Logic Chapter Ten
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THEN


Now here's a tricky one. The symbol "," called the "horseshoe" and pronounced "THEN," joins two statements together to make a new statement (called a "conditional") which is false only when the term to the left of the horseshoe (called the "antecedent" is true and the term to the right of the horseshoe (called the "consequent") is false. Lets go over that again slowly. Here's a diagram.


A conditional statement is true in all circumstances except when the antecedent is true and the consequent is false. (The words "antecedent" and "consequent" are not used for the terms of any other kind of statement.) Conjunctions and disjunctions don't have antecedents or consequents. We just use these words so we can talk about when a conditional is true and false. Here's a truth table. (Which worlds is P Q true in? Which world is it false in?)



For practice, work out the values of these conditionals in these worlds.


This is just saying "If we know that the antecedent ("P") is true, THEN we can infer that the consequent ("Q") is true." It doesn't say anything else! It doesn't say anything about what happens if we know that the consequent is true, and it doesn't say anything about what happens if we know that the antecedent is false. Test youself: Only one of the following three statements is true:

If we know that the antecedent is false, then we can infer that the consequent is false.
If we know that the antecedent is true, then we can infer that the consequent is true.
If we know that the consequent is true, then we can infer that the antecedent is true.

Here's its truth table again:


Notice again that P Q is only false where P is true and Q is false. This means that it's true when they're both false, true when they're both true and true when P is false and Q is true. This means that the only situation in which Q can't be false, is where P Q is true and P is true. This means that exactly two of the following four arguments are valid.


Can you figure out which ones are valid? If you can't, try filling in the following truth tables.


Now, look carefully at each completed truth table. Does that truth table have any lines in which the premises are true and the conclusion is false? If it does, that argument is invalid! If it doesn't, that argument is valid.

To check your answer, click on the valid one and you'll see a sign saying it's valid. Click on the others and you'll see pictures showing why they're invalid.

Why The Other Two Arguments Ain't Valid.

Let's define two deductive "Fallacies," which are argument forms that look sort of like good forms, but which are really terrible:

Affirming the Consequent occurs when an arguer takes a true "if P then Q" statement and reverses it, treating it as though it said "if Q then P"

If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. We do have Mars rocks on Earth. (This is true!) 
So NASA did send an expedition to Mars and back in 1974.
If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Mount Saint Helens did blow up, so Luxembourg did sent saboteurs to the US.
If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. Benjamin Franklin is dead, so the British did catch and execute Benjamin Franklin in 1777.

Denying the Antecedent takes a true "if P then Q" statement and treats it as though it said "if not P then not Q"

If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. NASA didn't send an expedition to Mars and back in 1974, so there are no Mars rocks on Earth.
If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Luxembourg has never sent saboteurs to the US, so Mount Saint Helens has never blown up.
If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. The British did not catch and execute Benjamin Franklin in 1777, so Benjamin Franklin is not dead.

Proving Conditionals

Sometime soon we will want to prove statements of the form P Q, where P and Q are any sentences. Notice that proving a conditional only requires us to prove that Q is true if P is true. It doesn't require us to prove Q true absolutely. Rather, it only requires us to prove Q under the assumption that P is true.
This allows us to create conditional statements. Remember that PQ means IF the antecedent, (P) is true, then the consequent (Q) is true. Which means that if we want to prove a conditional statement, all we have to prove is that IF we assume that the antecedent is true THEN the consequent will be true. So lets have a rule that lets us make that assumption that the antecedent is true, and which agrees that, if we've proved the consequent, then we've proved the whole conditional.

This allows us to create the following rule.

Rule 10. Conditional Proof (Abbreviated by "CP")

IF, and ONLY IF, you first write a show line of "Show PQ" you may write "P" as the next line, justifying it by "ACP," for "Assumption for Conditional Proof."

IF, and ONLY IF, you subsequently produce "Q" as an available line, and there is no other uncancelled "show" between that show line and that "Q", then that "show" may be cancelled, and a box may be drawn around that formula and any other lines there may be between it and it's show line. If the relevant show line is line #1, then the derivation is successfully finished. (Write "CP" by the show line.)

(Rules 7-13 are conveniently listed on Logic Rules Sheet Two )



Notice that we didn't need the Lc on line 2 for the rest of the derivation at all. It's just there because the rule says it gets to be there.



It's important to remember that line 3 can't be used until after the word "show" has been cancelled. Line 5 can't be justified by "3, 4, MP" because, when we write line 5, line 3 still has that uncancelled "show."

Now, what about this argument? Is it valid or not? Well, what do you think?

However:


Here's a little rule we might need sometime.
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Rule 11. Repetition (Abbreviated by "R")
If "P" appears as an available line, then "P" may be written as a new line.

(Rules 7-13 are conveniently listed on Logic Rules Sheet Two )

As usual, here is a whole s***load of arguments for you to practice on. At least some of the valid ones require conditional proof. Some of them need you to use conditional proof inside a derivation.


Isn't this fun?

Practice 10. Use your own paper or the answer sheet at practice.
This practice sheet requires you to derive at least four arguments.
Circle
the valid arguments, and cross out the invalid ones. Try to see if you can evaluate the arguments by just looking at them and thinking about the meanings of the formulas.
Then check your answers, and evaluate the other arguments, by truth table.
Finally, indentify the arguments that can be derived by conditional Proof, and derive them.
1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12.
13. 14. 15. 16. 17. 18.

Copyright 2007 by Martin C. Young

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