The computational processes of natural human language (C
HL) form a physical object. The first axiom of matrix syntax (MS), which investigates linguistic structures on the complex plane using quantum mechanics-based mathematical tools, converts the [±N] and [±V] values of the Chomsky matrix to [±1] and [±
i]. This study analyzes the potential developments of this axiom, which has been used previously for raising examples and to describe the grammaticality algebraically. Matrix calculations were performed on elaborate geometries of grammatical and ungrammatical raising examples. Standard analysis based on the minimal computation (MC) principle of the minimalist program (MP) is on the correct track, but it can be combined with mathematical analysis to answer a fundamental question: Why does the internal merge create argument chains (A-chains) in this way, and not in other logically possible ways? With the context-sensitivity hypothesis in MP, MS provides a new perspective on A-chains. Results indicate that a chain-forming geometry contains three areas with distinct growth factors: 1) occurrences (i.e., syntactic contexts of chain members) with a constant rate; 2) backbone-structure building with an exponential rate (i.e., larger φ ~ 1.618), and 3) superposed chain-member copies with a diminishing rate (i.e., little φ ~ -0.618 or σ
1 = 0.7071). The MP-MS collaboration is a viable approach to the structure-building dynamics of C
HL and approaches linguistic physics.
Context Sensitivity Fibonacci Numbers Internal Merge Matrix Syntax Minimalist Program Quantum Mechanics Superposition1. Introduction: How Matrix Syntax Can Collaborate with Minimalist Program
The principle of minimal computation (MC; perform efficient computations) (Chomsky, 2005) is a fundamental guideline of the minimalist program (MP) (Chomsky, 1995). The MC includes, for example, the shortest move constraint (SMC) (Chomsky & Lasnik, 1993), no procrastination (Chomsky, 1995: chap. 3), Greed containing last resort (LR) principle (Chomsky, 1986a; Chomsky, 1995): Move raises α only if morphological properties of αitself would not otherwise be satisfied in the derivation (Chomsky, 1995: 261); italics added by the author; move α whenever it is possible to eliminate (check off) uninterpretable features (uF; φ (person, number, gender) and case) in α, no look-ahead (Chomsky, 2000), goal-accessibility restriction (i.e., phase impenetrability condition (PIC) (Chomsky, 2000: 108; Chomsky, 2001: 13-14).1 Let us consider (1).
(1) a. Alice seems to like Bob.
b. * Alice seems that it is certain to like Bob.
Example (1a) is a grammatical A-internal merge (A-IM), whereas (1b) is an ungrammatical A-IM, namely, a “super-raising”.2 Let us assume that (1a) and (1b) have structures as shown in Figure 1 and Figure 2, respectively, where indexes are added for clarity, and waved lines indicating parts that are externalized by the sensorimotor (SM) system, i.e., pronounced.
Let us assume, with certain modifications, the essence of a probe-goal mechanism (Chomsky, 2000, 2001). In more detail, unlike the standard assumption, I assume that DP φs are uFs because they are arbitrary and lack natural (e.g., biological) bases.3 Therefore, I do not adopt a valuation, where DP φs value probe-uFs and remain. Let capital letters indicate probe-features, and small letters goal-features.
In (1a), Alice0 is externally merged at vP edge and assigned a θ role [experiencer]. The probe uF [NULL] (null case) in the embedded-clause tense head T1 (infinitival to) attracts the closest goal Alice0 containing a matching uF [null] (Chomsky & Lasnik, 1993; Martin, 2001), which obeys PIC. [NULL] and [null] match and delete. Note that [3, sg] in Alice0 is still present. The matrix-clause tense head T2 (a suffix s) bears a probe uF [III, SG], which attracts the closest goal Alice1 bearing matching uF [3, sg] to the TP2 edge, obeying PIC. The T2-probe uF [III, SG] and uF [3, sg] in all Alice copies are eliminated because
they match. Every [3, sg] in every Alice copy is deleted since the same thing happens in every copy, i.e., the same entity. The nominative case [nom] in Alice0 disappears as a side-effect of φ elimination.4 Importantly, by the end of the derivation, each of the three Alice copies has the same θ role, containing no uF. Each probe attracts the closest goal, obeying PIC, in (1a).
One might ask a fundamental question. If IM is motivated by the need to check and value features, then how is it possible that a chain contains identical members? Will the lowest link in the chain always have more uninterpretable features than the highest link? Does enforcing identity defeat the whole purpose of movement in the MP? I do not adopt the valuation, which causes a significant complication. All copies are identical at the final derivation step if all uFs in probes and goals eliminate via matching without valuation.
Let us consider (1b). The same algorithm applies until the matrix tense T3 externally merges with VP3, except that an expletive it externally merges with T2'. The T3-probe [III, SG] attempts to attract the closest goal that has matching φ [3, sg]. The T3-probe cannot attract the expletive it because it is frozen at the TP2 edge. If T3-probe attracted it forcefully, the output is ungrammatical: * It seems that is certain to like Bob. Given PIC-I, T3-probe can see C1 (that) and CP1 edge (none) only. That is, there is nothing T3-probe can attract. If an “expletive-support” (like do-support) is invoked as an LR, i.e., fetch an expletive from the lexicon and externally merge it at TP3 edge, the output is grammatical: It (there) seems that it is certain for Alice to like Bob, where a preposition for is inserted as an LR to eliminate uF in Alice1. However, the relevant derivation chose to attract a distant Alice1, which is already transferred to CI and SM and invisible to T3-probe, thereby violating PIC-I. Alternatively, Alice1 skipped a potential landing site, i.e., TP2 edge. If no expletive existed and Alice1 managed to reach TP2 edge, the output is grammatical: Alice seems to be certain to like Bob.
A point to be elaborated regarding (1a; Figure 1) is as follows. The occurrences (i.e., syntactic contexts) of Alice copies are v’, T1’, and T2’. They are distinct. Why do the same copies appear in different contexts? Suppose that the T2-probe [III, SG] attracted externally-merged Alice0 in Figure 1, which obeys PIC because no strong phase intervenes between TP2 and vP. Why can Alice0 not move to the TP2 edge? Furthermore, we cannot distinguish whether Alice1 or Alice0 has undergone IM, i.e., either externalization ends up “Alice seems to like Bob.” Therefore, we cannot tell which copy has undergone an IM by just by looking at (1a), given PIC.
