1. Introduction

2.Part I. Revisits the Current Reasoning Research

3. An Empirical Puzzle and The Quantum Theoretic Solution

4. Why Do We Need a Quantum Model?

4.2. Types of Experimentations

Given the limitations of experimental methodologies, both in the research of psychology and physics, there are generally certain degrees of disturbance in observation. Often, in books written to introduce nonclassical physics, such as quantum theory or theories of relativity to nonexperts, some cartoons or personified examples are given in order to illustrate how the mind perceive physical phenomenon beyond everyday intuition (e.g., Hawking, 1996, 2001). For example, when a father goes out to do yard work, his two boys are peacefully watching TV; when he returns to the house, the boys are still watching TV peacefully. In between these observations however, the boys have a fight while he works in the yard. This phenomenon can be seen as a type of quantum fluctuation (Greene, 1999). Similarly, different kinds of quantum fluctuations occur when inward observations are performed. For example, when it takes longer than a minute for a participant to complete an experimental trial, say solving an SAT item, there is generally a five-second mental daydreaming break or bout of confusion to disturb observation when the latency data is being collected. In certain types of experiments, this disturbance is small and can be ignored; in others however, it matters greatly. Taking this into account and following the insight of Dirac (1930/1958), we have:

Principle 4.2 (disturbance): The higher the degree of disturbance in the experiment, the smaller the world can be observed.

What kind of experiment has a low degree of disturbance, and what kind of experiment has a high degree of disturbance? The principle of disturbance provides a way of distinguishing two very general types of experiments. One can be characterized as the smooth type of experiment with a low degree of disturbance; the other can be characterized as the Yes/No type of experiment (von Neumann, 1955; Penrose, 2004), with a high degree of disturbance. The principle of disturbance allows us to draw the chart below.

Here, we provide an intuitive way to understand the levels of scientific observation. As an example, imagine a picture of an iceberg. This image allows us to distinguish three levels of research. The first level takes place on top of the iceberg and enables the researcher to make a map of the parts floating above the water line. This can be compared with behavioral research in psychology as well as in economics. For the second level, assume that a boat equipped with advanced measurement technologies is near the iceberg. Imagine that this technology allows for the possibility of simulations, aimed at approximating the properties of the iceberg beneath the water, to take place. This can be compared with cognitive computational modeling research in cognitive science. Both the first and the second levels of observation have a relatively low degree of disturbance and are accordingly treated as “smooth” experiments. For the third level, suppose that the boat is within a certain distance from the iceberg and that it only allows us to see the top of the frozen mass. This third level of observation carries a high degree of disturbance and shall be treated as the Yes/No type of experiment. It is this type of experimentation that this paper focuses on. Why is the third level of research as important as the first two levels?

5. The Wavefunction and The Measurement Problem

5.5. Mental Energy

By the intuitionistic logic (Bell, John, & Machover, 1977), each symbol needs to be constructed. For example, a propositional variable p is an atomic sentence in the propositional logic, and p demands a construction. Accordingly, $\neg p$ demands two constructions, one for p and another for $\neg p$. Thus, we have:

Postulate 5.4 (the mental energy quantization). We assume as our working hypothesis that in the reasoning process, mental energy is quantized and denoted by h.

We assume that the mental logic processes and the mental models processes share equal mental energy in the sense that they have the same representational demand. Some evidence for this assumption is based on the observations explained in Table 5.1 below. Table 5.1 lists 10 inference schemas. For each schema, the first line is the mental logic representation of the schema, and the second line is the mental model representation of the schema. The schema weight is generated from the mental logic research. It shows that the number of symbols divided by 10 is exactly the schema weights.

From the above, one can observe that the schema weights can be reduced to three levels: 0.5, 1.0, and 1.5. It suggests that the 10-parameter model may be reduced to a 3-parameter model. The statistical analyses show that the 3-parameter model has the same predictive power as the 10-parameter model. It hints that the mental energy in reasoning can be quantized into three levels, or in other words, it can be characterized by three and only three equivalent curves.

6. The R-Procedure and A Solution Of The Measurement Problem

7. Gauge Symmetries and Gauge Transformations

7.3. The Local Gauge Symmetry

At the local level, for all kinds of bounded rationality, the individual differences of reasoners are sensitive. We have

Postulate 7.3 (the bounded rational reasoners). We assume as our working hypothesis that there are individual differences of real reasoners. Each reasoner has different gauge potentials and gauge strength in reasoning.

