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| Deductive Reasoning Processes |
Deductive Reasoning Processes
Deductive processes are based on the precise use of language. The standard approach is to begin with a set of definitions and assumptions, and then to systematically follow the associated logical implications.
Deductive reasoning is an essential ingredient in the fields of logic, mathematics, and computer programming.
Logic
Logic is a powerful tool for building complex arguments from basic concepts, and for the reverse process of taking a complex argument apart to examine its essential components.
A logical argument begins with a few initial assumptions and definitions. Then more complex ideas can be formed by combining these simple assertive statements using the standard conjunctions of and, or, not, and if .–. then. If you begin with carefully crafted assumptions and definitions, your subsequent logical deductions can often lead to insights that were not initially apparent. (Note that interrogatives and emotional statements have no status within logical arguments.)
The simple-to-complex application of logic can be used to construct a philosophical argument or a mathematical theory. The truth of such an argument, or theory, is based on the truth of the original assumptions and the accuracy of the logical constructions.
The converse approach is to start with a complex argument and then work backwards towards the essential definitions and assumptions that form the underpinnings of the argument.
Strictly speaking, logic exists only in the minds of humans and is independent of reality. In practice, the assumptions and definitions used in logic often reflect the best available interpretations of reality – an inductive approach.
The abstract methods of logic can lead to very practical results. George Boole (1815 – 1864) used a logical process to develop a system of mathematics using just two numbers, 0 and 1. He was able to show that all of our mathematics using base-ten could be reduced to equivalent statements in base-two. Several decades after his death, Boole’s binary algebra was used to create electronic logic gates that now form the core of every digital computer in the world.
Deductive geometry
Geometry is a field of mathematics that deals with shapes and boundaries. The origin of geometry is lost in the mists of time, but by 2000 BCE geometry played a key role in the allocations of agricultural land after annual floods of the Euphrates and Tigris rivers in Mesopotamia, and the Nile River in Egypt. Geometric properties were also used in the design and construction of fortresses, temples, and pyramids.
About 300 BCE, the Greek mathematician Euclid, summarized and organized all that was known about geometry in a book called The Elements. This work was so thorough and complete that it was used as a standard geometry text for the next two thousand years. The Elements is more than a collection of geometric properties. As students study Euclidian geometry, they are also studying the process of deductive reasoning.
The six volumes on plane geometry begin with a set of just three postulates and two definitions, and then all the other geometric properties of plane figures are derived using deductive logic.
Algebra
About 825 CE, an Arabian mathematician in Baghdad summarized all the known techniques for solving equations in a book entitled Al-jabr wa’l muqabalah. Al- Khowarizmi’s work was so successful that a branch of mathematics is named after his book. Initially, algebra consisted of detailed instructions, or algorithms, for solving different types of equations. Over the past thousand years, the subject has grown and been refined so that algebra is now based on definitions, postulates, and rules of logic.
Using algebraic techniques, a mathematical statement can be rearranged into an infinite number of equivalent statements. Given a complex statement, a typical goal is to find the simplest possible representation that still contains all the information of the original statement. For example, the concept of equality can be used in combination with the operations of addition, subtraction, multiplication, and division to convert a linear equation into a variety of alternate forms. Thus, the equation 3x – 7 = 5 can be logically re-arranged to yield x = 4.
When similar mathematical problems reoccur in a number of situations, it is practical to devise a streamlined approach. An algebraic theorem can be created that reduces a whole sequence of logical steps to just one line. Quoting the theorem and skipping all the associated details then simplifies solving similar problems in the future.
Mathematicians like to use theorems so they can work more efficiently.
Unfortunately, these shortcuts can make formal solutions difficult for nonmathematicians to follow.
Algebra is not the only field of mathematics that utilizes proofs and theorems.
Currently, there are over two hundred recognized fields of mathematical research. In fact, you can create your own field of mathematics with just a few definitions and postulates. The challenge is to create a system that has interesting and possibly useful features.
Computer programming
Programming a computer is a challenging task in deductive reasoning. A computer language is highly structured, all the terms are carefully defined, and all processes are completely logical in nature. When you create a computer program – every word, every number, every comma, and every bracket must be used in accordance with the rules of the computer language you are using. A program will not run on a computer until every syntax error is corrected.
Even when all the syntax errors are eliminated, a computer program may still contain errors in logic that produce false results in unexpected situations.
Students new to computer programming often find the cold logic of a machine very frustrating. When results do not turn out as expected, one often hears the comment, ‘But I meant to instruct the computer to do this!’ A computer pays no attention to emotion or intention – it uses pure logic to do precisely what you tell it to do.
Deductive proof
Deductive reasoning flows from precise definitions for classes of objects, and to the operations that can be performed on those objects. There are only three basic strategies that lead to a deductive proof:
1. Direct proof – you create a step-by-step logical path from your starting information to a final statement. Then if your assumptions and starting information were correct, you have proven that your final statement must also be correct.
2. Indirect proof – in a limited number of cases, you can list all the possible outcomes, and then show that all but one of the outcomes leads to a logical contradiction. The one remaining outcome must then be true. It is like solving a murder mystery by proving all the possible suspects, except George, must be innocent. Therefore George must be guilty.
3. Mathematical induction – in a very limited number of cases involving patterns with natural numbers, this deductive technique can be used to prove that a numerical pattern must hold true within a given range of natural numbers. The term ‘mathematical induction’ is somewhat misleading. While finding the pattern in the first place is an inductive process, proving that the pattern holds true is actually a deductive process.
