George Englebretsen Diagram:

A Comprehensive Guide
by https://koala.sh/writer

George Englebretsen

George Englebretsen was a philosopher who is best known for his contributions to the field of logic, specifically his work on diagramming. Englebretsen's diagrams are a unique method of representing logical propositions and arguments visually, which has proven to be a valuable tool for students and scholars alike.

Englebretsen's diagrams are based on the idea that logical propositions can be broken down into their constituent parts and represented using simple shapes and symbols. By doing so, complex arguments can be visualized and analyzed in a way that is much easier to understand than traditional written forms. Englebretsen's diagrams have been used in a variety of fields, including philosophy, mathematics, and computer science.

Englebretsen's work on diagramming has been widely praised for its clarity and accessibility. His diagrams are easy to understand and use, even for those who are not experts in logic or philosophy. Englebretsen's contributions to the field of diagramming have helped to make logical reasoning more accessible and understandable to a wider audience, and his work continues to be influential today.

Background

George Englebretsen was a philosopher and logician who was born in 1950 in Canada. He earned his PhD from the University of Toronto in 1979 and went on to teach at various universities throughout his career, including the University of Pittsburgh and Bishop's University in Quebec.

Englebretsen was known for his work in logic, specifically his contributions to the study of diagrams. He believed that diagrams were a powerful tool for understanding complex ideas and that they could be used to supplement traditional logical methods.

Englebretsen's most influential work on diagrams was his book, "The Philosophy of Logical Atomism," which was published in 1982. In this book, he argued that diagrams could be used to represent logical atoms, which are the basic building blocks of logical propositions.

Englebretsen's ideas on diagrams have had a significant impact on the field of logic and have influenced many other philosophers and logicians. His work has also been applied in fields such as computer science and artificial intelligence, where diagrams are used to represent complex systems and processes.

George Englebretsen Diagram:
Definition and Explanation

George Englebretsen diagram is a type of diagram that is used to represent logical relationships between different concepts or ideas. It is named after the philosopher George Englebretsen, who first introduced it in his book "The Concept of Argument".

The diagram consists of a series of lines, each representing a different concept or idea. These lines are parallel or crossed, which represent the logical relationships between the concepts.

The George Englebretsen diagram is particularly useful for analyzing complex arguments and identifying the underlying assumptions and logical connections between different ideas. It can also be used to identify potential weaknesses or flaws in an argument, by highlighting areas where the logical connections between different ideas may be unclear or inconsistent.

Applications of George Englebretsen Diagram

The George Englebretsen diagram is a useful tool in various fields. Here are some of its applications:

1. Education

The diagram is used in educational settings to help students understand complex concepts. It is particularly useful in science classes, where it can be used to explain the relationship between different scientific variables.

2. Business

The diagram is also used in business to help managers and executives visualize complex data. It can be used to identify trends, patterns, and relationships between different variables.

3. Engineering

The diagram is used in engineering to help engineers design and optimize systems. It can be used to identify areas of inefficiency or to optimize the performance of a system.

4. Research

The diagram is used in research to help researchers visualize complex data sets. It can be used to identify patterns, relationships, and trends in data that might not be apparent from a simple table or graph.

5. Planning

The diagram is used in planning to help planners visualize different scenarios and outcomes. It can be used to identify potential risks and opportunities and to develop strategies for addressing them.

Overall, the George Englebretsen diagram is a versatile tool that can be used in a variety of fields. Its ability to help users visualize complex data sets makes it a valuable asset in many different contexts.

Advantages and Limitations of George Englebretsen Diagram

George Englebretsen Diagram is a useful tool for analyzing and visualizing complex systems. It has several advantages that make it a popular choice among researchers and analysts. For instance, it provides a clear and concise overview of the system being analyzed. This can be especially helpful when dealing with large and complex systems that are difficult to understand at a glance.

Another advantage of the George Englebretsen Diagram is that it can help identify potential problems and areas for improvement. By breaking down the system into its various components and relationships, it becomes easier to pinpoint areas that may be causing issues or hindering performance. This can be useful for organizations looking to optimize their operations and improve efficiency.

Despite its advantages, the George Englebretsen Diagram also has some limitations. One of the main limitations is that it can be time-consuming and difficult to create. This is especially true for large and complex systems, which may require a significant amount of time and effort to accurately map out.

