^{1}

This brief piece argues that it is desirable to reconceptualize the syntactic combinatorial mechanism Merge as a higher-order function that takes two functions (= a selector function and its ‘argument’ function) and yields a composite function in the context of I-language. On this functional characterization of Merge, all of the elements involved in Merge are conceived as functions as well: lexical items (LIs) as input of Merge and syntactic objects (SOs) as both input and output of Merge. It is claimed that this perspective of Merge is a bridging step toward further facilitating the mesoscopic-level (= dynome-level) investigation of the brain oscillatory nature of human language in the field of biolinguistics. In this framework, I make the case that it would be possible to analyze the brain oscillatory nature of Merge by appealing to the mathematical operation of the Fourier transform (FT) to the extent that Merge-related brain oscillations as physical waves can be captured by complex exponential functions/trigonometric functions in the temporal domain.

With the advent of the minimalist program (^{1}

Although MERGE in

Merge (X, Y) = {X, Y} (where X, Y = a lexical item (LI) or a syntactic object (SO))

This brief opinion piece aims to point out that, strictly speaking, the formulation such as (1) is not ideal from the perspective of the characterization of language as I-language in generative grammar, attempting to reconceptualize Merge in search for the link with brain oscillatory nature of language in biolinguistics as proposed by

First, I would like to point out that, strictly speaking, the standard view of Merge as a label-free binary unordered set-formation operation (

Now, if the characterization of human language as I-language is on the right track, every element of I-language, including the lexicon and the Merge-generated syntactic objects (SOs), is naturally expected to be intensional as well as internal and individual, i.e., being a function of some sort. Hence, structural descriptions (SDs) in the above quote should be intensional objects.

In (1), Merge is conceived as a recursive function, an intensional entity, which takes X and Y as inputs and produces an unordered set {X, Y} as its output, which in turn can be used as input to Merge recursively. Notice, however, that the output of Merge is a set {X, Y}, which can be considered as an extensional object rather than an intensional one. Given that the set {X, Y} generated by Merge could be employed as input to Merge in a recursive fashion, it would mean that the input set-theoretic element must be regarded as extensional as well. If we strictly keep to the view of human language as I-language, it would be more desirable to have not only Merge but also its inputs and outputs as specified in intension, i.e., functions.

Suppose that the above conjecture holds. Then, it means that every element that constitutes I-language is a function of some sort. Namely, X and Y as input elements and the output element SO with X and Y as its constituents are all to be functions. With this much in mind, while maintaining

Merge (X, Y) = Merge (f_{x}*, g_{y}) => [f_{x}*(g_{y})]_{z} (= w_{z}) (linear order irrelevant)

Merge as a higher-order function-creating function takes in X and Y as two functions, where X= f_{x}* and Y= g_{y}, and creates a new composite function [f_{x}*(g_{y})]_{z}. The subscripted x in f indicates that f is a function with a variable x and the subscripted y in g indicates that g is a function with a variable y (see the explanation below for what those variables range over). In applying Merge, it is crucial to assume that the statuses of f_{x} and g_{y} are not equal/symmetrical and the two must work differently: one of them, say, f_{x}, is taken to serve as a function that selects for the other g_{y} as an element that virtually plays the role of an ‘argument’ for the function. The effect of * in (2) virtually corresponds to _{x}* _{x} serves as a selector function and g_{y} without it indicates that it serves as an ‘argument’ for the selector function in a workspace (WS), creating a new complex function [f_{x}*(g_{y})]_{z} = w_{z}, which is to be properly interpreted at the C-I interface.

Note that the resulting function is w_{z} with a variable z with the same nature as the input functions and thus can naturally serve as a further input to Merge recursively. Furthermore, the mode of binary syntactic composition derives from the very nature of ‘binary function composition’. On this conception of Merge, the selector function f_{x}* will play the role of determining the label in syntax for the newly Merge-generated syntactic object [f_{x}*(g_{y})]_{z} (= w_{z}) (see, e.g.,

To be more specific, what do f_{x}* and g_{y} in (2) stand for in the context of syntactic computation? Carrying over