Each particular reasoner has different reasoning potential, including reasoning skills, language capacities, preparations for taking a reasoning test, etc. Accordingly, each individual reasoner may perform differently in taking a reasoning task. For example, the GRE is a high-profile selection test. Of which, the logic inference items are used to disclose the individual differences. These differences become more significant and sensitive. Thus, at the local level, we must pay attention to individual differences.

Recall the notion of the wavefunction $\psi \left(x\right),$where x

ranges over all the possible reasoners. Accordingly, when we consider the individual differences of the reasoners’ potential, the $\theta$is no longer a constant but becomes a phase function $\theta \left(x\right)$. Consequently, when the potential transforms from one state to another state $\psi \left(x\right)\to e^{i\theta \left(x\right)}\psi \left(x\right)$, we have the strength transformation,

$${\partial }_{\mu }\left(e^{i\theta \left(x\right)}\psi \right(x\left)\right)=\left(e^{i\theta \left(x\right)}i{\partial }_{\mu }\theta \right(x\left)\right)\psi \left(x\right)+e^{i\theta \left(x\right)}{\partial }_{\mu }\psi \left(x\right).$$

On the right side of the equation, the first term is not what we would expect there, so it makes the transformation no longer conformal. In other words, the standard derivative does not work at the local level. To solve this problem, we need to introduce the new mathematical tools (Bailin & Love, 1986/2019)., namely, the covariate derivative $D_{\mathsf{\mu }}$and the gauge field $A_{\mathsf{\mu }}$, which are defined as below, respectively,

$$D_{\mu } ={\partial }_{\mu }+𝑖𝑔A_{\mu },$$

$$A_{\mu }^{\text{'}}=A_{\mu }-{\displaystyle \frac{1}{g}} {\partial }_{\mu }.$$

Then, we can see below that the gauge transformation at the local level becomes conformal,

$$\psi \to {\psi }^{\text{'}}=e^{i\theta }\psi ,$$

$${D_{\mathsf{\mu }}{e}^{i\theta }\psi \to e}^{i\theta }{D}_{\mu }\psi .$$

This is called the gauge transformation of the second kind. For those readers who are not familiar with the gauge field theoretic modeling, the derivations below would be helpful. Note that for convenience, the individual variable x is omitted from both ψ and $\theta$. Then we have,

The above derivation is given step by step for the convenience of readers who are not familiar with gauge field theory.

8. The Language Cones and The Relativistic Phase Function

8.2. Interval and the Global Cone

Now we characterize the Minkowski spacetime in terms of reasoning. Each point stands for a permissible reasoner. The neighborhood of a point is called a reasoning event. The main structure is called the interval, which is defined as follows,

$$(∆{s)}^2=c^2{(∆t)}^2-{(∆x)}^2-{(∆y)}^2-{(∆z)}^2$$

with gauge, $g_{\mu \nu }=(+,-,-,-)$. The interval format is invariant. In special relativity, on the right side of this equation, the first term can be seen as the energy term. The next three terms are space position terms. The reasoning can be characterized suitably in terms of an interval. We discussed about the mental energy in reasoning in Section 4 and discussed about the relative difficulty of a reasoning take in Section 3. Now let $(∆{s)}^2$ stand for reasoning, $c^2{(∆t)}^2$ stand for the mental energy, and ${(∆x)}^2-{(∆y)}^2-{(∆z)}^2$ stand for the relative difficulty of a reasoning task, we may propose,

Postulate 8.1 (reasoning). Reasoning can be characterized as the mental energy of the reasoner minus the relative difficulty of a reasoning task. In other words, reasoning is a two-component structure, one is the reasoner, the other is a reasoning task.

We assume as our hypothesis that language travels with the highest but limited speed. Also, the timeframe allowed to solve a reasoning task may be longer or shorter, while the relative difficulty can be greater or smaller. Thus, the performance of reasoning can have three possible cases: $(∆{s)}^2=0,(∆{s)}^2>0,\mathrm{a}\mathrm{n}\mathrm{d}(∆{s)}^2<0.$ This situation can be characterized by a figure, called the light cone. In our context, it should be treated as the global language cone, which is given below.

