Deduction and arguments

We have seen that we can deduce new conditionals from old. But where do we get conditionals to start with? Fortunately, we can get conditionals from our three principles. Premise (5) in Part 1's LSAT example was shorthand for a collection of conditionals; in the same way, both non-contradiction and excluded middle provide us with several conditionals. For example, excluded middle gives us "if *P* is not true, then *P* is false" for anything we want to fill in the blank for *P*.

In addition to conditionals, these principles get us two more induction techniques beyond simple if-then arguments. In reality, these arguments could be converted to if-then deductions if we wanted to. These additional methods are just templates that help us speed up the process a bit.

In an argument by contradiction, we provisionally assume that a claim is false. We then deduce a premise is false. As a valid argument is an argument about true premises, this violates the principle of non-contradiction. Hence our deduction had to have made a mistake somewhere. The only possible mistake was our initial assumption that the claim is false, so it cannot be false. By the principle of excluded middle, this means this claim is true.

Returning to our LSAT example from Part 1, suppose we want to deduce the claim

One ofP,G, orScannot be cut.

To argue by contradiction, we start assuming that this claim is false. In other words, all three of them are cut. When *G* and *S* are cut, we can deduce from premise (2) that *W* is cut. By premise (5) we know that two of *L*, *M*, and *R* must be cut. Counting all these up, we get at least six areas that must be cut. This contradicts premise (1), so one of the initial three—*P*, *G*, or *S*—cannot be cut.

In an argument by cases, we start with a claim and create two arguments: one assuming the claim false, and another assuming the claim true. If we can deduce the same conclusion in both cases, then that conclusion is valid because of the principle of excluded middle.

Returning to our LSAT example, suppose we want to deduce the claim

IfRis not cut, thenGmust be cut.

We are deducing a conditional, so we start with the provisional assumption that *R* is not cut. We can immediately deduce, from premise (4), that *L* and *M* are cut. By premise (1), we need to reduce three more areas from *G*, *S*, *W*, and *N*. We now break our argument into two cases:

. This leaves us only three areas left to choose from, so*N*is not cut*G*must be cut.. We deduce from premise (3) that*N*is cut*S*is not cut. That leaves us only two areas to choose our remaining two cuts from, so*G*must be cut.

*G* is cut in both cases, so we have successfully argued "if *R* is not cut, then *G* must be cut."

In practice, these arguments rarely have a true/false breakdown. Instead, they start with an exhaustive list of possibilities and use them as the cases. For example, if we want to argue that tuition at Ivy League schools is too expensive, we can list all of the Ivy League schools and present a separate argument for each. Be warned that this approach has an unstated premise, namely that the list covers all of the possibilities. Ignoring this additional premise is the source of the false dilemma fallacy.

Please thank Professor White in the comments for sharing these articles!

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Thank you for the article Professor, immensely helpful!

ReplyDeleteThanks Professor White for this article and the previous one! So helpful!!

ReplyDeleteThis was beyond confusing

ReplyDeleteAnon, I agree. I have no idea what's going on over here...

ReplyDeleteI think the confusion arises from the Professor's use of the word "cut" in place of the term reduced. The word cut implies that it is cut from the selection of five items which are being reduced.

Essentially cut implies "not reduced" which is clearly the opposite of his intention.

I don't understand what numbers (1)-(5) represent. I read Part 1 of this article but still couldn't fully comprehend. Can someone explain them?

ReplyDeleteNumbers (1) - (5) represent the premises in prep test 20 game 2. number 1 is reducing 5 of 8 area's of expenditure, number 2 is if both G and S are reduced...etc. (posted below by accident).

DeleteThank you Professor White.

ReplyDeleteNumbers (1) - (5) represent the premises in prep test 20 game 2. number 1 is reducing 5 of 8 area's of expenditure, number 2 is if both G and S are reduced...etc.

ReplyDeleteSorry meant to post under Anonymous dated September 2, 2013

DeleteThat was very enlightening! Thank you for the article, Professor!

ReplyDeleteThanks so much Professor White and Steve for sharing it on here. Very helpful.

ReplyDeleteI'm following the one month study guide and this and the previous post reference the "principle of excluded middle," but none of the blog entries assigned on this study guide explain what that is, so it makes understanding these concepts difficult.

ReplyDelete