In the former, the head H as a categorizer (n, v, a, p) serves as the selector function f_{x}* and takes its complement XP (= a root element phrase √RP) as its ‘argument’ function g_{y}, with f_{x}* determining the label of the whole SO [f_{x}*(g_{y})]_{z}, roughly such as nominal, verbal, adjectival, prepositional/postpositional. As for the latter, when the Merge-generated composite function [f_{x}*(g_{y})]_{z} corresponds to {T, vP} or {C, TP} in the traditional notation, the functional category head T and C would be regarded as the selector function f_{x}* that receives an eventuality expressed by vP as its ‘argument’ function g_{y}, yielding a tensed eventuality (= a situation) as the resultant function [f_{x}*(g_{y})]_{z} in the former, and a selector function f_{x}* that accepts a tensed eventuality/situation expressed by TP as its ‘argument’ function g_{y}, returning a tensed eventuality/situation with a force (e.g., interrogative/non-interrogative) (= a proposition) as the output function [f_{x}*(g_{y})]_{z} in the latter, along the line of

There remains a theoretical complication concerning the root element √R. In _{x}* that has DP/nP or vP as its ‘argument’ function g_{y}. Given the special nature of the root element as label-free, I will tentatively assume that the root element is a special selector function that does not provide any label before its relation with a categorizer is established in the derivation, unlike other head H elements.

What about the case of {XP, YP} structures in the traditional notation such as the subject-predicate configuration {DP/nP, vP} or the configurations for {DP/nP, TP} (= subject-raising from the predicate-internal position) or {Wh-DP/nP, CP} (= the final landing-site of wh-movement)? First, for the case of {DP/nP, vP}, I will take it that vP is the selector function f_{x}* that selects DP/nP as its ‘argument’ function g_{y}. Second, for the cases of {DP/nP, TP} and {Wh-DP/nP, CP}, I regard TP and CP as serving as a kind of λ-expression,^{2}

See

_{x}* that takes DP/nP and Wh-DP/nP as its ‘argument’ function g

_{y}, providing the label for the whole SO [f

_{x}*(g

_{y})]

_{z}. At the same time, since the wh-phrase also serves as an operator (= a function) in the operator-variable construction on the standard view, I assume that the whole SO with the initial wh-phrase as the ‘argument’ function g

_{y}to the λ-expression selector function f

_{x}* (= CP) undergoing ‘reprojection’ or ‘relabeling’ in such a way that the wh-operator will be reinterpreted as the label of the whole SO with the original CP part turning into its argument as the nuclear scope (see, e.g.,

The proposed formulation of Merge in (2) has some implications not only for lexical items (LIs) and syntactic objects (SOs) as input of Merge and SOs as output of Merge but also for the domain of cognition in our species.

It is customary to posit that lexical items (LIs) in the technical sense are simplex conceptual atoms (containing bundles of features) and SOs are conceptual complexes (implicating possible rearrangement of such arrays of features) in generative grammar (_{x}*, g_{y}, and w_{z} in section 3 should mediate between two brain-internal representations: one is the brain-internal representation corresponding to the ‘concept/meaning’ of a lexical item or an SO, i.e., λ and the other is the brain-internal representation corresponding to the ‘form’ of externalization of such a lexical item or an SO, i.e., π in the minimalist framework (

Following _{x}*, g_{y}, and w_{z} as the category labels corresponding to lexical items like _{x}*, g_{y}, and w_{z} range over neurally coded brain internal representations for the ‘form’ (= π in the above sense) of the I-language objects like lexical items and SOs.

Note that Cat is defined as a mental process that involves purely brain-internal representations that correspond not only to individuals but also to anything else, such as properties, events, states, and propositions. By this definition of Cat, I would like to follow

The proposed view of Merge as a higher-order function that takes functions as inputs and produces a new function as output has some biolinguistic implications. Up until recently most of the neurobiological research on language has mainly focused on what

In the recent development of the study of brain oscillations in neuroscience, there has emerged the recognition that brain oscillations and their interactions will play a significant role in realizing various kinds of cognitive functions, language being no exception (see, e.g., ^{3}

Brain oscillations reflect synchronized activity of ensembles of cortical or subcortical neurons in the brain and their interactions via cross-frequency coupling (CFC) (see below) will serve an important role for inter-regional efficient communication, i.e., information transmission and exchange, in the brain within the globular space of the skull (see, e.g.,

Among a series of theses on brain oscillatory nature of language,

In my opinion, the shift in generative grammar from geometrical tree diagram syntactic representations in the pre-minimalist program to set-theoretical syntactic representations in the minimalist program can be considered a desirable theoretical step toward approaching the reality of syntactic computation in the brain. However, in order to make an attempt to solve _{x}*, g_{y}, and w_{z} in (2) as representing the mesoscopic level of dynome (

Various forms of waves in the physical world (including brain waves in the physical space of the brain) can be characterized by wave functions as a complex exponential function of roughly the form f(x, t) = ^{i(kx−ωt)}