A part to be sophisticated regarding (1b; Figure 2) is the following. The occurrences of internally-merged Alice copies are T1’ and T3’. The occurrence of externally-merged expletive it is T2’. The syntactic contexts of internally-merged Alice copies and expletive it are the same: T’. Why can Alice and it not appear in the same context? The probe uFs [III, SG] in T2 (finite tense to which a copula be is adjoined) are eliminated when they match with the goal uFs [3, sg] in the expletive it and its case [nom] is deleted as a side-effect of φ-elimination. Assume PIC-II. When T3-probe starts searching a matching goal, the strong phase CP2 is not formed yet. Furthermore, no strong phase intervenes between CP1 and TP1. Therefore, T3-probe should be able to see Alice1 and attract it. Why is it impossible? In contrast, the stricter PIC-I correctly bars T3-probe from attracting Alice1. Given the Greed-LR principle, why can Alice1 not move to the TP3 edge to eliminate its uF? Why is a Greed-LR unavailable given that it is another MP guideline? Why does PIC-I override the Greed-LR principle?
Furthermore, Martin et al. (2019: 29) asked fundamental questions regarding (1a). Why are all copies not pronounced/interpreted? Why is it the case that no more than one copy is available for interpretation at CI and for externalization at SM? Why do the objects behave that way, and not in other equally rational ways (all copies are interpreted, some copies are interpreted, …)?5 The MC account has explanatory power, but it can be collaborated with matrix syntax (MS), which uses mathematical tools employed in quantum mechanics (QM), to find a key to solve a hard problem (2), which is the fundamental question of the MS.
(2) Fundamental question of MS (Orús et al., 2017: 22).
What does it mean for such occurrences (i.e., syntactic contexts such as v', T1', T2' in Figure 1) to be a way to distribute an item like Alice over a phrase marker?
This paper offers a fresh look at the grammaticality of (1a) and (1b), using tools of the MS.6 When we investigate a chain, it is important to consider the occurrences of chain members because the occurrences are the syntactic contexts in which those member copies are allowed (1a) or disallowed (1b).7 Significantly, the MP has started to recast crucial concepts such as EPP, IM (movement), and phase in terms of context-sensitivity: “the status of X … changes depending on the syntactic contexts in which X occurs (Bošković, 2013, 2014). This study investigates a mathematical property of dynamics of chain-member copies (subsection 4.1) and the occurrences (syntactic contexts) of grammatical and ungrammatical raising examples (subsection 4.2). It attempts to find a clue to solve a difficult problem as (2).
This topic was chosen to enhance MP and science cooperation. The MP is a linguistic foundation of the MS, which employs QM tools. Orús et al. (2017: 3) explains why MS can be a crucial perspective: “We are not claiming to have discovered quantum reality in language. The claim, instead, is that our model of language, which we argue works more accurately and less stipulatively than alternatives, especially in providing a reasonable explanation of the behavior of chains, turns out to present mathematical conditions of the sort found in quantum mechanics. In our view, this is quite striking and merits public discussion.” The MS cooperates with MP to investigate a question (2) posed in Martin et al. (2019: 29). The MS uses the same mathematical tools as QM, the most successful physics framework to date. I believe that QM provides “more general tools to handle information” (Sato & Matsushita, 2007: 151) than just algorithms that investigate the behavior of electrons and photons. The MS should be able to provide a new perspective on the algorithms created by nature since the language system is a natural (physical) object that evolved in the human brain. More importantly, as Uriagereka (2019) stated, an MS approach enables us to perform a rigorous argument that can efficiently prove correct or incorrect based on simple mathematics.
This work is organized as follows. In Section 2, we introduce theoretical assumptions and experimental apparatus. In Section 3, we summarize our results. In Section 4, we discuss the fundamental question of MS, and propose mathematical images of a chain. The conclusion is presented in Section 5. Appendix A introduces a formal toolkit that is assumed in MS. Appendices B and C present matrix calculation of structure building of (1), which is the base of the argument in Sections 3 and 4.
2. Methods2.1. MS2.1.1. Axiom 1: The Fundamental Assumption of MS
Chomsky (1974) made a breakthrough when he used the distinctive features proposed by Jakobson and Halle (1971) to classify four parts of speech (adjective (A), verb (V), noun (N), and adposition (P). Table 1 shows the Chomsky matrix, where [N] represents nominal properties and [V] verbal properties; both N and V are “conceptually orthogonal” (Orús et al., 2017: 17). As Orús et al. (2017: 9) reminds the reader, “[i]t is important not to confuse the lexical attribute V with the lexical category V (which has the lexical attribute V in the positive and the lexical attribute N in the negative).”
Another breakthrough represented by MS (Uriagereka, 2016; Orús et al., 2017: 22; Martin et al., 2019), which attempts to connect CHL with QM; MS expresses Chomsky’s matrix using the complex number z = a + bi, by translating [N] into the real number 1 and [V] into the imaginary number i.8 In more detail, the numerical translations are A = 1 + i,N = 1 − i,V = −1 + i, and elsewhere category = −1 – i. I adopt Orús et al. (2017: 7), where P in Chomsky matrix is replaced by elsewhere category, although I use P as a mnemonic aid for elsewhere category. The fundamental assumption of MS is (3) (Orús et al., 2017: 7), Martin et al., 2019: 31).9
(3) Axiom 1 (fundamental assumption)
Lexical features take the values N = 1 and V = i.
Chomsky matrix
[+N]
[−N]
[+V]
A
V
[−V]
N
P
We can restate the Chomsky matrices (Chomsky, 1974) as shown in (4).
