"Some x's are not y's"

Both readings this week focused on the differences in language processing in children and adults, specifically in deriving scalar implicature. They both define some scale or spectrum of descriptive words (e.g. "none", "some", "all") and show that children and adults process these words differently -- children using their logical meanings and adults using pragmatics. In other words, if we asked a child whether "some elephants have trunks", they would most likely respond true (because the logical definition of the word "some" encompasses the logical definition for the word "all"). However, if we asked an adult the same question, they would be more likely to respond false, because the weak statement "some" implies the negation of the stronger statement "all" -- thus, this statement means to adults that some elephants have trunks, but not all elephants have trunks.

The use of the example words "some", "all", and "none" brings me back to the first few weeks of CS103, in which we learned the basics of set theory and first-order logic. In set theory, the concept of a subset illustrates the distinction between the logical definitions of "some" versus "all": a set A is a subset of another set B if every element in A is also an element of B. We can say that there are some elements of B that are not in A, and the subset relation would still hold true between A and B (imagine a Venn Diagram in which circle A is completely contained within circle B, but A and B are not the same circle). However, we can also say that A is a subset of B if A and B are the same set (the same circle), that is, if all elements of B are also in A.

Much of our time in class was spent learning these definitions, but just as much was spent unlearning the pragmatic definition of "some" and relearning its logical definition (at least in the context of set theory). We ran into similar problems when negating statements in first-order logic. For example, consider the statement "all x's are y's." Pragmatically (colloquially/outside the context of a logic class), one would think that the negation of this statement is "no x's are y's." But logically, this is not a correct negation (the correct negation is that "there exists an x which is not y", or "some x is not y"). In fact, many statements in English (pragmatically) seemed inconsistent with their corresponding mathematical definitions, simply because we were creating and defining a system where the meaning of every word unit must be clearly defined and unambiguous. Because math allows for no ambiguities in language, we had to unlearn the pragmatic definitions of many terms, essentially revert to a more "childlike" way of thinking, taking into account only logical, not pragmatic, definitions.

This conversion between logical and pragmatic interpretations of language also calls into question the notion of the computational mind. Although young children process language mainly using the logical meanings of words, they are constantly taking in cues from their environment and learning the pragmatic meanings of the words in their vocabulary. How does one reconcile the lessening of specificity in language understanding with the precise, deterministic view of computation? Moreover, how do we ensure that the definition of any word that we first learn is the logical one and not a pragmatic one?