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Logical Reasoning Logical Reasoning Proposition or Statement is a declarative sentence that is either True or False. Argument: An argument is a group of statements including one or more premises and one and only one conclusion. Validity and soundness are properties of an argument. Premise: A premise is a statement in an argument that provides reason or support for the conclusion. There can be one or many premises in a single argument. Truth and falsity are properties of premises. Conclusion: A conclusion is a statement in an argument that indicates of what the arguer is trying to convince the reader/listener. What is the argument trying to prove? There can be only one conclusion in a single argument. Distinction between premises and conclusions: The foolproof way to do this is to ask yourself what the author of the argument is trying to get you to believe. The answer to this question is the conclusion. There must also be at least one reason and possibly many. These are your premises. Reasoning: the process of thinking about something in a logical way in order to form a conclusion or judgment. Inference: An inference is a process of drawing conclusions based on the evidence. On the basis of some evidence or a “premise,” you infer a conclusion. Consider the following example. Premise: Weather forecast says 80% chance of thunderstorms. Inference: It’s a good idea to bring an umbrella Logic: It is the systematic study of applying reasoning that leads to the acceptance of one proposition, the conclusion, on the basis of a set of other propositions, the premises. In other word, logic is the science that evaluates arguments. Analytical Reasoning: Refers to the ability to look at information, be it qualitative or quantitative in nature, and discern patterns within the information. In other words, analytical reasoning is the act of carefully considering a problem, claim, question or situation in order to determine the best solution. Applying analytical reasoning also means seeing things from your point view, there may be some subjectivity also. It also represents judgments made upon statements that are based on the virtue of the statement's own content. Analytical reasoning involves deductive reasoning with no specialised knowledge, such as: comprehending the basic structure of a set of relationships; recognizing logically equivalent statements; and inferring what could be true or must be true from given facts and rules. Analytical reasoning is axiomatic in that its truth is self-evident. Logical Reasoning is the process of using a rational, systematic series of steps based on sound mathematical procedures and given statements to arrive at a conclusion, without any ambiguity. Logical reasoning can be categorised in three branches: deduction, induction and abduction. Deductive reasoning determines whether the truth of a conclusion can be determined for that rule, based solely on the truth of the premises. Deduction is an inference based on logical certainty. It usually starts from a general principle and then infers something about specific cases. Example: Grapes are poisonous to all dogs. This allows you to infer that grapes are poisonous for your dog, too. If the premise is true, then the conclusion has to be true. There’s no other possibility. Notice, however, that this doesn’t really tell you anything new: once you say, “grapes are poisonous to all dogs,” you already know that grapes are poisonous for your specific dog. Deduction has the advantage of certainty, but it doesn’t generate new knowledge. Mathematical logic and philosophical logic are commonly associated with this type of reasoning. Deductive inference is further categorised into “immediate“ in which conclusion is drawn from a single statement and “mediate” in which conclusion is drawn from two statements, known as “syllogism” Inductive reasoning attempts to support a determination of the rule. It hypothesizes a rule after numerous examples are taken to be a conclusion that follows from a precondition in terms of such a rule. Induction is an inference based on probability. It usually starts from specific information and then infers the more general principle. Example: For the last two years, Amanda has woken up at 6 am every day” This allows you to infer that Amanda will probably wake up at 6 am tomorrow, too. You would probably be right, and it’s a reasonable inference but it’s not certain! Tomorrow could be the first day that Amanda decides to sleep in. While they may be persuasive, these arguments are not deductively valid. Despite this uncertainty, however, induction does offer the possibility of predicting future events and creating new knowledge. Science is associated with this type of reasoning. Abductive reasoning infers to the best explanation, selects a cogent set of preconditions. Given a true conclusion and a rule, it attempts to select some possible premises that, if true also, can support the conclusion, though not uniquely. Example: "When it rains, the grass gets wet. The grass is wet. Therefore, it might have rained." This kind of reasoning can be used to develop a hypothesis, which in turn can be tested by additional reasoning or data. Diagnosticians, detectives, and scientists often use this type of reasoning. Within the context of a mathematical model, the three kinds of reasoning can be described as follows. The construction/creation of the structure of the model is abduction. Assigning values (or probability distributions) to the parameters of the model is induction. Executing/running the model is deduction. Structure of Arguments (Deduction and Induction) The following is an indicative representation of the structure of a typical argument. Validity and Soundness of Arguments: • A deductive argument is said to be valid if and only if premises are true and the conclusion is also true. In other words, a valid argument cannot have true premises and false conclusion. Otherwise, a deductive argument is said to be invalid. • In effect, an argument is valid if the truth of the premises logically guarantees the truth of the conclusion. The following argument is valid, because it is impossible for the premises to be true and the conclusion nevertheless to be false: Anil owns either a Honda or a Saturn. Anil does not own a Honda. Therefore, Anil owns a Saturn. • It is important to stress that the premises of an argument do not have actually to be true in order for the argument to be valid. An argument is valid if the premises and conclusion are related to each other in the right way so that if the premises were true, then the conclusion would have to be true as well. In other words, the validity of argument depends, however, not on the actual truth or falsity of the premises and conclusion, but solely on whether argument has a valid logical form. Consider, the following argument: All toasters are items made of gold. All items made of gold are time-travel devices. Therefore, all toasters are time-travel devices. A deductive argument is sound if and only if it is both valid, and all its premises are actually true. Otherwise, a deductive argument is unsound. The following argument is both valid and sound: In some states, no criminals are eligible voters. In those states, some professional athletes are criminals. Therefore, in some states, some professional athletes are not eligible voters. • In short, a deductive argument must be evaluated in two ways. First, one must ask if the premises provide support for the conclusion by examining the form of the argument. If they do, then the argument is valid. Then, one must ask whether the premises are true or false in actuality. Only if an argument passes both these tests is it sound. However, if an argument does not pass these tests, its conclusion may still be true, despite that no support for its truth is given by the argument. • An inductive argument is an argument that is intended by the arguer to be strong enough that, if the premises were to be true, then it would be unlikely that the conclusion is false. So, an inductive argument’s success or strength is a matter of degree, unlike with deductive arguments. There is no standard term for a successful inductive argument, but this article uses the term “strong.” Inductive arguments that are not strong are said to be weak. Here is a mildly strong inductive argument: Every time I’ve walked by that dog, it hasn’t tried to bite me. So, the next time I walk by that dog it won’t try to bite me. • Analogical Argument is a special type of inductive argument, whereby perceived similarities are used as a basis to infer some further similarity that has yet to be observed. Analogical reasoning is one of the most common methods by which human beings attempt to understand the world and make decisions. • The following diagram gives pictorial depiction of validity and soundness of various kind of arguments. Analysis of A Proposition A proposition is a sentence that makes a statement and gives relation between two or more terms. • A proposition is assumed to be true and from which a conclusion can be drawn. • The statement, ‘all books are apples’ is assumed to be true as a proposition (or premise), but actually we all know that books and apples are entirely different entities. In standard form of a proposition consists of four parts as shown below: Quantifier + Subject + Copula + Predicate 1. Quantifier: The words 'all', 'no' and 'some' are called quantifiers because they specify a quantity 'All' and 'no' are universal quantifiers because they refer to every object in a certain set, while the quantifier 'some' is a particular quantifier because it refers to at least one existing object in a certain set.
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