(4) Axiom 2 (Chomsky matrices)
The four lexical categories are equivalent to diagonal matrices
N = [ 1 0 0 − i ] , V = [ − 1 0 0 i ] , A = [ 1 0 0 i ] , elsewhere = [ − 1 0 0 − i ]
Any structure building in any system must start somewhere. The standard assumption in MP is that a structure building proceeds bottom up in CHL. MS postulates that the starting point is a noun and that only nouns self-merge (Axiom 6 (Guimarães))10, a compound of which is a cognitive anchor in CHL.
Thus, N × N = [ 1 0 0 − i ] ⋅ [ 1 0 0 − i ] = [ 1 0 0 − 1 ] = NP = Z , i.e., one of the Pauli
matrices, which is the most elegant matrix.11 MS introduces the Jarret graph (Figure 3), indicating how phrases (nodes, e.g., NP and VP) are connected through operators (edges, e.g., N ^ and v ^ ) by matrix multiplication (first merge).12 The determiner (det) of each twin (plus and minus) node is also
shown as well, e.g., the twins ± NP = ± [ 1 0 0 − 1 ] have the determiner det = −1.
MS claims that a phrase det is the phrase label.
For instance, an operator (function) v ^ acts on (undergoes external merge (EM) with) an NP (input) and outputs VP. More elaborately, an operator v ^ = [ − 1 0 0 i ] acts on NP, i.e.,
v ^ × NP = [ − 1 0 0 i ] ⋅ [ 1 0 0 − 1 ] = [ − 1 0 0 − i ] = − [ 1 0 0 i ] = VP , the twin of which is [ 1 0 0 i ] = VP . Thus, the det of VP is i.
One might argue against the current analysis that employs the Chomsky matrix, too simple. One might criticize that for an approach that aims at explanation,
the basis of the system remains deeply stipulative. One might point out that MS faces criticisms of the Chomsky matrix for its excessive rigidity and empirical inadequacies, which is a step back in linguistic research.
Regarding P, I strictly adopt Orús et al. (2017: 7) here, which names [−N, −V] output as elsewhere category. I admit that the Chomsky matrix is simple, i.e., a sentential structure containing functional elements such as light verb v, tense T, and complementizer C cannot be captured by only four categories. In fact, Orús et al. (2017: 27-32) mentioned possible matrices for functional elements.13 However, I want to limit this analysis to the smallest possible set, which Orús et al. (2017: 17-18) calls G8 (the “magnificent eight” group) as shown in (5) (ibid., 18).
(5) G 8 = { ± I , ± Z , ± C 1 , ± C 2 } , where
I = [ 1 0 0 1 ] , Z = [ 1 0 0 − 1 ] , C 1 = [ 1 0 0 − i ] , C 2 = [ 1 0 0 i ] .
The subsets { ± I , ± Z } are from the Pauli group (Hermitian), and { ± C 1 , ± C 2 } are Chomsky matrices (non-Hermitian), which have complex entries on the diagonal. G8 is “abelian (commutative) because all matrices are diagonal and therefore mutually commute [ x ∘ y = y ∘ x ] (ibid.)”.14 I limit the building-block matrices to G8 because (a) it is consistent with the strong minimalist thesis (SMT), i.e., CHL must provide the simplest possible solutions to legibility problems, (b) G8 “allows us to explain the basic features of the MS model” (ibid.), and (c) it is cost effective to integrate T (suffix) and C into elsewhere. Although this is a preliminary analysis that can only accommodate the most basic tools, I admit that in the future, I will need to investigate and elaborate on the tools and system.
The linear-algebraic Chomsky matrices are diagonal and eigenvalue matrices Λ that have eigenvalues λ1 and λ2 on their diagonal.15 In all cases, λ1 fixes its eigenvector s1 = (1, 0), and λ2 determines its eigenvector s2 = (0, 1). The four lexical categories have an eigenvalue either λ =i or λ = −i, indicating that they share a rotational property: the unit 90˚-rotation matrix Q. A and N have an eigenvalue λ = 1, which suggests that A, N, and the projection matrix share a property of perfect symmetry, i.e., everything is left unchanged. V and elsewhere category have an eigenvalue λ = −1, which shows that a reversion property characterizes them.
2.1.2. Chain Vector as Superposition
“An image is a large matrix of grayscale values, one for each pixel and color. When nearby pixels are correlated (not random) the image can be compressed (Strang, 2016: 364).” A linguistic structure is similar to a large matrix that can be compressed by correlating nearby nodes. We calculated matrices on chain (CH) structures that are created by IMs using the matrix method proposed by Orús et al. (2017) and Martin et al. (2019). They proposed that a CH is mathematically isomorphic to superposition in QM. See Appendix A for a formal toolkit that assumed in MS. The CH for Alice in (1a) is expressed in Equation (1), where v', T1', and T2' are the occurrences (i.e., syntactic contexts) in which Alice appears. SM externalizes Alice2.
C H = ( ( A l i c e 0 , v ′ ) , ( A l i c e 1 , T 1 ′ ) , ( A l i c e 2 , T 2 ′ ) ) (1)
As the three copies of Alice are superposed, the three occurrences are as well. By factoring Alice out, we obtain Equation (2).
C H = ( A l i c e ( v ′ , T 1 ′ , T 2 ′ ) ) (2)
More generally, let αdenote an NP undergoing IM, and let Kand Λ denote occurrences (contexts or sisters that the NP is merged with) of a higher and the original NP copy, respectively. The normalized vector | CH 〉 of the NP chain state is Equation (3) (Orús et al., 2017: 23).
Orús et al. (2017: 23) proposed that a linguistic chain is a superposition of at least two different states, postulating the axiom 8 shown in (6).
(6) Axiom 8 (Hilbert space)
Linguistic chains are normalized sums of vectors from a Hilbert space.
Martin et al. (2019: 30) claimed that syntactic computation “collapses” into interface observables (i.e., a classical linguistic reality) when “syntax acts on some Hilbert space, by way of linear operations.”
Let us focus on the first IM step of the entire CH of Alice | A l i c e 〉 : CH = (Alice0, Alice1, Alice2). The first subset of a normalized vector of chain CH (Alice) is in the superposed of two occurrence states (i.e., v' and T1'). | CH ( A l i c e ) ^ 〉 is indicated in Equation (4).