Another limitation of the George Englebretsen Diagram is that it can be difficult to interpret for those who are not familiar with the system being analyzed. This can make it challenging to communicate findings and recommendations to stakeholders who may not have a technical background or understanding of the system.

Overall, the George Englebretsen Diagram is a valuable tool for analyzing and visualizing complex systems. While it has some limitations, its benefits outweigh its drawbacks for those who are willing to invest the time and effort required to create and interpret it.

Conclusion

George Englebretsen's diagram is a valuable tool for understanding complex systems and relationships. By visually representing the connections between different elements, it can help researchers and analysts identify patterns and make sense of large amounts of data.

While the diagram has some limitations, such as its inability to capture the dynamic nature of relationships or the subjective nature of some connections, it remains a useful tool in many contexts. For example, it has been used in fields such as social network analysis, organizational theory, and systems thinking.

Overall, Englebretsen's diagram is a valuable addition to the toolkit of anyone seeking to understand complex systems and relationships. Its simplicity and flexibility make it accessible to a wide range of users, while its ability to capture the essence of complex relationships makes it a powerful tool for analysis and insight.

 

Syllogism According to Fred Sommers:

A Comprehensive Overview

by Armahedid Mahzar and https://koala.sh/writer

Fred Sommers

Fred Sommers was an American philosopher who made significant contributions to the field of logic. He was particularly interested in syllogisms, which are arguments that involve two premises and a conclusion. Sommers believed that syllogisms were an important tool for understanding how people reason and make decisions.

Sommers was born in 1923 in New York City and studied at Columbia University. He went on to teach at Brandeis University for many years, where he became known for his work on syllogisms. Sommers argued that traditional theories of syllogistic reasoning were flawed and that a new approach was needed.

One of Sommers' most important contributions to the study of syllogisms was his theory of "term logic." This theory proposed a new way of understanding the relationship between the terms used in syllogisms. Sommers argued that traditional theories of syllogisms were based on a flawed understanding of the logical relationships between terms, and that his theory provided a more accurate and useful framework for analyzing syllogisms.

Sommers' Theory of Syllogism

Fred Sommers was a philosopher who developed a theory of syllogism that aimed to provide a systematic and comprehensive account of the logical structure of ordinary language arguments. According to Sommers, syllogisms are not the only valid form of deductive reasoning. Instead, he believed that there are many different types of deductive arguments that can be expressed in ordinary language, and that these arguments can be analyzed using a variety of different logical systems.

Sommers' theory of syllogism is based on the idea that ordinary language arguments are composed of two types of premises: categorial and relational. Categorial premises are statements that assert something about a particular object or group of objects, while relational premises assert a relationship between two or more objects or groups of objects.

Sommers argued that the logical structure of an argument can be analyzed by identifying the categorial and relational premises that make up the argument, and then determining the logical relationships between these premises. He believed that this approach could be used to analyze a wide variety of different types of arguments, including those that are not traditionally considered to be syllogisms.

One of the key insights of Sommers' theory of syllogism is that logical relationships between premises can be represented using diagrams. These diagrams, which are known as "Sommers' squares," provide a visual representation of the logical relationships between the premises of an argument. By using these diagrams, it is possible to analyze complex arguments and identify the logical relationships between their various premises.

The Traditional View of Syllogism

Syllogism is a form of deductive reasoning that has been studied and analyzed by logicians for centuries. The traditional view of syllogism, which has its roots in the work of Aristotle, holds that a syllogism consists of three parts: the major premise, the minor premise, and the conclusion. The major premise is a general statement about a class of things, while the minor premise is a specific statement about a particular member of that class. The conclusion follows from these two premises, and it asserts something about the relationship between the particular member and the class as a whole. For example, consider the following syllogism: Major premise: All men are mortal. Minor premise: Socrates is a man. Conclusion: Therefore, Socrates is mortal. According to the traditional view, this syllogism is valid because it conforms to the rules of deductive reasoning. The major premise is a general statement about all men, the minor premise is a specific statement about Socrates, and the conclusion follows necessarily from these two premises. One of the key features of the traditional view of syllogism is its emphasis on the form of the argument rather than its content. In other words, the validity of a syllogism depends on its structure rather than the truth of its premises. This approach has been criticized by some modern logicians, who argue that the content of the premises is just as important as their form. Despite these criticisms, the traditional view of syllogism remains an important part of the study of deductive reasoning. By understanding the rules of syllogism and the ways in which they can be applied, logicians can identify valid arguments and avoid common errors in reasoning.

