80-110: The Nature of Mathematical Reasoning

Spring 1998

Class 15: 3/5/98

 

1.Validity and Possible Worlds

An argument is valid if it is impossible for the premises to be true and the conclusion false simultaneously. This definition of validity, which we have used all term, uses the notion of "impossible" as a primitive concept. To use this definition, you have to know how to assess when it is impossible for each member of a set of sentences to have a truth value simultaneously. Claims about possibility are called "modal" claims, and they are extremely difficult to make sense of.

Consider, for example, the following sentences:

S1: The instructor for 80-110 in Sp97 at CMU is Robert Scheines.

S2: The earth has 1/100th of its current amount of water.

S3: I sent a message to Pluto and got a response 2 seconds later.

S4: 2+2 = 5

S5: 2+2 ? 5.

The first sentence (S1) might be true or false in the actual world. We might consider possible but not actual worlds in which it were true however. Imagine a world just like this one, but my first name is Robert, not Richard. My birth certificate might look different, and my personal paper’s and file’s, but suppose that I were in all other respects the same. I can imagine such a world, and it is not so different from the actual world. The second sentence S2 is a little more difficult. If we went to a possible world in which it were true, things would be very different. The weather would be enormously different, and thus the landscape, the vegetation, the fish, etc. It is possible, however. It could be. The third sentence is strange, in that it seems possible to Star Trek fans, but not to Physicists who believe in the theory of Special Relativity, which says nothing can travel faster than the speed of light and that Pluto is way more distant than 1 light second. These people say that S3 is logically possible, but not physically possible. That is, there are possible worlds we can imagine in which the physical laws that seem to govern our universe simply do not hold. By logic, however, there are no worlds in which S4 and S5 are both true.

In Tarski’s World, sentences are true of a world. Possible world analyses work the same way. A sentence like: My name has an "R" in it – is true in many possible worlds, including the actual one.

An argument is valid, then, if there are no possible worlds whatsoever in which the premises are true and the conclusion false. From this you can see why it is you can’t tell whether an argument is valid when you are told the premises are true and conclusion is true in the actual world. You can’t tell unless you can visit all possible worlds, not just the actual world.

Tarski's World provides a good environment within which to understand all of this. Consider the following argument:

Argument 1

Premise 1: tet(a)

Premise 2: a = b

Conclusion: tet(b)

Clearly the argument is valid in Tarski's World (TW), but how could you establish that it is so? There are two basic techniques we could use, one semantic and one rule based. In the semantic approach, we could visit all the possible worlds one could express in TW, which might be alot but is clearly finite, and check the truth of the premises and the conclusion in each of these worlds. As soon as we found a world in which the premises were true and the conclusion false, we could stop and demonstrate that the argument is invalid by displaying this world as a counterexample. If we traverse all worlds, and find none in which the premises were true and the conclusion false, then we would have demonstrated the argument to be valid. This might take a long time, but if we were able to program Tarski's World to generate all worlds, then we could do it. We could take a short cut for argument 1 by only considering worlds in which a is a tetrahedron, etc. Because for validity, we don't care about worlds in which the premises aren't true.

The rule based technique takes advantage of the fact that all arguments of a certain form, regardless of their content, should be valid. For example, the Aristotelian syllogism:

All A are B

All B are C

Therefore, All A are C

is valid no matter what we use for A, B, and C. So we need a rule to cover argument 1. For example:

Indiscernibility of Identicals: If "a" and "b" name individual objects, S is any sentence involving the term "a," and S' the sentence resulting from replacing "a" with "b" wherever it occurs in S, then from the sentences S and a=b, conclude S'.

 


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