The scalar 1 2 indicates that the resulting vector is set according to the
normalization condition, which dictates that every state’s square sum of probability amplitude must be equal to one. The probability that a copy of Alice is observed at vP edge and that the same copy is observed at the TP1 edge are identical, i.e., 50%, respectively.16
3. Results
Let us consider how we can differentiate between grammatical (1a) from ungrammatical (1b) linear algebraically. See Appendix B for the matrix calculation for structure building of the grammatical example (1a), the geometry of which is shown in Figure 1, and Appendix C for that of ungrammatical (1b), the geometry of which is shown in Figure 2.
3.1. SVD of Occurrence Matrices3.1.1. SVD of Occurrence Matrices of Chain Head Alice in (1a)
I focus on the occurrence (i.e., sister) matrices of (potential) chain heads in question: substantive DP (Alice) for (1a) and substantive DP (Alice) and expletive (it) for (1b). For (1a), the relevant occurrences are v', T1', and T2'; (1b), they are v', T1', T2', and T3'. MATLAB returns the singular value decomposition (SVD) result for each occurrence matrix.17 The SVD results of occurrence matrices of Alice in (1a) are shown in (7).
(7) SVD of occurrence matrix of chain head Alice in grammatical (1a)
a). v ′ = [ 1 0 0 i ] , and its [ U , S , V ] = svd ( v ′ ) = [ 1 0 0 − i ] [ 1 0 0 1 ] [ 1 0 0 − 1 ] .
b). T 1 ′ = [ − 1 0 0 − i ] , and its [ U , S , V ] = svd ( T 1 ′ ) = [ 1 0 0 i ] [ 1 0 0 1 ] [ − 1 0 0 − 1 ] .
c). T 2 ′ = [ − 1 0 0 i ] , and its [ U , S , V ] = svd ( T 2 ′ ) = [ 1 0 0 − i ] [ 1 0 0 1 ] [ − 1 0 0 − 1 ] .
The diagonal matrix S indicates the fundamental characteristics of the matrix. All occurrence matrices show that S is the identity matrix I, as shown in (7), implying that the relevant occurrence does not change. For explanatory purposes, the boldface matrices highlight a point in question.
3.1.2. SVD of Occurrence Matrices of Alice and Expletive It in (1b)
The SVD results of occurrence matrices of potential chain heads Alice and an expletive it in the ungrammatical example (1b) are shown in (8) below.
(8) SVD of occurrence matrix of chain heads Alice and it in ungrammatical (1b)
a). v ′ = [ 1 0 0 i ] , and its [ U , S , V ] = svd ( v ′ ) = [ 1 0 0 − i ] [ 1 0 0 1 ] [ 1 0 0 − 1 ] .
b). T 1 ′ = [ − 1 0 0 − i ] , and its [ U , S , V ] = svd ( T 1 ′ ) = [ 1 0 0 i ] [ 1 0 0 1 ] [ − 1 0 0 − 1 ] .
c). T 2 ′ = [ − 1 0 0 − 1 ] , and its [ U , S , V ] = svd ( T 2 ′ ) = [ 1 0 0 1 ] [ 1 0 0 1 ] [ − 1 0 0 − 1 ] .
d). T 3 ′ = [ − 1 0 0 i ] , and its [ U , S , V ] = svd ( T 3 ′ ) = [ 1 0 0 − i ] [ 1 0 0 1 ] [ − 1 0 0 − 1 ] .
All occurrence matrices show that S is the identity matrix I, implying that the relevant occurrence does not change. Unlike other U’s, the first orthogonal matrix U in the SVD result of T2' occurrence matrix (8c) is a real matrix (indicated by a boldface matrix). Later I will propose that the heterogeneous property of U of occurrence T2’ of the expletive it in (8c) is responsible for a quantum collapse failure.
3.2. Normalized Vectors and Wave Function of Alice Chain in (1a)
Let us consider the normalized vectors of an Alice chain in the grammatical raising example in (1a). Let us recapitulate on the general formula for a normalized vector of a set of chain members. Let | K ^ 〉 and | Λ ^ 〉 be normalized vectors of two the Alice occurrences. A normalized vector of Alice chain is shown in Equation (5) below.
| CH ( A l i c e ) ^ 〉 = 1 2 | A l i c e ^ 〉 ⊗ ( | K ^ 〉 + | Λ ^ 〉 ) (5)
In Equation (5), | A l i c e ^ 〉 is factored out of ( | K ^ 〉 + | Λ ^ 〉 ) , which means that the occurrences | K ^ 〉 and | Λ ^ 〉 share the same chain-member, i.e., | A l i c e ^ 〉 . Each step of chain formation in the grammatical raising (1a) is expressed as a vector formula in Equation (5). Incidentally, Orús et al. (2017: 25) introduced an ungrammatical vector called a “weird” vector as shown in Equation (6).
| W e i r d 〉 = 1 2 ( a | α ^ 〉 ⊗ | K ^ 〉 + b | β ^ 〉 ⊗ | Λ ^ 〉 ) (6)
When a , b ≠ 0 , | α ^ 〉 and | β ^ 〉 cannot be factored out, which is called an entangled state. Equation (6) expresses a chain formation step in the ungrammatical (1b), where | α ^ 〉 = | A l i c e ^ 〉 , and | β ^ 〉 = | i t ^ 〉 .18 I leave the entanglement for future research.
Now, let us consider Figure 1 of (1a) a grammatical geometry. Let us calculate a normalized vector of the first sub-step of the entire Alice chain, which is shown in (9).