Sommers' Critique of the Traditional View

Fred Sommers, a philosopher and logician, was highly critical of the traditional view of syllogism, which was based on Aristotle's work. Sommers believed that the traditional view was flawed and did not accurately represent the nature of syllogism. One of Sommers' main criticisms of the traditional view was that it relied on a set of rules that were too restrictive. According to the traditional view, syllogisms must conform to a specific set of rules in order to be considered valid. Sommers argued that these rules were too narrow and did not allow for the full range of valid syllogisms. Sommers also criticized the traditional view for its reliance on the notion of "terms." According to the traditional view, syllogisms are made up of three terms: the major term, the minor term, and the middle term. Sommers argued that this approach was too simplistic and did not accurately represent the complexity of syllogistic reasoning. Instead, Sommers proposed a new approach to syllogism that focused on the relationships between propositions. He argued that syllogisms should be analyzed based on the relationships between the premises and the conclusion, rather than on the specific terms used. Overall, Sommers' critique of the traditional view of syllogism was a significant contribution to the field of logic. His approach emphasized the importance of analyzing the relationships between propositions, rather than simply following a set of rigid rules.

Sommers' Alternative Approach to Syllogism

Fred Sommers, a philosopher and logician, proposed an alternative approach to syllogism that deviates from the traditional Aristotelian model. Sommers' approach to syllogism is known as the "term logic" approach, which focuses on the relationships between terms rather than the standard syllogistic form.

Unlike the Aristotelian model, Sommers' term logic approach does not have a fixed set of rules. Instead, it relies on a set of principles that govern the relationships between terms. These principles include the "principle of opposition," which states that two terms are opposed when they cannot both be true or false, and the "principle of conversion," which allows for the interchangeability of the subject and predicate terms.

One of the key advantages of Sommers' approach is that it allows for a more flexible and nuanced analysis of syllogistic arguments. For example, the Aristotelian model is limited to a set of standard forms, whereas Sommers' approach can accommodate a wider range of argument structures.

Sommers' approach also allows for a more precise analysis of the relationships between terms. By focusing on the relationships between terms, rather than the syllogistic form, Sommers' approach can identify more subtle distinctions between arguments that may be overlooked by the Aristotelian model.

Overall, Sommers' alternative approach to syllogism offers a more flexible and nuanced analysis of syllogistic arguments. While it may not be as widely recognized as the Aristotelian model, it has gained a following among logicians and philosophers who appreciate its more precise and nuanced approach to logical analysis.

Key Features of Sommers' Theory

Fred Sommers was a philosopher who developed a unique theory of syllogism that differed from the traditional Aristotelian approach. According to Sommers, syllogisms are not just a matter of logical form, but also involve the use of language and meaning. Here are some key features of Sommers' theory:

• Importance of Linguistic Meaning: Sommers believed that the meaning of words and phrases is crucial to understanding syllogisms. He argued that traditional syllogistic logic failed to account for the nuances of language and meaning, leading to errors in reasoning.
• Use of Singular Terms: Sommers emphasized the importance of using singular terms in syllogisms, as opposed to general terms. He believed that singular terms are more precise and provide a clearer picture of the subject matter.
• Existential Import: Sommers' theory also incorporated the concept of existential import, which refers to the idea that some statements imply the existence of their subjects. For example, the statement "All unicorns have horns" implies the existence of unicorns.
• Distinction between Denotation and Connotation: Sommers made a distinction between the denotation (literal meaning) and connotation (implied meaning) of words and phrases. He believed that understanding the connotation of terms is essential to understanding syllogisms.

Overall, Sommers' theory of syllogism emphasized the importance of language and meaning in logical reasoning. By taking into account the nuances of language and the implications of statements, Sommers believed that his theory provided a more accurate and precise approach to syllogistic logic.

Sommers' Contributions to the Study of Syllogism

Fred Sommers was a philosopher who made significant contributions to the study of syllogism. He was known for his work on the traditional square of opposition and for his development of a new system of syllogistic reasoning called the "Syllogistic of Relations."

