thoughts' and nonsensical sequences. The sentence 'Mark Twain wrote Huckleberry Finn' has its own denotation and is made up of a sequence of tokens that also have individual

• denotations. We should expect an intelligent agent to be

comprehending both, whether aided by word2vec to understand a token's significance or by building its own functional representation

• f the full sentence. How do we convince the agent that the

sentence 'Mark Twain wrote I like Potatoes' is not a sensical thought, even though its individual components-- and its separate halves-- do have senses? Artificial agents do not really care (unless we tell them to) whether the sentence should make sense to a human; should they?

Discussion! Vague “thoughts” are fine, but they why isn’t it just “anything goes”?

• The vectors for "Mark Twain" and "Samuel Clemens"

would presumably be very different, even though they refer to the same entity. Does that pose any issue for computers to interpret the truthfulness of a sentence like "John believes that Samuel Clemens wrote Huckleberry Finn." ?

• It seems like humans build a sense for words through their

lived experiences. Most of our NLP algorithms however simply learn mappings from one set of words to another. Can we ever hope to capture abstract connotations of words with our current approach as NLP practitioners? What mathematical models better align with Frege's concept of semantics?

Discussion!

• VSMs. What would Frege do?

• Why does Frege (and this SEP entry's author) hold the Principle
• f Identity Substitution so dearly? Sure, the sense and

denotation distinction is a useful perspective; but why should we assume transitivity must hold for languages at the first place?

• To what extent should the ‘sense’ of a name need to vary from

speaker to speaker to be able to effectively convey language? Does every new context also bring a new ‘sense’ of a name? And ultimately, does it matter if people have very different senses

• f the same name, or is the only relevant fact for language

whether two speakers agree on the corresponding reference

• f a name (however different their senses of the same name)?"

Discussion! Is human language logic? What is the end game of modeling language?