(9) Normalized vector of the first sub-step of Alice chain in (1a)
v ′ = [ 1 0 0 i ] . Since [ 1 0 0 i ] ⋅ [ 1 0 0 − i ] = [ 1 0 0 1 ] , the trace of which is 2, normalized vector is | v ′ ^ 〉 = 1 2 [ 1 0 0 i ] .Similarly, T 1' = [ − 1 0 0 − i ] . Since
[ − 1 0 0 − i ] ⋅ [ − 1 0 0 i ] = [ 1 0 0 1 ] , the trace of which is 2, the normalized vector is | T 1 ′ ^ 〉 = 1 2 [ − 1 0 0 − i ] . Therefore, | v ′ ^ 〉 + | T 1 ′ ^ 〉 = 1 2 [ 1 + ( − 1 ) 0 0 i + ( − i ) ] = [ 0 0 0 0 ] . Therefore, | CH ( A l i c e ) ^ 〉 = 1 2 | A l i c e ^ 〉 ⊗ 0 = 0 .
In the grammatical geometry (Figure 1), the normalized vector of the first sub-step occurrence of the Alice chain is zero. Next, let us calculate a normalized vector of the second sub-step occurrence of the entire Alice chain in Figure 1, which is shown in (10).
(10) Normalized vector of the second sub-step occurrence of Alice chain in (1a)
T 1' = [ − 1 0 0 − i ] . Since [ − 1 0 0 − i ] ⋅ [ − 1 0 0 i ] = [ 1 0 0 1 ] , the trace of which is 2, the normalized vector is | T 1 ′ ^ 〉 = 1 2 [ − 1 0 0 − i ] .Similarly, T 2 ′ = [ − 1 0 0 i ] . Since [ − 1 0 0 i ] ⋅ [ − 1 0 0 − i ] = [ 1 0 0 1 ] , the trace of which is 2, normalized vector is | T 2 ′ ^ 〉 = 1 2 [ − 1 0 0 i ] . Therefore, | T 1 ′ ^ 〉 + | T 2 ′ ^ 〉 = 1 2 [ − 2 0 0 0 ] = [ − 2 0 0 0 ] . Now, let us calculate a normalized vector of Alice chain. Since A l i c e = [ 1 0 0 − i ] , and [ 1 0 0 − i ] ⋅ [ 1 0 0 i ] = [ 1 0 0 1 ] , the trace of which is 2, a normalized vector is | A l i c e ^ 〉 = 1 2 [ 1 0 0 − i ] . Therefore,
| CH ( A l i c e ) ^ 〉 = 1 2 | A l i c e ^ 〉 ⊗ ( | T 1 ′ ^ 〉 + | T 2 ′ ^ 〉 ) = 1 2 [ 1 0 0 − i ] ⊗ [ − 2 0 0 0 ] = 1 2 [ 1 ⋅ [ − 2 0 0 0 ] 0 ⋅ [ − 2 0 0 0 ] 0 ⋅ [ − 2 0 0 0 ] − i ⋅ [ − 2 0 0 0 ] ] = 1 2 [ − 2 0 0 0 0 0 0 0 0 0 2 i 0 0 0 0 0 ] = [ − 2 2 0 0 0 0 0 0 0 0 0 2 2 i 0 0 0 0 0 ]
Using MATLAB to apply SVD, we obtain
[ U , S , V ] = svd ( CH ( A l i c e ) ^ ) = [ 1 0 0 0 0 0 1 0 0 − i 0 0 0 0 0 0 ] [ 0 . 7 0 7 1 0 0 0 0 0 . 7 0 7 1 0 0 0 0 0 0 0 0 0 0 ] [ − 1 0 0 0 0 0 1 0 0 − 1 0 0 0 0 0 1 ]
Significantly, the scaling matrix S (indicated by a boldface matrix) in the SVD matrix is not I, and its (1, 1) (i.e., row 1, column 1) and (2, 2) positions have values that are smaller than one, but not zero. This means that the normalized vector of the second sub-step of the Alice chain tends to diminish. The grammatical (1a) shows a wave function ψof Alice chain vector consisting of a 4 by 4 complex matrix in Equation (7).
Calculating its squared length ‖ ψ ‖ 2 , we obtain a probability as shown in Equation (8) regarding where the plural copies of Alice are observed in Figure 1.
The probability matrix in Equation (8) contains 1 2 at (1, 1) and (3, 3) positions. There is a 50% probability of Alice being in the TP1 or TP2 edge; in other words, the same NP simultaneously exists in two different positions. The superposed states collapse into one in SM and CI. In Figure 1, the superposed state collapses into Alice at TP2 edge. If a collapse happens at the TP1 edge, SM externalizes (11).
(11) It seems for Alice to like Bob.
In (11), when Alice has reached at TP1 edge, a distinct derivation is realized, where a preposition for is inserted as a last resort to eliminate Alice’s case, i.e., a probe uF [ACC] (accusative feature) in for eliminates a goal uF [acc] of Alice.19
4. Discussion4.1. A Mathematical Structure of Grammatical Chain Formation
A normalized vector of a grammatical chain of Alice in (1a) has three significant properties, as shown in (12).
(12) a). The SVD-diagonal matrix (i.e., the middle matrix Sin [U, S, V]) of occurrence matrices of Alice, as shown in each SVD matrix in (7), is the identity matrix I.
b). The value of the first chain sub-step is zero, as shown in the last line of (9).
c). The S value of the second sub-step is 0.7071, as shown in the last line of (10).
The basic property of occurrence matrices, according to property (12a), indicates that they provide a stable mode. Regarding property (12b), I argue that the zero value of the first sub-step of CH corresponds to the fact that the SM does not externalize the intermediate chain-member copies, i.e., Alice0 and Alice1. This is notable because it is a linear-algebraic proof that intermediate copies are not externalized. Property (12c) indicates that chain-member copies in the final IM step are in a decaying mode given that the infinite power of 0.7071 approaches zero but never reaches zero.
Let’s use geometry to express these properties and tendencies. Regarding (12a), let us assume that the geometrical form of I is a unit circle, i.e., a basic form. The geometrical occurrence forms (syntactic contexts) remain in the same unit circle. As to (12c), the value 0.7071 shows that the geometrical form of Alice chain copies approaches zero-dimensional point infinitesimally toward the chain head.20Figure 4 (to be modified) shows the geometrical image schematically. The chain geometry teaches us that the chain copies undergo a dimensional drop from 2 to 0, whereas the occurrences (syntactic contexts) are stable, i.e., they remain 2 dimensions. A direction of dimension drop equals a direction of chain realization toward the highest chain member, a point at which SM externalizes an Alice copy, just as a direction of potential drop equals a direction in which a current flows (Strang, 2016: 458).