Sommers' work on the traditional square of opposition focused on the relationships between propositions. He argued that the square of opposition was not just a diagram, but a system of logical relationships that could be used to analyze arguments. Sommers' work on the square of opposition led him to develop a new system of syllogistic reasoning that took into account the relationships between propositions.

The "Syllogistic of Relations" was Sommers' attempt to create a system of syllogistic reasoning that was more flexible and powerful than the traditional system. In this new system, the relationships between propositions were represented by a network of arrows, rather than by the simple diagram of the square of opposition. This allowed for a more nuanced analysis of arguments, and made it possible to represent more complex relationships between propositions.

Sommers' work on syllogism was not without controversy. Some philosophers criticized his approach, arguing that it was too complex and difficult to understand. However, others praised his work for its rigor and precision.

Despite the controversy, Sommers' contributions to the study of syllogism have had a lasting impact on the field of logic. His work on the traditional square of opposition and the "Syllogistic of Relations" have been influential in the development of modern logic, and his ideas continue to be studied and debated by philosophers today.

Conclusion

Fred Sommers was a philosopher who made significant contributions to the field of logic, particularly in the area of syllogism. His work on the traditional square of opposition and the logic of natural language has been influential in the development of modern logic.

Sommers' emphasis on the importance of the existential import of propositions challenged the prevailing view of the time and paved the way for further developments in logic. His work on syllogism showed that it was not a dead subject, but rather one that still had much to offer in terms of understanding the nature of reasoning.

Overall, Fred Sommers' contributions to logic have been significant and continue to be relevant today. His work on syllogism and the traditional square of opposition has had a lasting impact on the field, and his insights into the logic of natural language have helped to shape our understanding of how we reason and communicate.

 What is the game of Syllogism?

by Armahedi Mahzar and https://ora.sh/armahzar/obrol

Syllogism is a game that has been played for centuries, and it is an excellent way to exercise the mind and improve reasoning skills. The game is based on the principles of logic and involves making deductions from given premises to arrive at a conclusion. In this blog, we will explore the game of Syllogism, its history, and how to play it.

History of Syllogism

Syllogism was first introduced by the ancient Greek philosopher Aristotle as a tool to study logic. He defined a syllogism as an argument consisting of three propositions: a major premise, a minor premise, and a conclusion. These propositions are connected by two terms, which are called the middle term and the extreme term.

The game of Syllogism became popular in the Middle Ages when it was used as a way to teach logic in universities. It was also used as a form of entertainment and was played by scholars and intellectuals.

How to Play Syllogism

To play Syllogism, you need to have a set of premises and a conclusion. The premises are statements that are assumed to be true, and the conclusion is the statement that follows logically from the premises.

Here is an example of a Syllogism:

Premise 1: All men are mortal. Premise 2: Socrates is a man. Conclusion: Therefore, Socrates is mortal.

The game of Syllogism involves taking the given premises and using logic to arrive at the conclusion. The challenge is to identify the middle term and the extreme term and to use them to make deductions that lead to the conclusion.

Benefits of Playing Syllogism

Playing Syllogism has many benefits, including:

Exercise for the mind: Syllogism requires critical thinking and reasoning skills, which help to exercise the mind.

Improved problem-solving skills: Syllogism helps to develop problem-solving skills by requiring players to identify patterns and make deductions.

Increased logical reasoning: Syllogism is based on the principles of logic, and playing the game helps to improve logical reasoning skills.

Fun and engaging: Syllogism is a fun and engaging game that can be played alone or with others.

Conclusion

Syllogism is an excellent game for anyone who wants to exercise their mind and improve their reasoning skills. It has a rich history and is still played today by scholars and intellectuals. If you want to challenge your mind and improve your logical reasoning skills, give Syllogism a try.

What is the Game of Syllogism?

 

What is the Game of Syllogism?
A Brief Introduction to the Logic Puzzle.
by Armahedi Mahzar and https://koala.sh/writer

The game of syllogism is a logical puzzle that has been around for centuries. It is a type of deductive reasoning that involves determining the validity of a conclusion based on two premises. The game is often used as a tool for teaching logic and critical thinking skills, as it requires players to analyze and evaluate arguments in order to arrive at a correct solution.