In reality, brain waves as in other physical waves propagate spatially and oscillate temporally (^{4}

In FT, it is theoretically assumed that the distribution of various frequencies of the original wave in the temporal domain remains constant. However, in reality, such a distribution of various frequencies can change over time in the original wave, which cannot be captured in FT. In order to tackle this problem, various methods such as Discrete Fast Fourier transform and Wavelet transform are employed in analyzing oscillations in the field (see, e.g.,

Roughly, FT is a mathematical operation of integral calculus for spectral analysis in terms of a spectrum of frequencies that converts the original function (= f(t)) corresponding to a wave in the temporal domain into the frequency domain representation (= F(ω)) of the wave in question. As a result of FT, the information will be revealed as to how many waves of different frequencies (ω) are involved in the original wave expressed by the function f(t) in the temporal domain. e^{-iωt} in the formula represents the decomposed waves in complex exponential function notation in the first line, which can be equally expressed as in the second line with the combination of cosine and sine functions due to Euler’s formula. Accordingly, in theory, if we could identify a relevant brain oscillation for a cognitive process via EEG/MEG in the temporal domain, we could analyze the original brain wave by decomposing it into the component waves of different frequencies, converting from the temporal domain to the frequency domain by FT (see, e.g.,

If

Given that CFC/PAC involves embedding an oscillation with a higher frequency into another oscillation with a lower frequency, yielding a brain oscillatory pattern implicating a modulated amplitude of the higher frequency wave by the phase of the lower frequency wave (see, e.g., _{x}*, g_{y}, and w_{z} in (2) could be viewed as presumably corresponding to CFC/PAC, a lower frequency oscillation, a higher frequency oscillation, and the whole modified oscillatory pattern implicating the two oscillations, respectively. Then, in theory, you could employ gedCFC and FT to analyze EEG/MEG data for obtaining the functions f_{x}*, g_{y}, and w_{z} in (2) in order to obtain the frequency spectral properties of those relevant brain oscillations characterized by those functions, if the “functional characterization hypothesis of Merge” proposed in this piece is theoretically adopted.^{5}

Notice that I am NOT claiming that all the information about relevant linguistic features (morpho-phonological, syntactic, and semantic, etc.) in lexical items (LIs) and syntactic objects (SOs) generated by Merge is encoded by the functions f_{x}*, g_{y}, and w_{z}. The gist of FT is that any complex brain oscillations can be decomposed into a combination of a variety of patterns of sine and cosine waves. Thus, in principle, each of the relevant linguistic features can be encoded and carried by a specific sine/cosine oscillation, which could lead to producing a virtually unlimited possibility of encoding feature-sets, given that, on top of such various patterns of combination of sine and cosine waves, parameters such as amplitude, frequency, and phase for the sine/cosine oscillation functions can assume a variety of values in a certain neuro-biologically determined range. On this view, the functions f_{x}*, g_{y}, and w_{z} in (2) should be regarded as being constituted by a multitude of ‘hidden’ sine/cosine waves in this sense.

What kind of a cognitive task should be instrumental in discovering the presumed relevant functions indicated by f_{x}*, g_{y}, and w_{z} in (2) in my framework? While specifying the mathematical formulas for f_{x}*, g_{y}, and w_{z} in concrete terms is beyond the scope of this short piece and my ability at this juncture,

Thus, a word corresponds to the superposition brain oscillation _{i}_{i}_{j}, ω_{j},_{j}_{x}* and the ‘argument’ function g_{y} and if we could formulate the mathematics for CFC/PAC, we would be able to obtain the resultant function w_{z}.

In this connection, _{x}* and the adjective as g_{y} and the Merged derived noun phrase as w_{z} in principle, if we could somehow combine the CFC/PAC detection technique and FFT as briefly discussed above.

In conclusion, this brief piece argued that it is desirable to reconceptualize the syntactic combinatorial mechanism Merge as a higher-order function that takes two functions (= a selector function and its ‘argument’ function) and produces a composite function. On this functional characterization of Merge, all of the elements involved in Merge are regarded as functions as well: lexical items (LIs) as input of Merge and syntactic objects (SOs) as both input and output of Merge. I claimed that this perspective of Merge is a bridging step toward further facilitating the biolinguistic mesoscopic-level investigation of the brain oscillatory nature of human language, originally launched by

I am grateful for the reviewer’s constructive and useful comments on an earlier version of this article. I would also like to thank Kleanthes K. Grohmann for editorial support.

The author has no funding to report.

The author has declared that no competing interests exist.