Medeiros (2012: 244-245) calculated the growth factor of the spine, i.e., the head-complement structure, as shown in (13), where PSR stands for phrase structure rule, a characteristic polynomial is used to find eigenvalues, which describes the fundamental property of the relevant matrix, the black circle corresponds to a head, and the blank triangle indicates a phrase.
(13) The Spine (adapted from Medeiros (2012: 244))
PSR: 1 → 0 1
Matrix: [ 1 0 ]
Characteristic polynomial: x– 1
Growth factor: 1
Tree:
Given that Medeiros’ spines correspond to chain occurrences, i.e., v’, T1’, and T2’ in Figure 1, I believe it is significant that the spine’s growth factor is 1 (i.e., nothing changes; equals the identity matrix I), which corresponds to our finding that the basic property of occurrence matrices of the grammatical chain formation in (1a) and Figure 1 is that they are stable, as shown in (13), which is geometrically described as the check unit circles in Figure 4.
Regarding the growth factor of the sentential geometry using IMs, I adopt Medeiros’ result that the sentential-structure-building growth factor is φ ~ 1.618, as shown in (14).21
(14) X-bar (adapted from Medeiros (2012: 245))
PSR: 2 → 2 1, 1 → 0 2
Matrix: [ 1 1 1 0 ]
Characteristic polynomial: x2 – x – 1
Growth factor: φ ~ 1.618
Tree:
The growth factor φ ~ 1.618 is a positive larger root of the x2 − x − 1 = 0 characteristic polynomial. The matrix in (14) is the Fibonacci (F) matrix, which describes F sequence. Medeiros (2012) and Medeiros and Piattelli-Palmarini (2018) persuasively argued that the golden rule determines the growth factor of geometry-creation in CHL. As indicated in Carnie and Medeiros (2005: 52), Medeiros (2008: 161), and Medeiros (2012: 220), it is significant that CHL creates a geometry whose basic property is governed by the golden rule. The diagram in Figure 5 is adapted from Medeiros (2012: 220), which indicates an idealized linguistic tree, where its edge and complement are maximized, and how F numbers emerge in the tree.
(15) F emerging in Figure 5
a). XP = {1, 1, 2, 3, 5, 8, …} = Fib(n)
b). X’ = {0, 1, 1, 2, 3, 5, …} = Fib(n − 1)
c). X = {0, 0, 1, 1, 2, 3, …} = Fib(n − 2)
The complement XP and the edge XP are maximized in Figure 5. Count the number of XP, X’, and X0 for each derivational line, F sequence defined by the linear recurrence equation Fn = Fn–1 + Fn–2, where F1 = F2 = 1 and F0 = 0, appears22, i.e., F = {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …}, as indicated in (15).
A new finding of this paper is that the chain copy members, Alice copies here, tend to diminish, approaching zero, but never reach zero, as shown by S in the last line of (10), which is geometrically described as diminishing circles toward TP2 edge. Here it is tempting to propose a hypothesis that the emergence of diminishing value 0.7071 corresponds to the little φ ~ −0.618, which is the other solution (eigenvalue) of x2 − x − 1 = 0, i.e., infinite power of both numbers approaches zero, but never reaches zero. If this logic is true, the hypothesis that chain-member copies diminish is probable. Following Medeiros (2012) and Medeiros and Piattelli-Palmarini (2018), assume that the basic property of CHL-geometry building is the golden rule, which determines the growth factor of F sequence. Strictly speaking, the growth factor of F numbers consists of both the greater φ ~ 1.618 and little φ ~ −0.618, where the infinite power makes the former grow exponentially, and the latter becomes infinitesimally small; approaching zero, but never reaching zero. This is the reason why the little φ is
often ignored. Therefore, it is reasonable that the chain-formation geometry contains both exponential boost and infinitesimal shrinkage. Thus, we modify Figures 4 to Figure 6, which is more accurate. In Figure 6, the identity matrix I is expressed by the constant unit circles, the greater φ ~ 1.618 by the enlarging circles, and the little φ ~ −0.618 by the diminishing circles.
The fundamental MP assumptions and network theory support Figure 6. A DP (NP) with a set of uF externally merges at a position where it is assigned a θ role. The driving force of IM is uF elimination, which means that the uF number decreases as the DP undergoes successive IM bottom-up. The currents (i.e., edge information) flow in the direction of a potential (i.e., node information) drop, according to network theory (Strang, 2016: 458). A set of chain-head DPs reduces uF numbers, corresponding to diminishing circles in Figure 6. The MP dictates that a sentential-structure building proceeds bottom-up, accumulating information, which theoretically corresponds to the exponential growth factor of a positive golden ratio. However, CHL uses IM edges to create loops and the loops lead to equilibrium, avoiding a computational explosion (Arikawa, 2019). A spine ( X ¯ ) is formed by externally merging a head X and its complement YP, which is transferred to the interfaces as soon as possible. A higher probe targets X and its edges but not the spine X ¯ . A spine X ¯ is invisible to CHL and does not affect the computational result, which is due to the spine’s extreme symmetry, as expressed by the identity matrix (i.e., unit circle).
4.2. A Mathematical Structure of Ungrammatical Chain Formation
Now consider a CH (Alice) in an ungrammatical (super-raising; (1b)) geometry, as shown in Figure 2. The basic property of ungrammatical (1b) occurrence matrices is the same as that of grammatical (1a), as indicated by the diagonal matrices S in (8) and those in (7), respectively. Consequently, the matrix calculation yields the same value of normalized vector for ungrammatical super-raising and grammatical raising geometries. Therefore, this analysis faces a problem (16).
(16) A problem
We cannot distinguish the grammatical chain from the ungrammatical one in terms of values of normalized vectors. Is there a way to algebraically differentiate between the two types of CH?
