The ubiquity of logic

The ubiquity of logic One common example of reasoning If I take an umbrella, I can prevent getting wet by rain I don’t want to get myself wet by rain Therefore, I need to come back home to take an umbrella Logic is the study of the methods and principles used to distinguish good from bad reasoning

Logic as a skill Only the student of logic can reason well Excellent athletes who know nothing about physics and physiology A sport science professor who risks his dignity

Practicing logic A person who has studied logic is more likely to reason correctly The proper study of logic will approach it as an art as well as a science  Practice makes perfect A large part of the study of logic is the examination and analysis of fallacies The study of logic will give students techniques and methods for testing the correctness of different kinds of logical reasoning

Declarative sentence Declarative sentences are either true or false Questions to be asked Commands to be given Exclamations to be uttered

Argument Inference is a process by which one sentence is arrived at and affirmed on the basis of one or more other sentences accepted as the starting point of the process There is an argument that corresponds to every possible inference Group of sentences of which one is claimed to follow from the others If I am a superman, I can have laser beam come out of my eyes I cannot have laser beam come out of my eyes Premises Therefore, I am not a superman Conclusion

If the Athenians condemned Socrates to death, they would condemn Plate to death as well Premises But the Athenians did not condemn Plato to death Therefore, the Athenians did not condemn Socrates to death Conclusion The conclusion of an argument is that proposition which is affirmed on the basis of the other propositions of the argument The premises of an argument are the propositions meant to provide evidence or reasons for the conclusion

Deductive and inductive arguments A deductive argument involves the claim that its premises provide conclusive evidence All humans are mortal Socrates is human Therefore, Socrates is mortal All humans are mortal Socrates is human Therefore, Socrates is a philosopher

Differences What is the difference between good and bad arguments Given the truth of the premises, the conclusion must be true Given the truth of the premises, the conclusion couldn’t be otherwise Good arguments are valid but bad arguments are invalid

The validity of deductive argument A argument is valid iff (if and only if) Given the truth of the premises, the conclusion must be true The premises provide (conclusive, decisive, highest possible, strongest possible) (evidence, reason, ground, basis) for the conclusion The truth of the premises (necessitates, entails, implies) the truth of the conclusion It is (contradictory, absolutely impossible, logically impossible) that the premises are all true but the conclusion is false The conclusion (deductively follows, logically follows, is deducible, is derivable) from the premises5) P provides (conclusive or decisive) (evidence, reason, ground, basis) for Q We cannot have true premises and false conclusion at the same time The truth of the premises is (inconsistent, incompatible) with the falsehood of the conclusion

Examples In what sense is this argument valid? All humans are mortal Socrates is human Therefore, Socrates is mortal In what sense is this argument invalid? All humans are mortal Socrates is human Therefore, Socrates is a philosopher

Validity and Contradiction An argument is valid iff it is (contradictory, absolutely impossible, logically impossible) that the premises are all true but the conclusion is false Suppose that the following four sentences are jointly contradictory: A, B, C, and D Then the following arguments are valid A, B, C; therefore ~D A, B, D; therefore, ~C A, C, D; therefore, ~B B, C, D; therefore, ~A

Examples The following sentences are jointly contradictory x is smaller than 2 or x is greater than 2 x is not smaller than 2 x is not greater than 2 The following arguments are valid x is smaller than 2 or x is greater than 2 x is not smaller than 2 It is not the case that x is not greater than 2 x is smaller than 2 x is not greater than 2 It is not the case that x is not smaller than 2

The task of deductive logic The claim that the premises provide conclusive evidence for the truth of the conclusion The task of deductive logic is to clarify the nature of the relation between premises and conclusion in valid arguments, and thus to allow us to discriminate valid from invalid arguments

Inductive argument An inductive argument involves the claim that its premises give inconclusive grounds for the truth of its conclusion Socrates is human and is mortal Plato is human and is mortal Aristotle is human and is mortal Therefore, all humans are mortal Deductively invalid It is not the case that given that the premises are true, the conclusion must be true The truth of the premises doesn’t provide conclusive ground for the truth of the conclusion

Given that the premises are all true, the conclusion is more likely to be true than false An inductively good argument is an argument where the truth of its premises gives some support, albeit not decisive support, to the truth of the conclusion

The strength of support We can meaningfully speak of the strength of the support in terms of the likelihood of the conclusion conditional upon the truth of the premises The strength of the support given by the premises to the conclusion varies from one inductive argument to another Socrates is human and is mortal Plato is human and is mortal Therefore, all humans are mortal The support given to the conclusion by the premises in this argument is weaker than the one in the original Socrates argument

The conclusion of this argument is less likely to be true conditional upon its premises than the conclusion of the original Socrates argument is true conditional upon its premises By omitting the third premise of the original Socrates argument, we have weakened the resulting argument Socrates is human and is mortal Plato is human and is mortal Aristotle is human and is mortal Choi is human and is mortal Therefore, all humans are mortal By adding the fourth premise of the new argument, we have strengthened the resulting argument

Evaluating inductive arguments Inductive arguments may be evaluated as better or worse, according to the strength of the support given to their conclusions by their premises

Distinction One way to distinguish the two types of argument: In a deductive argument, the premises are claimed to furnish outright support or basis for the truth of the conclusion. But in an inductive argument, the premises are claimed to furnish some inconclusive support or basis for the truth of the conclusion. Another way? One suggestion: in a deductive argument we infer a particular sentence from universal sentences, while in an inductive argument we infer a universal sentence from particular sentences An intuitive appeal of the suggestion

Distinction Not universally applicable It is wrong both about deductive arguments and about inductive arguments It is not always the case that in deductive arguments, particular conclusions are inferred from universal premises If the Athenians condemned Socrates to death, they would condemn Plate to death as well But the Athenians did not condemn Plato to death Therefore, the Athenians did not condemn Socrates to death

It is not always the case that, in inductive arguments, universal conclusions are inferred from particular premises All cows are mammals and have lungs All horses are mammals and have lungs All humans are mammals and have lungs Therefore, all mammals have lungs Hilter was a dictator and was callous Stalin was a dictator and was callous Mugabe is a dictator Therefore, Mugabe is callous The criterion for the distinction between deductive and inductive argument in terms of generality and particularity of the sentences involved are mistaken

The ubiquity of logic