Reasoning is the mental process by which the relationship between two premises, ideas, concepts, facts, or data is established to generate information and create a conclusion; the conclusion of this reasoning can be true or false, and depending on the criterion applied to it, it is valid or not. Reasoning allows you to analyze and form your own criteria, expanding the knowledge acquired without resorting to experience. Likewise, it represents the middle point between instinct and thought.
Depending on the type of mechanisms employed during this process, the reasoning can be divided into:
In logical reasoning, the conclusion is implicit in the premises that compose it. Likewise, depending on the form or structure of the premises, different forms of logical reasoning can be found.
In non-logical reasoning, the conclusion is not necessarily implicit in the premises, so it is necessary to resort to context and/or experience to obtain a conclusion. [
It is an inferential reasoning method that uses two premises to generate a conclusion, as can be seen in the following example:
Major premise: All humans are mortal
Minor premise: Frank is human
Conclusion: Frank is mortal
These premises are composed of three terms (major, middle, and minor) whose order and presence within the premises allow us to reach a conclusion that consists of a third premise in which the middle term will not appear. Below, the distribution of the terms in the previous example is shown.
Both premises and terms are subject to rules that are described below:
The major premise contains a major term and a middle term. The minor premise contains a minor term and a middle term. The conclusion contains the minor term and the major term in that order.
The syllogism consists of 3 terms: Major, Middle, and Minor
No term can be more extensive in conclusion than in premises. The middle term should not appear in the conclusion. The middle term should appear in the major and minor premise.
The relationship between the different terms within the premises is established using judgments based on prepositions with which it is possible to evaluate the validity of the conclusions generated. These mechanisms are described by the operators of the prepositional logic that will be addressed in detail in the next section. These mechanisms must adhere to the following rules.
Two negative premises do not give a conclusion.
Two affirmative premises do not give a negative conclusion. Two particular premises do not give a conclusion. Conclusion always is next to the weaker party.
Through these we build the different figures of the syllogism (deductive, inductive, and abductive), and we find four different forms as can be seen in the following tables: [
Prepositional logic is a formal system that allows us to perform operations on statements or sets of data using logical operators that allow us to obtain the degree of veracity of these, as shown in the following table:
Negation
¬
A = Socrates is mortal. ¬A = Socrates is not mortal.
Conjunction
Λ
A = Socrates is mortal. B= Zeus is immortal. AVB = Sócrates is mortal, or Zeus is immortal.
Disjunction
V
A = Socrates is mortal. B= Zeus is immortal. AVB = Sócrates is mortal, or Zeus is immortal
Implication
→
A= Today is Monday. B=Tomorrow is Tuesday. A→B= If today is Monday, then tomorrow is Tuesday.
Biconditional
↔
A= Today is Monday. B= Tomorrow is Tuesday. A↔B= Today is Monday if and only if tomorrow is Tuesday.
Likewise, with these operators we can build more complex definitions, which can be used to evaluate the type of reasoning we are working with, provided that the rules defined for it (prepositions, premises, and terms) are met. [
As was described in the previous section, the implication is a prepositional operator that allows us to establish the cause-effect relationship (if, then) of two premises. This in turn can be defined in different ways within prepositional logic. However, any definition can be reduced in the following fundamental ways:
Jauch:
Sasaki Hook:
Dishkant:
Kalmbach:
Non-Tollens:
Relevance:
With the conjunction, negation, and disjunction operators, we can develop any operation using these definitions. For the present work we will focus on the implication using the Jauch form. [
Combinational logic circuits are a set of logical gates that execute the logical operators corresponding to negation, union, and disjunction. These allow complex operations to be performed using the values zero and one, true and false, or another binary system.
Negation
Conjunction
Disjunction
These have technical applications within electronics and computing and are currently the basis of the operations on microprocessors.
With this we can define the implication in its Jauch form as the following combinational circuit:
Quantum logic is a formal system composed of operators that allow us to perform experiments using the principles of quantum mechanics used in quantum information theory that are presented below:
Quantum entanglement. This describes how two particles that are generated in the same event maintain the same state. If one changes, the other will change simultaneously, acquiring the same state as the other without the need of any physical connection and regardless of the distance between them. [
With these principles, a new computational system was proposed that could surpass the capabilities of classical computing by taking advantage of these properties.
Also called Bra-Ket notation because of the name of its elements, it denotes the quantum states of a system.
