reasoning methods and argumentation

8/6/2019 Reasoning Methods and Argumentation

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Reasoning methods and argumentation

Reasoning and proof have many meaning in mathematics; logical thinking, critical thinking,conjecturing, estimating, justifying, and explaining are some terms related to reasoning andproof. All emphasize thinking in mathematics.

One approach to the study of reasoning is to identify various forms of reasoning that may be

used to support or justify conclusions. The main division between forms of reasoning that ismade in philosophy is between deductive reasoning and inductive reasoning. Formal logic has

been described as 'the science of deduction'.[13]

The study of inductive reasoning is generallycarried out within the field known as informal logic orcritical thinking.

Deductive reasoning

Reasoning in an argument is valid if the argument's conclusion must be true when the premises(the reasons given to support that conclusion) are true. One classic example of deductive

reasoning is that found in syllogisms like the following:

Premise 1: All humans are mortal.Premise 2: Socrates is a human.

Conclusion: Socrates is mortal.

The reasoning in this argument is valid, because there is no way in which the premises, 1 and 2,

could be true and the conclusion, 3, be false.

Validity is a property of the reasoning in the argument, not a property of the premises in theargument or the argument as a whole. In fact, the truth or falsity of the premises and the

conclusion is irrelevant to the validity of the reasoning in the argument. The following argument,with a false premise and a false conclusion, is also valid, (it has the form of reasoning known as

modus ponens).

Premise 1: If green is a color, then grass poisons cows.Premise 2: Green is a color.

Conclusion: Grass poisons cows.

Again, if the premises in this argument were true, the reasoning is such that the conclusion would

also have to be true.

In a deductive argument with valid reasoning the conclusion contains no more information thanis contained in the premises. Therefore, deductive reasoning does not increase one's knowledge

base, and so is said to be non-ampliative.

Within the field offormal logic, a variety of different forms of deductive reasoning have beendeveloped. These involve abstract reasoning using symbols, logical operators and a set of rules

that specify what processes may be followed to arrive at a conclusion. These forms of reasoning


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include Aristotelian logic, also known as syllogistic logic,propositional logic,predicate logic,and modal logic.

Inductive reasoning

Induction is a form of inference producing propositions about unobserved objects or types, either

specifically or generally, based on previous observation. It is used to ascribeproperties orrelations to objects ortypes based onprevious observations or experiences, or to formulate

general statements or laws based on limited observations of recurringphenomenal patterns.

Inductive reasoning contrasts strongly with deductive reasoning in that, even in the best, or

strongest, cases of inductive reasoning, the truth of the premises does not guarantee the truth ofthe conclusion. Instead, the conclusion of an inductive argument follows with some degree of

probability. Relatedly, the conclusion of an inductive argument contains more information than

is already contained in the premises. Thus, this method of reasoning is ampliative.

A classic example of inductive reasoning comes from the empiricistDavid Hume:

Premise: The sun has risen in the east every morning up until now.

Conclusion: The sun will also rise in the east tomorrow.

. For example, even a first grader can use an informal proof by contradiction to argue that thenumber 0 is even: "If 0 were odd, then 0 and 1 would be two odd numbers in a row. But evenand odd numbers alternate. So 0 must be even."

Abductive reasoning

Abductive reasoning, or argument to the best explanation, is a form of inductive reasoning, since

the conclusion in an abductive argument does not follow with certainty from its premises andconcerns something unobserved. What distinguishes abduction from the other forms of reasoning

is an attempt to favor one conclusion above others, by attempting to falsify alternativeexplanations or by demonstrating the likelihood of the favored conclusion, given a set of more or

less disputable assumptions. For example, when a patient displays certain symptoms, there mightbe various possible causes, but one of these is preferred above others as being more probable.

Analogical reasoning

Analogical reasoning is reasoning from the particular to the particular. An example follows:

Premise 1: Socrates is human and Socrates died.Premise 2: Plato is human.

Conclusion: Plato will die.


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Analogical reasoning can be viewed as a form of inductive reasoning, since the truth of thepremises does not guarantee the truth of the conclusion. However, the traditional view is that

inductive reasoning is reasoning from the particular to the general, and thus analogical reasoningis distinct from inductive reasoning.[14] An example of inductive reasoning from the particular to

the general follows:

Premise 1: Socrates is human and Socrates died.Premise 2: Plato is human and Plato died.

Premise 3: Aristotle is human and Aristotle died.Conclusion: All humans die.

It has been argued that deductive, inductive, and abductive reasoning are all based on a

foundation of analogical reasoning.[15]

Fallacious reasoning

Flawed reasoning in arguments is known as fallacious reasoning. Reasoning within argumentscan be bad because it commits either a formal fallacy or an informal fallacy.

Formal fallacies

Formal fallacies occur when there is a problem with the form, or structure, of the argument. Theword 'formal' refers to this link to the form of the argument. An argument that contains a formal

fallacy will always be invalid. Consider, for example, the following argument:

1. If a drink is made with boiling water, it will be hot.2. This drink was not made with boiling water.

3. This drink is not hot.

The reasoning in this argument is bad, because the antecedent (first part) of the conditional (the

'if..., then...' statement) can be false without the consequent (second half) of the conditional beingtrue. In this example, the drink could have been made with boiling milk, or heated in the

microwave, and so be hot in spite of the truth of statement 2. This particular formal fallacy isknown as denying the antecedent.

Informal fallacies

An informal fallacy is an error in reasoning that occurs due to a problem with the content, rather

than mere struc

ture, of the argument. Reasoning that commits an informal fallacy often occurs inan argument that is invalid, that is, contains a formal fallacy. One example of such reasoning is ared herring argument.

An argument can be valid, that is, contain no formal reasoning fallacies, and yet still contain an

informal fallacy. The clearest examples of this occur when an argument contains circularreasoning, also known asbegging the question.


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Other than that, There are three kinds of rule-based intelligence in mathematics, logic and mostpattern-based subjects.

The first kind met in primary school arithmetic consists of skills with repeatable, reproducible

and therefore verifiable results - results that are then considered right or wrong.

The second kind also met in primary school consists of pattern or rule recognition. Thedevelopment or exploitation of the ability to recognize or suggest simply patterns in order to

predict the next element in a sequence. If the prediction fails, another pattern is required.

The third kind, assumption-based, deductive reason, appears after inductive mastery of logic, that

is mastery of implication rules If A then B and their use. The third kind follows the use ofimplication rules and definitions and assumptions, one at a time and one after another, to arrive

at logical conclusions. Here chains of reason how to be posed in a readable, repeatable,reproducible and therefore verifiable manner.

reasoning methods and argumentation

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