Two possible algebraic ways to differentiate grammaticality are shown in (17).
(17) Two possible solutions
a). Use the basic property (i.e., I) of occurrence matrices.
b). Use the left orthogonal matrix U.
Consider the diagonal S (i.e., Σ) of SVD of these occurrences for the initial solution (17a). As shown in (8), the occurrence matrices of the ungrammatical chain is I. SVD shows that σ1 (upper-left corner) of diagonal matrix S is equal to one. Thus, the maximum growth factor equals one (Strang, 2016: 382), indicating that the occurrence is stable.23 The identity property of occurrence matrices causes all chain-member copies of CH to be identical. Let us adopt the fundamental concept of context-sensitivity (Bošković, 2013, 2014) and propose the context-sensitivity hypothesis as shown in (18).
(18) The context-sensitivity hypothesis (original version, ibid.)
If the syntactic contexts of X vary, so does its status.
Suppose that the inverse of the conditional proposition in (18) is true, as shown in (19).
(19) The inverse of context-sensitivity hypothesis
If the syntactic contexts of X are identical, X’s status is identical.
As indicated by all S’s (i.e., the growth factor matrices) in (8), the syntactic contexts are the same, implying that their sisters are identical. However, the DP Alice and the expletive it are not identical, which contradicts (19). Alternatively, in (1b), the geometry of which is shown in Figure 2, the absolute CH uniformity requires Alice = it, which is a syntactic, semantic, and phonetic contradiction. Thus, the potential CH in Figure 2 contains a linear-algebraic contradiction, leading to ungrammaticality.24 However, in Figure 1, a geometry of (1a), the three CH-heads are identical, i.e., Alice0 = Alice1 = Alice2; and the sentence is grammatical, conforming to (19). The members of the relevant chain are completely uniform.
According to MP, as stated in section 1, (1b) is ungrammatical because Alice skipped over a potential-intermediate landing site, the TP2 edge, on its way to the final landing site, the TP3 edge, violating MC. However, given the Greed-LR principle in MP, the solution must be elaborated. Why is the Greed-LR principle not invoked in such a way that Alice is forced to skip the potential landing site if the expletive it is frozen at the TP2 edge and does not move out of the way? Why does MC take precedence over the Greed-LR principle? Another way to account for (1b) ungrammaticality using MS is to use a broader concept of chain that includes a potential landing site, i.e., potential CH = {TP3-edge, TP2-edge, TP1-edge, vP-edge}. The significant finding is that the occurrence (syntactic context) matrix calculation (i.e., the growth factor matrix S in SVD in (8)) reveals that the identity matrix I (see also (13)) describes the basic property of these occurrences (Alice-occurrence nodes = v', T1', T3' and it-occurrence node = T2' in Figure 2). If these occurrences (syntactic contexts) are identical, then their sisters must also be identical as well (19). It forces us to conclude that Alice = it, despite breaking every rule of syntax, semantics, and phonetics. Example (1b) is ungrammatical because the sentential geometry matrix calculation forces Alice to be equated to it, resulting in crashes at CI and SM.
Consider how the left orthogonal matrices U can be used in the second solution (17b). As shown in (8c), one distinguishing feature of the occurrence matrix T2' of expletive it is that its U is a real matrix, unlike other U’s, which are complex. Consider the following fact, on which SVD is entirely dependent (Strang, 2016: 365).
(20) Grayscale fact
Grayscales of nearby pixels are generally similar.
An image is a large matrix (ibid.). Similarly, a sentential structure is a large matrix. The entry aij describes the grayscale of each pixel in the image. It is difficult to compress different grayscales. A grammatical set of chain-member contexts constitute grayscales that can be compressed. However, a super-raising structure has a different context, i.e., SVD results with a real-matrix U. The different context causes the compression impossible, resulting in the ungrammatical status of (1b). An SVD factorizes a matrix A as A = U Σ V T , which has the geometrical meaning of which is (rotation) times (stretching) times (rotation). A real matrix represents stretching, whereas a complex matrix represents a rotation and stretching. A super-raising structure, as shown in (8c), contains complex-U syntactic contexts for DP Alice, and real-U context for the expletive it. An informal formula for a super-raising quantum state is shown in Equation (9), where a tensor-product symbol ⊗ represents IM and dot product ⋅ represents EM.
In Equation (9), unlike other occurrences (i.e., syntactic contexts) with complex-U’s, the occurrence | T 2 ′ ^ 〉 of expletive | i t ^ 〉 contains a real-U. As mentioned in Subsection 3.1.2, the heterogeneous property of U of occurrence T2’ of the expletive it in (8c) may result in quantum collapse failure at CI and SM, i.e., Alice cannot be interpreted or externalized at a single position when a speaker makes an observation, i.e., she is aware of grammaticality.
5. Conclusion
An insight about MS is that its conditions are consistent with QM, which provides more general tools to handle and investigate information in general. The MP CHL study can be integrated with complex dynamics using MS and linear algebra. The Merge has a property of simple iterations, resulting in a complex butterfly effect with discrete infinity. I list the findings of this research.
(21) Chain properties are calculable linear algebraically, and it can be used as a key to solving hard problems of MP and MS: Why does internal merge create A-chains in this way, and not in other logically possible ways? What does it mean for such occurrences (i.e., syntactic contexts such as v', T1', T2' in Figure 1) to be a way to distribute an item like Alice over a phrase marker? The MP and MS can collaborate to tackle the hard problem.
(22) LetCH = (X3, X2, X1) a grammatical chain, where X1 is an EM-copy of X, and X2 and X3 are IM-copies of X.
(a) The normalized vector value of the first sub-step of a chain is equal to zero in the scaling matrix S (the last line of (9)), showing that the SM does not externalize X2, i.e., the kinetic energy is zero.
(b) The normalized-vector S value of the second sub-step is 0.7071 (the last line in (10)), indicating that the chain-member copies (X3, X2, X1) tend to diminish in the direction from X1 to X3. 2D area approaches zero-dimensional point infinitesimally, which corresponds to a potential drop in graph theory, causing information to flow toward the drop. SM externalizes the highest chain-member copy X3, a zero-dimensional point where a computation converges.