Figure 1 shows three possible cases. Case 1, the event is within the cone, $(∆{s)}^2>0$, called the time-like event. It means that the reasoner has more than enough language energy to solve a reasoning problem with certain degree of the relative difficulty. Case 2, the event is on the surface of the cone, $(∆{s)}^2=0$, called the null event. It means that the reasoner has just enough (no more, no less) language energy to solve a reasoning problem with certain degree of the relative difficulty. Case 3, the event is within the cone, $(∆{s)}^2<0$, called the space-like event. It means that the reasoner does not have enough language energy to solve a reasoning problem with certain degree of the relative difficulty. Note that in the global language cone, the cone phase is an any given constant; $\mathrm{i}.\mathrm{e}.,\theta =C.$It is not concerned with individual differences.

8.3. The Proper Language Cones and The Individual Differences

Language skills play a sensitive role in reasoning. At the local level, there is a lot of the language related individual differences in human reasoning. The reasoner may have different degree of language influence, e.g., the native English speakers have certain advantages over most non-native English speakers in solving logical inference items of GRE or SAT. The reasoners may have different logic skills in solving reasoning problems in a reasoning experiment as well as in everyday life. In many cases, reasoning is sensitive to the timeframe allowed. This is significant to the in-class exams or to any standard educational tests. For a given reasoning problem with certain degree of relative difficulty, such as the lobster problem given in Section 3, the stronger the language capacity a reasoner has, the faster this reasoner can solve the problem. In other words, the faster a reasoner can solve a reasoning problem, the shorter the time needed. This time is called the proper time, denoted by $\tau$, which is also called the clock time. Different from the absolute time t shared by everyone, the proper time is individualized, so it shall be denoted by ${\tau }_i$ when necessary. The proper time is defined by $-d{\tau }^2=d{s}^2$; thus, the proper time is inverse proportional to the speed of solving a problem. Using proper time, we can define the notion of momentum, also called the four-velocity vector denoted by u as follows

$$u={\displaystyle \frac{{(∆S)}^2}{{\tau }_i}}={[c}^2(∆t{)}^2]/{\tau }_i-({∆x)}^2/{\tau }_i-({∆y)}^2/{\tau }_i-({∆z)}^2/{\tau }_i$$

This idea, borrowed from the special relativity theory, enables us to draw the proper language cones with different shapes for different reasoners, showing below,

In Figure 2a, the individual reasoner has stronger language skills, the shape of the language cone becomes flatter and wider, so it is with a smaller phase ${\theta }_1$. The wider $d_1$ indicates that for a given time frame, this reasoner has a greater chance to solve the problem correctly. In Figure 2b, the individual reasoner has weaker language skills, the shape of the language cone becomes thinner and longer, so it is with a greater phase ${\theta }_2$. Inversely, the narrower $d_2$ indicates that for the same time frame, this reasoner has a lesser chance to solve the problem correctly. Note, here the key point is that we have generated a relativistic phase function, $\theta \left(x\right)$, where the variable x ranges over all the possible individual reasoners.

8.4. Wittgenstein and the Language Game

Wittgenstein characterizes the communication as playing the language game. He also realizes that the language game is based on some psychological game. What psychological game do people play for reasoning? In psychology of deductive reasoning, we may image that the game is like the masterpiece, “The Servant of Two Masters” , by Carlo Goldoni.

The natural language has its syntax (grammar) and semantics. The formal language of a logic system has its formal syntax and formal semantics. In psychology of reasoning, as we mentioned in Section 2, the mental logic theory claims that people reason by applying inference schemas akin to inference rules. This can be classified as a syntactic approach. Call it the Master Synon, denoted by $e^{-}$. The mental models theory claims that people reason by constructing mental models based on the understanding of the meaning of premises. This can be classified as a semantic approach. Called it the Master Semon, denoted by $e^{+}$. Here the language naturally plays the role of the servant, called the Servant Langon, denoted by $\mathsf{\lambda }.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{f}$three starts can be shown in Figures 3a and 3b below.

The meaning of the two figures are the called the Feynman diagrams (Feynman, 1985/2006). In quantum theory, the spin is an intrinsic property of particles. The syntax and the semantics form the two-leg structure of language. Imagine that when the reasoning process is running, one leg is forward and the other is backward, and they reciprocate. Briefly, on one hand, this means that each leg, either Master Synon or Master Semon spins one second ($\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})$. On the other hand, no matter which leg is moving forward, the language is always moving forward. It means that the Servant Langon spins one (spin 1).

9. General Discussion

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