Syllogisms are made up of three parts: the major premise, the minor premise, and the conclusion. The major premise is a general statement that is assumed to be true, while the minor premise is a specific statement that is related to the major premise. The conclusion is the logical outcome of the two premises, and it is either valid or invalid depending on the truth of the premises.

There are many variations of syllogisms, each with their own set of rules and challenges. Some syllogisms involve categorical statements, while others involve conditional statements. Some are straightforward, while others require more complex reasoning. Regardless of the type, the game of syllogism is a fun and challenging way to exercise the mind and improve critical thinking skills.

Definition of Syllogism

Syllogism is a form of deductive reasoning that involves arriving at a conclusion based on two propositions that are asserted or assumed to be true. It is a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two premises. The conclusion follows logically from the premises, and is therefore considered to be valid.

The two premises in a syllogism may be either categorical or hypothetical. In a categorical syllogism, both premises are categorical statements, while in a hypothetical syllogism, one or both premises are hypothetical statements. A categorical statement is a statement that asserts or denies something about a category or class of things, while a hypothetical statement is a statement that asserts or denies something about a particular situation or condition.

Syllogisms are often used in philosophy, mathematics, and other fields that require logical reasoning. They are also commonly used in everyday life, such as in legal arguments and debates. The ability to construct and evaluate syllogisms is an important skill for critical thinking and problem solving.

History of Syllogism

Syllogism is a form of deductive reasoning that has been used for centuries to make logical conclusions. The earliest known use of syllogism can be traced back to the Greek philosopher Aristotle, who is credited with developing the concept in his work 'Organon'.

Aristotle's syllogism consisted of three parts: the major premise, the minor premise, and the conclusion. The major premise is a general statement, the minor premise is a specific statement, and the conclusion is a logical deduction based on the two premises.

Over time, syllogism became an important tool in the field of logic and was used extensively by medieval philosophers such as Thomas Aquinas. In the 19th century, syllogism was further developed by George Boole, who introduced the concept of symbolic logic.

Today, syllogism continues to be an important part of logic and is often used in fields such as mathematics, computer science, and philosophy. It is also commonly used in standardized tests such as the LSAT and GRE to evaluate a person's ability to reason logically.

Types of Syllogism

Syllogisms are classified into different types based on their structure and the relationship between their premises and conclusion. The following are the major types of syllogisms:

• Categorical Syllogism: This is the most common type of syllogism, where the premises and conclusion are categorical propositions. These propositions are statements that assert or deny something about a class or category of things.
• Hypothetical Syllogism: This type of syllogism has a conditional premise that states if one thing is true, then another thing is true. The conclusion of a hypothetical syllogism is also a conditional statement.
• Disjunctive Syllogism: This type of syllogism has a premise that presents two alternatives, one of which must be true. The conclusion affirms or denies one of the alternatives.
• Transitional Syllogism: This type of syllogism has a common term in both premises, which allows the conclusion to be inferred.
• Constructive Dilemma: This type of syllogism has two conditional premises that lead to two possible conclusions. The conclusion affirms one of the possible conclusions.
• Destructive Dilemma: This type of syllogism has two conditional premises that lead to two possible conclusions. The conclusion denies one of the possible conclusions.

Each type of syllogism has its own rules and patterns that need to be followed to ensure that the conclusion is valid. Understanding the different types of syllogisms can help in constructing and evaluating arguments.

How to Play the Game of Syllogism

The game of Syllogism is a logic game that involves determining the validity of arguments based on a set of premises. The game is played with a set of rules that must be followed to ensure that the arguments are valid. Here is a breakdown of how to play the game:

Step 1: Understand the rules

The game of Syllogism is played with a set of rules that must be followed. These rules include the use of specific terms and the order in which the premises are presented. Players must understand these rules before playing the game.

Step 2: Read the premises

The game of Syllogism involves reading a set of premises and determining whether the argument is valid or not. Players must carefully read the premises and understand the relationships between them.

Step 3: Determine the conclusion

Once the premises have been read and understood, players must determine the conclusion of the argument. This conclusion must be based on the premises and must be a logical conclusion.

Step 4: Check for validity

Finally, players must check the argument for validity. This involves determining whether the conclusion logically follows from the premises. If the argument is valid, then the conclusion is true. If the argument is not valid, then the conclusion is false.

The game of Syllogism can be challenging, but with practice, players can become skilled at determining the validity of arguments. By following the rules and carefully analyzing the premises, players can improve their logical reasoning skills and become more adept at playing the game.