The operators are denoted by a representative letter with some emphasis:
Therefore, an operation on a base state is described by the operator:
The projection of a state onto its own dual is defined as follows, where the result will always be equal to 1:
On the other hand, if the projection of a state is made on the dual of a different one, the result will always be 0:
Associated with any isolated physical system, there is a complex vector space with a defined internal product (Hilbert) or space of states of the system which are defined by its state vector. This vector is formalized in the space of the system. We find two types of states with which we define the physical system:
Using Dirac notation we can describe fundamental states as follows:
Where:
↑
The above example shows the notation for a two-state system. The notation of the basis states can be defined as appropriate depending on their number.
The formalized vector in our Hilbert space is defined as follows:
The qubit is the fundamental unit of quantum information. It corresponds analogously to the bit used in classical computing. However, while the bit corresponds to electrical impulses that are interpreted in a binary way as 1 or 0, the qubit corresponds to a physical system of two basis states and has a representation similar to that of classical computing. It is denoted as:
As qubits are entangled, their vector representation will grow to a ratio where n is the number of qubits. They are written by obtaining the tensor product of the qubits involved.
It is graphically represented as a sphere-shaped Hilbert space known as the Bloch sphere, and its state will determine the value of the qubit.
Quantum gates allow modifying the state of one or more qubits in a way analogous to logic gates. The following table shows the fundamental gates:
Pauli-X
Pauli-Y
Pauli-Z
Phase change
Hadamard
Controlled gate
Quantum algorithms are analogous to a classical logic circuit; a quantum algorithm is a circuit that is composed of a set of quantum gates that operate on one or more qubits in one or more scenarios. [
These are represented as horizontal lines in which we can add different quantum gates that will allow us to modify the state of the qubit. Depending on whether we use a controlled quantum gate or not, we can find the next kinds of qubits:
Scenarios are represented as vertical lines that intersect the qubits and correspond to the instants of time in which we can add a gate on a qubit. [
Terminals indicate what will be done in the last scenario of each qubit. There are two cases. [
Quantum simulators are represented as vertical lines that intersect the qubits and correspond to the instants of time in which we can add a gate on a qubit. [
Q-Team is a quantum circuit simulator developed as part of the master's thesis of the author of this project at the University of Guadalajara. It has a graphical interface that allows the circuits to be developed without the need to write a single line of code. Likewise, a programming framework can be used for users who feel more comfortable in a development environment. It also has three methods of operation that allow obtaining results from the generated circuits through the manual entry of the inputs, by a list of inputs, or using a module that obtains all possible input states. [
Although in both quantum logic and prepositional logic we find that operators are used to modify the state or the information that we have at the beginning, there is no one-to-one analogy between these operators with the exception of negation and the Pauli X gate. However, quantum circuits can be built to perform analogous operations to those found in prepositional logic, as long as the following rules are fulfilled.
The circuit must have the same number of inputs as the number of inputs in the logic circuit, not counting the objective qubits. The measurement of results must be made on the same number of qubits as the outputs of the combinational logic circuit. The qubits to be observed must be identified. Target qubits will always be initialized in the |0> basis state. There should be a one-to-one correspondence between the base states |0> and 1>, and binary states 0 and 1, respectively, both in the inputs and outputs.
Negation
Conjunction
Disjunction
We can observe that for cases where there are two inputs and an output, there is a one-to-one relationship between inputs A and B with the inputs of the qubits of Q1 and Q2. Likewise there is a one-to-one relationship between the output of the operator with the output measurement of the qubit Q3.
In cases where there is only one input and one output, the qubit α serves as an input, and it is also on this that the output is measured.
With the above we can define the quantum circuit analogous to the combinational of implication in its Jauch form as follows:
As previously mentioned, through prepositional logic we can build any definition or algorithm from basic logical operations, so to develop the hypothetical syllogism the following definition will be used from implications:
Where P, Q, and R are terms that can be ordered according to the way in which we are working. With this we can build a definition based on combinational logic circuits as follows:
With the following associated truth table:
1
1
1
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
0
1
0
1
0
0
1
1
0
0
0
1
Using the methodology mentioned above to build a quantum circuit from classical definitions results in the following quantum algorithm:
Where: Q0=P; Q1=Q; Q2=R
From the above, we take as inputs only those where the qubits Q3, Q4, Q5, Q6, and Q7 are initialized in the base state 0, so our analogous truth table is as follows:
Where Q7 is our observation qubit, and we see that both the classical and quantum outputs are tautological, so the circuit is analog.