Deductive processes are based on the precise use of language. The standard approach is to begin with a set of definitions and assumptions, and then to systematically follow the associated logical implications.
Deductive reasoning is an essential ingredient in the fields of logic, mathematics, and computer programming.
Logic
Logic is a powerful tool for building complex arguments from basic concepts, and for the reverse process of taking a complex argument apart to examine its essential components.
A logical argument begins with a few initial assumptions and definitions. Then more complex ideas can be formed by combining these simple assertive statements using the standard conjunctions of and, or, not, and if .–. then. If you begin with carefully crafted assumptions and definitions, your subsequent logical deductions can often lead to insights that were not initially apparent. (Note that interrogatives and emotional statements have no status within logical arguments.)
The simple-to-complex application of logic can be used to construct a philosophical argument or a mathematical theory. The truth of such an argument, or theory, is based on the truth of the original assumptions and the accuracy of the logical constructions.
The converse approach is to start with a complex argument and then work backwards towards the essential definitions and assumptions that form the underpinnings of the argument.
Strictly speaking, logic exists only in the minds of humans and is independent of reality. In practice, the assumptions and definitions used in logic often reflect the best available interpretations of reality – an inductive approach.
The abstract methods of logic can lead to very practical results. George Boole (1815 – 1864) used a logical process to develop a system of mathematics using just two numbers, 0 and 1. He was able to show that all of our mathematics using base-ten could be reduced to equivalent statements in base-two. Several decades after his death, Boole’s binary algebra was used to create electronic logic gates that now form the core of every digital computer in the world.
Deductive geometry
Geometry is a field of mathematics that deals with shapes and boundaries. The origin of geometry is lost in the mists of time, but by 2000 BCE geometry played a key role in the allocations of agricultural land after annual floods of the Euphrates and Tigris rivers in Mesopotamia, and the Nile River in Egypt. Geometric properties were also used in the design and construction of fortresses, temples, and pyramids.
About 300 BCE, the Greek mathematician Euclid, summarized and organized all that was known about geometry in a book called The Elements. This work was so thorough and complete that it was used as a standard geometry text for the next two thousand years. The Elements is more than a collection of geometric properties. As students study Euclidian geometry, they are also studying the process of deductive reasoning.
The six volumes on plane geometry begin with a set of just three postulates and two definitions, and then all the other geometric properties of plane figures are derived using deductive logic.
Algebra
About 825 CE, an Arabian mathematician in Baghdad summarized all the known techniques for solving equations in a book entitled Al-jabr wa’l muqabalah. Al- Khowarizmi’s work was so successful that a branch of mathematics is named after his book. Initially, algebra consisted of detailed instructions, or algorithms, for solving different types of equations. Over the past thousand years, the subject has grown and been refined so that algebra is now based on definitions, postulates, and rules of logic.
Using algebraic techniques, a mathematical statement can be rearranged into an infinite number of equivalent statements. Given a complex statement, a typical goal is to find the simplest possible representation that still contains all the information of the original statement. For example, the concept of equality can be used in combination with the operations of addition, subtraction, multiplication, and division to convert a linear equation into a variety of alternate forms. Thus, the equation 3x – 7 = 5 can be logically re-arranged to yield x = 4.
When similar mathematical problems reoccur in a number of situations, it is practical to devise a streamlined approach. An algebraic theorem can be created that reduces a whole sequence of logical steps to just one line. Quoting the theorem and skipping all the associated details then simplifies solving similar problems in the future.
Mathematicians like to use theorems so they can work more efficiently.
Unfortunately, these shortcuts can make formal solutions difficult for nonmathematicians to follow.
Algebra is not the only field of mathematics that utilizes proofs and theorems.
Currently, there are over two hundred recognized fields of mathematical research. In fact, you can create your own field of mathematics with just a few definitions and postulates. The challenge is to create a system that has interesting and possibly useful features.
Computer programming
Programming a computer is a challenging task in deductive reasoning. A computer language is highly structured, all the terms are carefully defined, and all processes are completely logical in nature. When you create a computer program – every word, every number, every comma, and every bracket must be used in accordance with the rules of the computer language you are using. A program will not run on a computer until every syntax error is corrected.
Even when all the syntax errors are eliminated, a computer program may still contain errors in logic that produce false results in unexpected situations.
Students new to computer programming often find the cold logic of a machine very frustrating. When results do not turn out as expected, one often hears the comment, ‘But I meant to instruct the computer to do this!’ A computer pays no attention to emotion or intention – it uses pure logic to do precisely what you tell it to do.
Deductive proof
Deductive reasoning flows from precise definitions for classes of objects, and to the operations that can be performed on those objects. There are only three basic strategies that lead to a deductive proof:
1. Direct proof – you create a step-by-step logical path from your starting information to a final statement. Then if your assumptions and starting information were correct, you have proven that your final statement must also be correct.
2. Indirect proof – in a limited number of cases, you can list all the possible outcomes, and then show that all but one of the outcomes leads to a logical contradiction. The one remaining outcome must then be true. It is like solving a murder mystery by proving all the possible suspects, except George, must be innocent. Therefore George must be guilty.
3. Mathematical induction – in a very limited number of cases involving patterns with natural numbers, this deductive technique can be used to prove that a numerical pattern must hold true within a given range of natural numbers. The term ‘mathematical induction’ is somewhat misleading. While finding the pattern in the first place is an inductive process, proving that the pattern holds true is actually a deductive process.
