# Chapter 7. Logic with Vectors: Ambiguous Words and Quantum States

As we try to learn and reason about language and its meaning by measuring the distributions of points and regions in geometric models, a variety of powerful logical operators become available. Many such operators correspond to well-known logic gates in computer science, and to familiar options in standard user interfaces.

A typical example is shown in Figure 7.1. In many browsers and user interfaces (in the year 2004), if you click on one icon and then click on another, the second will become selected and this will cancel the selection of the first, and if you click on more icons, the one most recently selected becomes the unique active icon, overiding previous selections. However, if you hold down the CTRL key and then click on two different icons, both icons will be selected. And if you hold down the SHIFT key and then click, both icons and those in between will be selected. When using these features, previous selections are not discarded --- all the icons selected while holding down the correct key are combined to give a special set of selected items. The processes of combination correspond to disjunctions (OR operators or joins) in different logics.

The 'CTRL + click' option is essentially a discrete approach which builds a set whose elements are two isolated points. The 'SHIFT + click' option is a more continuous approach, which uses the geometry of the surrounding region, building a set consisting of all points contained in the rectangle spanned by the two icons the user selected.
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 Holding down CTRL and clicking on Folder 1 and Folder 6 highlights these two folders only Making the same two clicks while pressing SHIFT highlights Folders 2 to 5 as well Figure 7.1: The first version is a Boolean disjunction, the second is a kind of quantum disjunction

This chapter describes these two brands of logic, using the algebra of George Boole as the classical example of a discrete (and indeed, binary) approach to logic, and the quantum logic of Garrett Birkhoff and John von Neumann as the standard example of a continuous logic. Geometrically, the differences between these approaches arise from the differences between set theory and vector spaces, which form the underlying spatial models for Boolean logic and quantum logic respectively.

Because vector spaces provide the spatial model for both quantum logic and the Infomap WORDSPACE, it turns out that we can use the same logical operators in both. For me, this story began when I implemented a vector negation operator in WORDSPACE using the concept of orthogonality, just as described by Birkhoff and von Neumann in 1936. This precedent was happily brought to my attention by Professor Lawrence Moss of Indiana University, as the research evolved.

Exploring such operations, comparing their effects on document retrieval, and piecing together the scientific context of the WORDSPACE model proved an interesting journey. Many historic branches of logic, geometry, and the physical sciences became woven into the story, and I shall explain how these traditions have been used in search engines, especially in WORDSPACE.

The 'quantum' thread in this chapter stretches from Aristotle's description in the Metaphysics, through geometric implementations by Euclid, Descartes, and Grassmann, the birth of quantum mechanics itself in the early 1900s, and the outward stimulus this has provided to areas of logic, philosophy, computation, and word sense discrimination. The quantum idea, and the spatial extensions with which it is modelled, is a useful tool both for describing the way a particle may be represented by a combination of possible pure states, and for describing the way an ambiguous word may be represented by a combination of available 'pure' meanings.

First, for necessary context and contrast, we explore some of the traditional combination operations in discrete logical models, focussing on the work of George Boole.

## Sections

### 1. Boolean Logic and Algebra

George Boole was a schoolteacher in Lincolnshire, but although he couldn't afford to go to college, his research attracted attention and he eventually became a Professor in Ireland. He worked on describing abstract "Laws of Thought", and would probably have regarded himself as more of a cognitive than a computer scientist. Nonetheless, in laying the foundations for modern set-theory, he also devised an algebra that used only the numbers 0 and 1. His argument was that only these two numbers satisfy the equation x2=x, and so only these numbers can be used in an algebra of sets where the intersection of any set with itself is simply the set you started with.

About 100 years later, electronic machines were built that could make Boole's algebra a reality. Computers were built that could store these 0s and 1s electronically, and perform logical operations on them exactly as Boole described. These discrete binary operations are called Boolean operators to this day, and you can perform a "Boolean search" in most information retrieval systems, often using the "Advanced Search" feature of an internet search engine. Boolean "logic gates" have become so familiar through electronics and computer science that it's easy to believe that Boolean logic is logic, just as until the 1830's, people believed that Euclidean geometry was geometry. But just as there are differently shaped spaces, there are differently shaped logics.

### 2. Vector Negation in WORDSPACE

People have often pointed out that "bag-of-words" methods such as those used to build WORDSPACE don't normally enable you to negate or deny anything, so one our first new challenges for performing logic in WORDSPACE was to remove the meaning of an unwanted term. Take the following example. You want to find out about formal clothing in New York Times articles, and you try the query-term suit. You find that suit has the following neighbours in WORDSPACE:
 Keywords: NegativeKeywords:

 Add to query Subtract from query Term Similarity suit 1.000000 lawsuit 0.868791 suits 0.807798 plaintiff 0.717156 sued 0.706158

These words are all about the legal meaning of suit, not the formal dress information that you were looking for. To get rid of the unwanted legal information, you click on the Subtract from query checkbox next to "lawsuit", and send the information again. This time you get results about the "formal dress" meaning you were looking for, as you can see below.
 Keywords: NegativeKeywords:

 Add to query Subtract from query Term Similarity pants 0.810573 shirt 0.807780 jacket 0.795674 silk 0.781623 dress 0.778841 trousers 0.771312

Notice that this hasn't just removed the single word "suit" from the results (which would be the effect of a Boolean negation operation). It has removed all sorts of words from this area of meaning, and even deprecated the highly ambiguous word "suit" below the less ambiguous "shirt". If you select the "Retrieve Documents" option you'll find that the topics covered are exactly those you were looking for - formal dress - and that the occurences of the word "suit" in these documents is much more likely to be the meaning you were looking for.