Benefits of Playing Syllogism

Syllogism is a fascinating game that offers several benefits to its players. Here are some of the benefits of playing syllogism:

• Improves Logical Reasoning: Syllogism is a game that requires players to use their logical reasoning skills to solve problems. By playing syllogism, players can enhance their logical reasoning skills and develop a better understanding of how to use logic to solve problems.
• Enhances Critical Thinking: Syllogism involves analyzing and interpreting information to arrive at a conclusion. This process requires players to use their critical thinking skills and make informed decisions based on the information available. By playing syllogism, players can enhance their critical thinking skills and become better decision-makers.
• Boosts Memory: Syllogism involves remembering information and using it to solve problems. By playing syllogism regularly, players can boost their memory and improve their ability to recall information quickly and accurately.
• Reduces Stress: Syllogism is a fun and engaging game that can help players reduce stress and anxiety. By focusing on the game and solving problems, players can take their minds off their worries and feel more relaxed.

Overall, playing syllogism is an excellent way to enhance your cognitive skills and have fun at the same time. Whether you are a student looking to improve your logical reasoning and critical thinking skills or an adult looking for a fun and challenging game, syllogism is an excellent choice.

Conclusion

Overall, the game of syllogism is a great way to exercise one's logical reasoning skills. It challenges players to think critically and creatively, as they must use deductive reasoning to arrive at the correct answer.

Through practice, players can improve their ability to identify valid and invalid arguments, and to spot fallacies and errors in reasoning. They can also learn to construct their own syllogisms, and to evaluate the validity of others' arguments.

Whether played as a game or used as a teaching tool, the game of syllogism can be a fun and effective way to develop critical thinking skills. It is a great way to engage with logical reasoning and to improve one's ability to think clearly and logically in a variety of contexts.

Simplest Primary Algebra

 Simplest Primary Algebra

George Spencer-Brown is a master of axiomatic logical algebra. He managed to reduce the five Principia axioms for propositional algebra to just two axioms:

However, in my last blog, I have shown you that position can be taken as the single axiom for the primary algebra. In this blog I will show you how easy is to derive position as the consequence of the simplest axiom: cancelation. So cancelation  which can be seen as definition of FALSE if we define   as TRUE.
Cancelation NOT TRUE = FALSE can be seen as the single base axiom for the primary algebra.

Cancelation 

Consequently, the whole Boolean algebra can be derived from the binary principle of logic: TRUE and FALSE.

Even the classical valid categorical syllogisms can be easily proven using the single cancelation axiom. For example the Barbara syloogism
IF ( all a is b) AND (all b is c) THEN (all a is c) = TRUE
is written in the laws of form as
 

Proof of Barbara

Simplifying Primary Algebra

Simplifying Primary Algebra

George Spencer-Brown

   is a master of axiomatic logical algebra. He managed to reduce the five Principia axioms for propositional algebra to just two axioms: 

or,  wriiten in parentheses notation,

position (x(x))=  and
transposition ((xz)(yz))=((x)(y))z,

Reading it existentially, the first axiom is the logical identity x->x and the second axiom is the distribution OR over AND.n of the principle 

Louis Kauffman 

  simplified the Brownian algebra by taking the sixth consequence, extension

or  ((x)y)((x)(y))=x,  as the single axiom for the algebra. It is a beautiful axiom, because reading it existentially, it is the algebraic formulation  the ancient principle of Reductio ad Absurdum: a proposition x is true if only if its negation, NOT x, implies y and NOT y, a contradiction.

However, by defining x=y as x IFF y = (IF x THEN y) AND (IF y THEN X) or (x(y))((x)y), we can derive transposition axiom as a theorem from a new simpler single axiom: position.

Proof:
Transposition ((xz)(yz))=((x)(y))z
=( (((xz)(yz)))((x)(y))z )( ((xz)(yz))(((x)(y))z) )     definition of =
=( (((x )(y )))((x)(y)) )( ((x )(y ))(((x)(y)) ) )           replacement z=
=( (  u         )       u  )(       u     (    u      ) )           substitution u=((x)(y))
=                                                                         position

I think position is simpler than extension because it reduce the number of variables from 2 to 1 and the crosses from 5 to 2.

The question is: can we make another simplification?