Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22A valid argument is one where the conclusion is a logical consequence of its premises. This means that if the premises are true, the conclusion must be true. Consider …Socrates is a man.All men are mortal.Socrates is mortal. valid or invalid?Lucretius is mortal.All men are mortal.Lucretius is a man. valid or invalid?goldfish?So then, an invalid argument may have true premises . . .(assume the goldfish interpretation)Lucretius is mortal. (true . . . fish die)All men are mortal. (true . . . people die)Lucretius is a man.(false . . . Lucretius is a fish, not a man)Indeed, an invalid argument may be built on not only true premises but on a true conclusion as well:The moon revolves around the earth.The earth revolves around the sun.Therefore, Mars revolves around the sun.But the conclusion is NOT a logical consequence of the two premises, and so the argument is NOT valid. Put differently, there is nothing about the truth of the premises that makes it necessary that the conclusion be true. The ‘therefore’ introducing the conclusion is empty of any force here. Imagine: what if ‘Mars’ were to name a planet in some other solar system instead of being a planet in our system?All true in our worldConversely, an argument may be valid even if one or more of its premises and even if the conclusion is false in some worlds. This is because validity is determined only by the form/structure of the argument, not the truth values of its statements in any particular world.The moon is made entirely of blue cheese.Blue cheese smells bad.Therefore, the moon smells bad. In our real world, the first premise above and the conclusion are both false, but the argument is still valid because of its form. That is, if it were indeed true that the moon is made of blue cheese and that such cheese smells bad, then it would necessarily be true that the moon smells bad.False in our worldTrue (?) in our worldFalse in our worldEven more to the extreme, it is possible to think of valid arguments that don’t have a single true statement in them (i.e., true as evaluated in regard to some particular world):#The moon is made entirely of blue cheese.Blue cheese always glows pink.Therefore, the moon glows pink.Every line of the above argument is false in the real world, and yet the argument is valid because the conclusion is a logical consequence of the two premises (i.e., if the premises were true, then the conclusion would necessarily be true too because of the way the sentences are related to each other).All false in our worldA sound argument: a valid argument with true premises and true conclusion as evaluated in some particular world.*a valid argument may be either sound or unsound, depending on the facts in the particular world.*an invalid argument is automatically unsound relative to all worlds, regardless whether the individual premises and conclusion are true in some worlds.Notice that validity is a feature of an argument itself without regard for particular worlds. (Another way to say this is that an argument is simply either valid or invalid for all worlds.) Soundness, in contrast, is always relative to a world. The same valid argument may be sound in one world but unsound in another.Exercise 2.1, p. 44(Tarski’s World with Socrates’ Sentences and Socrates’ World files open)Evaluate several of the arguments found in Socrates’ Sentences as valid or invalid. . .Then evaluate the same arguments as sound or unsound in Socrates’ World.More practice: Is the following argument valid? Is it sound in Hamilton’s Bizarre Little World?P1: SameCol(d,b)P2: RightOf(c,d)P3: RightOf(a,b)C: RightOf(c,a)bacdUsing counterexample interpretation to prove that an argument is invalid:T Lucretius is mortal. ‘Lucretius’ = goldfishT All men are mortal.F Lucretius is a man.The fact that the premisescan be true while the conclusion is false proves there is no logical consequence relation here.How to prove that an argument is valid?By building a ‘proof’: “a step-by-step demonstration that a conclusion follows from some premises” (p. 46).The steps of the proof are a “series of intermediate conclusions,” each built on the premises and conclusions that come before it and leading toward the final conclusion. As long as each step along the way builds on the preceding steps strictly according to the laws of logic, your conclusion is guaranteed, and you’ll have proven the validity of the argument.Suppose we want to show that the last sentence below is a logical consequence of the first four:Socrates is a man, All men are mortal, No mortal lives forever, Everyone who will eventually die sometimes worries about it. Therefore, Socrates sometimes worries about dying Informal proof: Since Socrates is a man and all men are mortal, it follows that Socrates is mortal. But all mortals will eventually die, since that is what it means to be mortal. So Socrates will eventually die. But we are given that everyone who will eventually die sometimes worries about it. Hence Socrates sometimes worries about dying. (pp. 47-48)A sample formal proof (a Fitch-style deductive system; p. 49):1. Cube(c) each line is numbered2. c = b each line is an FOL sentence ‘Fitch bar’ separates premises from rest of proof3. Cube(b) = Elim: 1, 2Any step that is not a premise must be justified with a rule and step citationsIdentity elimination rule: = Elim aka “the identity of indiscernibles”This rule justifies our saying that if two names refer to the same identical object, then we can substitute one of those names for the other in a proof. 1. Cube (c)2. c = b3. Cube(b) = Elim: 1, 2To make a substitution using the identity elimination rule, you must have two components: (1) ‘instructions’ telling you what can be substituted for what … like: c = b(2) a place or ‘template’ into which you make the substitution … like: Cube(c)Identity introduction rule: = Introaka “the reflexivity of identity”This rule allows us to state at any time in a proof that an object is equal to itself. For example: a = aWe can use the preceding two identity rules to prove another principle of identity (one which doesn’t have its own formal rule): symmetry of identity (e.g., if a = b, then b = a)Informal
View Full Document
Unlocking...