All of this is happening by manipulating vectors in WORDSPACE so that they are perpendicular or orthogonal to one another. Move the vectors so that they are at right angles to one another, and you effectively make the meanings of two word-vectors irrelevant to one another.

### 3. Ambiguity and the Quantum

 A single particle emitted from E can interfere with itself as though it is a wave that has passed through the slits at both A and B. But if you observe which slit the particle has gone through, this interference doesn't happen!
In language, a written word like "suit" could potentially be signifying one of several meanings and until you see the word in context, it doesn't make sense to ask "Which meaning is the true one?"

In quantum mechanics, a single particle could potentially be in one of several pure states, and until you make an observation of the particle, it doesn't make sense to claim that it is in one state or another (though you can predict that some states are on the whole more likely than others).

A famous experiment that demonstrated this amazing behaviour is the double-slit experiment, which is described briefly in the diagram on the right. It appears that the particle behaves like a wave, one component of which went through A and another through B. Until you observe the particle, all you can predict is that the particle will be represented by some linear combination of pure states, which is expressed as a vector &lambda A + &mu B.

The equations that are used to represent these particles are exactly the same as those used to represent an ambiguous word in WORDSPACE, and the projection equation used to remove an unwanted meaning from a word in Section 2 is exactly the same as the equation used in quantum mechanics to obtain the pure states of a particle from the observed interference patterns.

### 4. From points to lines and planes

This section explains the mathematics needed to take this model from two states, one of which we want to remove, to many states, several of which we want to remove. The geometry needed to remove 2 unwanted meanings was described by Euclid as early as 300 BC. The unwanted meanings generate a plane, and the final query must be orthogonal to this plane. Algebraically, the expression &lambda A + &mu B we met in the previous section can be used to describe any of the points in such a plane, and by implementing operations based upon such equations we can implement Euclid's ancient instructions directly in WORDSPACE.

This enables a user to strip away unwanted meanings in succession and zoom in on a region in WORDSPACE. When tested in document retrieval experiments, this method removed not only the unwanted keywords, but many more of their neighbours in WORDSPACE and their synonyms in WordNet than was achieved by the alternative Boolean operators.

### 5. Subspaces of higher dimension

The futher techniques and the extra dimensions need to remove any number of meanings are all described in the 1862 Ausdehnungslehre (Extension Theory) of Hermann Grassmann, an amazing German schoolteacher. His mathematical ideas were neglected by professionals at the time, and while this was a loss for mathematics (many of Grassmann's discoveries were ignored at the time and rediscovered only later), it was a gain for linguistics. Frustrated by stuffy formalism, Grassmann turned his attention to Indo-European linguistics, compiled a dictionary of Sanskrit that is still used by German-speaking scholars today, and discovered certain laws of sound-change in language that demonstrated that no currently existing language was the unique root of the Indo-European group. Around the same time that Darwin discovered that all species undergo change, Grassmann demonstrated that the same is true for languages.

### 6. Quantum Logic and beyond

This final section discusses the full picture of Quantum Logic. This logic was introduced by Garrett Birkhoff, the founder of modern lattice theory, and John von Neumann, the founder of modern computer architecture, in an little-known paper they published in 1935. AND, OR, and NOT operators can all be defined and built in a vector space such as WORDSPACE, and these are the natural logical operations in this space, just as the Boolean logic gates are the natural logical operations for set theory. Classical set theory is binary and discrete, and vector spaces are smooth and continuous, and these properties are reflected in the logics that arise in these spaces.

You find this difference between continuous and discrete interpretations of logic in very simple sentences. If somebody says "you should take bus 60 or bus 70", and you later discover that they meant you to take bus 67, you would think you had been very badly advised. If, however, you were told that the journey would take "60 or 70 minutes", and it took 67 minutes, you would agree that you had been advised perfectly wisely. This is just like the difference between 'CTRL + click' and 'SHIFT + click' in the diagram at the beginning of this chapter.

### Conclusion

The logic of WORDSPACE (described recently) and the logic of quantum mechanics (described in 1935) both arise from the same model, the vector spaces that Grassmann introduced in 1862.

## Demo

Quantum logical operations including vector negation can be experimented with using the "Remove from query" or "Negative keywords" options on the Infomap demo (no longer operational) and the Semantic Vectors package.

## References

One book I wish had been published at the time I was writing Geometry and Meaning is The Geometry of Information Retrieval, by C.J. van Rijsbergen. For those wishing to delve into the topics of vector spaces, observations, and the relationship between information retrieval and quantum theory on a deeper level, I recommend this book.

The following paper describes the quantum connectives and their semantic properties in more detail.

Dominic Widdows and Stanley Peters. Word Vectors and Quantum Logic: Experiments with negation and disjunction. Eighth Mathematics of Language Conference, Bloomington, Indiana, June 20-22, 2003, pages 141-154.

The following paper describes the implementation of the quantum negation operators in WORDSPACE and evaluates their statistical performance in document retrieval experiments.

Dominic Widdows. Orthogonal Negation in Vector Spaces for Modelling Word-Meanings and Document Retrieval. 41st Annual Meeting of the Association for Computational Linguistics, Sapporo, Japan, July 7-12, 2003, pages 136-143.

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