(c) The S of the occurrences (sister node of chain copies; the syntactic context of chain members) is the identity matrix I, indicating that the occurrences are in steady mode.
(23) The calculation leads to the dynamics of chain-forming geometry in CHL (Figure 6). A chain-forming geometry contains three areas with distinct growth factors: (a) occurrences (spine) in a steady mode, (b) the entire structure building in a developing mode at an exponential rate (i.e., greater φ ~ 1.618), and (c) the chain-member copies in a decaying mode (i.e., little φ ~ −0.618 or σ1= 0.7071).
(24) Two possible ways to distinguish linear-algebraically grammatical from ungrammatical raising are:
(a) MATLAB returns the SVD result of occurrence (i.e., syntactic context) of potential-chain members, where every scaling matrix (S = Σ) is I. The inverse of context-sensitivity hypothesis demands that the members of a potential chain must be identical. In an ungrammatical raising (super-raising) geometry, one of the potential-chain members is an expletive, forcing “Alice = it,” which is a contradiction.
(b) Unlike the occurrence matrices of the chain member Alice, where the rotational matrices U’s are all complex, the SVD result is peculiar for the occurrence matrix of the expletive it, where the rotational matrix U is real. The heterogeneous property of U of occurrence T2' is responsible for failures of compression and quantum collapse, i.e., Alice copy cannot be externalized at one position in ungrammatical super-raising when a speaker makes grammaticality observation.
A practical implication based on the current research results is that this paper may enhance collaboration among the MP linguists, mathematicians, and physicists, which was initiated by Orús et al. (2017) and Martin et al. (2019), as well as a scientific investigation into CHL.
Regarding the following studies, my perusal is necessary for Orús et al. (2017) and Martin et al. (2019) to continue the MS project in detail. My MS learning must parallel with the MP investigation, which is the linguistic foundation of MS. Simultaneously, I need to continue studying linear algebra by reading a textbook, such as Strang (2016), because linear algebra is a fundamental mathematical tool used in MS.
Acknowledgements
I want to thank an anonymous reviewer for taking the time and effort necessary to review the manuscript. I sincerely appreciate all the valuable comments and suggestions that have helped me improve the manuscript’s quality. This work was supported by JSPS KAKENHI Grant Number JP21K00491 and the Momoyama Gakuin (St. Andrew’s) University Fund for Research Support. The author wants to thank Enago (http://www.enago.jp) for the English language review.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Arikawa, K. (2022). Internal Merge: Why Does It Work This Way? A Matrix Syntactic Approach to Argument Chain. Open Journal of Modern Linguistics, 12, 336-365. https://doi.org/10.4236/ojml.2022.123025
Appendices
A). MS toolkit
Orús et al. (2017) and Martin et al. (2019) adopted the standard Dirac (bra-ket) notation as shown in (1) below to define a scalar product.
〈 A | B 〉 = T r ( A † B ) (1)
In (1), the ket | B 〉 is a system state, the bra 〈 A | is a scalar, tr is the matrix trace, i.e., the sum of its diagonal entries, which equals to the sum of eigenvalues, and A † (read as “A dagger”) is the conjugate transpose of a complex matrix A.25
Pierce (2021) provides a concise statement regarding the reason why we use bra-ket notation. Namely, “[t]he bra-ket notation ⟨bra|ket⟩ is a way of writing special vectors used in QM. A ket is a quantum state, the values of which are complex numbers, and it can have any number of dimensions, including infinite dimensions. The bra is similar, but the values are in a row, and each element is the complex conjugate of the ket’s elements. We can ‘map’ some real-world case (usually one with probabilities) onto a well-defined mathematical basis. This then gives us the power to use all the math tools to study it (ibid).” The norm N (also called size, length, modulus, magnitude, or absolute value) of a vector | z 〉 is shown in (2).
N = | 〈 z | z 〉 | = | T r ( z † z ) | (2)
When obtaining inner products of complex vectors, we must use Hermitian matrices to avoid contradictions.26 A vector zwith norm N = 1 is called a normalized vector, which corresponds to z ^ (read “z-hat”), as indicated in (3).
N ( | z ^ 〉 ) = 1 〈 z | z 〉 | z 〉 = | z | | z | = 1 (3)
For example, NP ^ of NP = [ 1 0 0 − 1 ] is shown in (4), where 〈 NP | NP 〉 = 2 .
See Orús et al. (2017: 19) and Martin et al. (2019) for relevant definitions.
B). Structure building of grammatical (1a)
In Orús et al. (2017: 10), Axiom 3 (multiplication) states that 1st MERGE (M) is matrix multiplication, and Axiom 4 (tensor product) elsewhere M [non-first MERGE] is matrix tensor product. The former corresponds to EM, and the latter to IM.27 The matrix calculation of the entire structure building of grammatical (1a) is shown in (1) below.
[ − 1 0 0 − i ] × [ 1 0 0 1 ] = [ − 1 0 0 − i ] = T 1 ′
5) NP (Alice) IMs with T1'.
[ 1 0 0 − i ] ⊗ [ − 1 0 0 − i ] = [ 1 [ − 1 0 0 − i ] 0 [ − 1 0 0 − i ] 0 [ − 1 0 0 − i ] − i [ − 1 0 0 − i ] ] = [ − 1 0 0 0 0 − i 0 0 0 0 i 0 0 0 0 − 1 ] = T P 1
6) SVD compresses TP1 matrix to TP1 = UΣVT, where diagonal matrix Σ (scaling factor; singular value) indicates how axes stretch, and orthogonal (unitary) matrices U and VT indicate how vectors rotate (UTU = UUT = I, VTV = VVT = I). MATLAB returns the following result of SVD for TP1.
T P 1 = U Σ V T = [ 1 0 0 0 0 i 0 0 0 0 − i 0 0 0 0 1 ] [ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 ] [ − 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ]
Singular value Σ is I. Therefore, 4-by-4-TP1 matrix is compressed into 2-by-2 I.
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