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In traditional Aristotelian logic, Deductive reasoning is reasoning in which the conclusion is necessitated by, or reached from, previously known facts. 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. Deductive reasoning may also be defined as inference in which the conclusion is of no greater generality than the premises or inference in which the conclusion is just as certain as the premises.
How it works Somebody could say, "Since it is raining, the street must be wet". However, there is a hidden argument in this statement: "If it's raining then the street gets wet." Using the premise "If it's raining then the street gets wet" one could argue that "Since it's raining the street is wet" but not "the street is wet so it must be raining". This is because the wet street is an unavoidable product created by the rain but the wet street does not have to be caused by rain. The basic statement "if something then something else" could logically be followed by "something is; so something else must be" and "something else is not; so something else cannot be". These are the first two basic valid reasoning types. A few examples follow: Valid: Since Socrates is a man, and since all men are mortal, Socrates is mortal. Since the picture is above the desk, and since the desk is above the floor, the picture is above the floor. Since a cardinal is a bird, and since all birds have wings, a cardinal has wings. Invalid: A truly left wing politician does not tolerate animal cruelty. G. Houseman thinks hitting a dog is wrong. G. Houseman is a truly left wing politician. Every criminal opposes the government. Everyone in the opposition party opposes the government; therefore, everyone in the opposition party is a criminal. These are invalid because the premises fail to establish commonality between hitting a dog and being a left wing politician, and membership in the opposition party and being a criminal, respectively. This is the famous fallacy of the undistributed middle. Symbolic logic Axiomatization In formal terms, a deduction is a sequence of statements such that each statement can be derived from the preceding one. This leaves open the question of how to prove the first sentence (since it has no predecessor). Axiomatic propositional logic solves this by requiring the following conditions for a proof: A proof of α from an ensemble Σ of well-formed formulas (wffs) is a finite sequence of wffs: β1,...,βi,...,βn where βn = α and for each βi (1 ≤ i ≤ n), either or or
Different versions of axiomatic propositional logics contain a few axioms, usually three or more, in addition to one or more inference rules. For instance, Gottlob Frege's axiomatization of propositional logic, which is also the first instance of such an attempt, has six propositional axioms and two rules. Bertrand Russell and Alfred North Whitehead also suggested a system with five axioms. For instance a version of axiomatic propositional logic due to Jan Lukasiewicz (1878-1956) has a set A of axioms adopted as follows:
and it has the set R of Rules of inference with one rule in it that is Modu Ponendo Ponens as follows:
The inference rule(s) allows people to derive the statements following the axioms or given wffs of the ensemble Σ. Natural deductive logic One version of natural deductive logic has no axioms. System L, developed by E.J. Lemmon, has only nine primitive rules that govern the syntax of a proof. The nine primitive rules of system L are In system L, a proof has a definition with the following conditions: Then if no premise is given, the sequent is called theorem. Therefore, the definitions of a theorem in system L are An example of the proof of a sequent (Modus Tollendo Tollens in this case): An example of the proof of a sequent (a theorem in this case): Each rule of system L has its own requirements for the type of input(s) or entry(es) that it can accept and has its own way of treating and calculating the assumptions used by its inputs. See also | ||||||||
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