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DEDUCTION PRESENTED TO MADAM FOZIA PRESENTED BY MOAZAM IFTIKHAR

DEDUCTION  In

mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it "naturally" occurs.

Arguments and Logic 

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Philosophy is based on critical thinking. Critical thinking simply means that you assume any and every claim is false unless and until it’s proven true. To get a critical thinker to agree that your claim is true, you need to provide evidence for that claim. The claim and its supporting evidence constitute an argument. Every argument has premises and a conclusion. The conclusion is the claim that the argument is trying to prove; the premises are the supporting evidence. There are two general types of arguments: deduction and induction. They differ in their concepts of “proof”, the kinds of evidence they allow, and the strength of their conclusions

MOTIVATION  Natural

deduction grew out of a context of dissatisfaction with sentential axiomatizations common to the systems of Hilbert, Frege, and Russell (see e.g. Hilbert-style deduction system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica.

Judgments and propositions A judgment is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it. Thus "it is raining" is a judgment, which is evident for the one who knows that it is actually raining; in this case one may readily find evidence for the judgment by looking outside the window or stepping out of the house. In mathematical logic however, evidence is often not as directly observable, but rather deduced from more basic evident judgments. The process of deduction is what constitutes a proof; in other words, a judgment is evident if one has a proof for it.

Deductive reasoning Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). If the premises are true, the conclusion must be true. This is distinguished from abductive and inductive reasoning, where the premises may predict a high probability of the conclusion, but do not ensure that the conclusion is true. For instance, beginning with the premises "sharks are fish" and "all fish have fins", you may conclude that "sharks have fins".

Deductive argument A deductive argument is one in which it is claimed that it is impossible for the premises to be true but the conclusion false. Thus, the conclusion follows necessarily from the premises and inferences. In this way, it is supposed to be a definitive proof of the truth of the claim (conclusion). Example:  All birds have feathers,  Socrates has no feathers,  Therefore Socrates is no bird. In symbols, this is: A→B, not B, not A.

Example of a simple syllogism Premise 1: All left-handed people are short. (All X are Y). Premise 2: waseem is left-handed. (A is X). Conclusion: waseem is short. (A is Y).

Example of a simple syllogism  

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Premises provide the evidence or reasons upon which a conclusion is based. We can thus define a 'conclusion' as a statement that is asserted to be true on the basis of the premises of the argument. However, the terms 'premise' and 'conclusion' are relative terms. That is, a conclusion from one argument can become the premise of other arguments. Alternatively, a statement can be a premise in one argument and a conclusion of another.

Consistency, completeness, and normal forms A theory is said to be consistent if falsehood is not provable (from no assumptions) and is complete if every theorem is provable using the inference rules of the logic. These are statements about the entire logic, and are usually tied to some notion of a model. However, there are local notions of consistency and completeness that are purely syntactic checks on the inference rules, and require no appeals to models. The first of these is local consistency, also known as local reducibility, which says that any derivation containing an introduction of a connective followed immediately by its elimination can be turned into an equivalent derivation without this detour. It is a check on the strength of elimination rules: they must not be so strong that they include knowledge not already contained in its premisses. As an example, consider conjunctions.

Philosophical point of view: Distinction in logic between types of reasoning, arguments, or inferences. In a deductive argument, the truth of the premises is supposed to guarantee the truth of the conclusion; in an inductive argument, the truth of the premises merely makes it probable that the conclusion is true.

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