logic an introductory course

[...]... consider any possible circumstances in which the premises are all true and ask if the conclusion is true in just those circumstances EXERCISES 1 Give an example of a valid argument in which both the premises and the conclusion are false and an example in which both are true Logic and language 5 2 Give an example of an invalid argument in which both the premises and the conclusion are false and an example... and ‘Q’ is true, ‘P and Q’ is false And if ‘P’ is false and ‘Q’ is false, ‘P and Q’ is false It is clear that we have covered all the possibilities for truth and falsity with regard to ‘P’ and ‘Q’ Writing ‘T’ for ‘true’ and ‘F’ for ‘false’ we can represent the possibilities as follows: P T T F F Q T F T F Logic 14 We based our determination of the truth-value and ‘P and Q’ on our intuitive understanding... have spin +1/2 is less easily explained Of course you have to know that they are in fact sentences of English And, as we have seen, you do Logic 6 have to know the meaning of certain key words such as ‘and’ and ‘all’ To see the importance of these key words which we will call logical constants, replace ‘and’ by ‘or’ in list I and ‘all’ by ‘some’ in list II and examine the resulting arguments for validity... ‘if’ is called the antecedent and the sentence following the ‘then’ is called the consequent This case is of crucial importance to the development of logic and we cannot avoid the controversy It will be helpful to consider first an important difference in the ways in which we can evaluate assertions We can ask if someone’s assertion is true or false We can also consider whether an assertion is misleading... treatment of logic without treating the conditional as a truth-function There are logics that do not do this However, one cannot run logically without first walking logically and we ought to begin at the beginning with a simple logic Having mastered it the diligent student can go on to study more sophisticated logics in which there is no crude equation of the conditional in English with the logician’s material... and not on content that we can aspire to develop a systematic study of valid arguments We can describe the form of a given valid argument and show that all arguments of that form (there will be an indefinitely large number of such arguments) are valid And it is this fact, the fact that validity depends on form and not content, that licenses us to introduce symbols into our logic For instance, we can... proposition to express what the English sentence and its translation into another language have in common: we will say that the sentences express the same proposition This notion of a proposition applies within a language as well For instance, we recognize that ‘Caesar stabbed Brutus’ and ‘Brutus was stabbed by Caesar’ have the same meaning and we can convey this by saying that they express the same... schizophrenic and turns out not to have a vitamin B excess And certainly we would not want to put an F here To see this consider the conditional above about the liquid in the glass and suppose that the glass is empty Then both the antecedent and the consequent are false but we would count the conditional as true Logicians introduce a symbol ‘→’ which is called the material conditional and give it the... indicates a form and can be converted into an assertion if ‘A’ and ‘B’ are replaced by terms expressing particular propositions, just as replacing xs and ys in algebraic equations by terms for particular numbers yields an assertion Let ‘P’ and ‘Q’ be understood by the interpretation given above ‘P and Q’ is true just in case ‘P’ is true, ‘Q’ is true If ‘P’ is false and ‘Q’ is true, ‘P and Q’ is false... distinguishing between valid and invalid arguments To the extent that we are successful we will be able to offer an answer to the question: in virtue of what is it that one can recognize an argument as valid? That is, we will develop through the study of logic a technique for doing explicitly and reflectively something that we can do reasonably well for simple arguments implicitly and without reflection . Oxford. I thank the Master and Fellows for this. Andrew Boucher and Martin Dale provided detailed comments on the manuscript at an early stage and their help has been invaluable. I thank, too,. to know the meaning of certain key words such as ‘and’ and ‘all’. To see the importance of these key words which we will call logical constants, replace ‘and’ by ‘or’ in list I and ‘all’ by. argument and to deny the conclusion? Logic and language 9 interested in logic even if it will not turn us into demons of rationality. This will be done using an analogy from linguistics. Any reader