I have a logic license |

My “Logic” chariot, crunched from behind before my travels, you might recall (blogpost Nov. 15, “Logic Takes a Bit of a Hit”), has been robustly repaired and beautifully corrected, all in my absence!^{1} So here’s a little bit of blog logic….

In a couple of the early posts (e.g., Sept. 9 post), some logical terms were noted (e.g., the valid form of *modus tollens*); but it can’t hurt to review them with a mind toward the specific patterns of arguments that arise in the Birnbaum case.

An argument is deductively valid if it’s impossible for all of its premises to be true and its conclusion false at the same time (on pain of logical contradiction). By validity, in this post, I will always mean deductively valid. The conclusion is what is inferred from the premises that purport to provide evidence for its truth.^{2} To say an argument is valid is not to say its premises or its conclusion are true, but only that *if all the premises are true, then it’s conclusion would also have to be true*. It’s an if-then claim.

A valid argument can have a false conclusion, but if it does, then at least one of the premises must be false.

Deductive validity is a matter of pure form (hence the term formal logic). If an argument (argument form) is valid, then any argument with that same form is also valid. Likewise for arguments that are invalid.

For an argument to be sound, however, it must have true premises as well as be formally valid. Since we want to infer the conclusion, i.e., detach it from its premises, we would like sound arguments. Making an argument valid (by adding premises) is easy; ensuring those premises are true is hard. In evaluating an argument for validity, we do not add premises, but evaluate it just as given. (A different activity could be to consider what additional premises, if added, would convert an invalid argument into a valid one.)

EXAMPLE 1: Here’s a valid argument:

Premises:

1. Any two entrees ordered off the special Dec. 25 menu M* have the same price.

2. The duck and the mahi mahi were both ordered off the special Dec. 25 menu M*.

3. The price of the duck entrée is $29.99.

Conclusion:, the price of the mahi mahi entree is $29.99.

To give a partial symbolization with (hopefully) obvious assignments to the abbreviations,we might have:

1. For all x, y, If M*(x) and M*(y), then $(x) = $(y)

2. The duck and the mahi mahi have property M*, that is,

M*(duck) and M*(mahi mahi)

3. $(duck) = $29.99

Conclusion: Therefore, $(mahi mahi) = $29.99.

Note: M* is a property (of entrees), $ is a function (from entrees to prices).

If the bill shows your mahi mahi order is not $29.99, then one of the three premises is false. Once you discover, say, that the mahi mahi was actually not ordered off the special menu M*, but rather off the regular menu M’, (and you were charged the mahi mahi price listed in menu M’), then you know the second premise is false. The argument is valid but unsound. For an argument to be sound, it must have true premises as well as be formally valid.

EXAMPLE 2: An Invalid argument

1. Any two entrees ordered off the special Dec. 25 menu M* have the same price.

2. Duck and mahi mahi were ordered, and the duck was ordered off the special Dec. 25 menu M*.

3. The price of the duck is $29.99.

Conclusion: the price of the mahi mahi is $29.99.

It’s easy to see that the premises of example 2 can be all true and yet the conclusion false. The way we actually show truth values of sentences requires assigning interpretations to the elements of the argument: the domain over which variables x, y range, the names of objects in the domain, and the various properties, relations, and functions on the domain.^{3} An interpretation that makes all the premises true is a “model” of those statements.

EXAMPLE #3: Exercise: Valid or invalid? (Try to symbolize)

1. For any married couple (x,y) filing federal taxes jointly (in the U.S.), x and y have the same tax liability; namely, the amount in the “married filing jointly” column.

2. If a married couple in the U.S. does not file jointly but each files separately, then each owes the amount in the “married, filing separately” column.

3. Deborah and George are a married couple in the U.S.

4. If Deborah files separately, then the amount of tax Deborah owes equals d.

5. If George files separately, then the amount of tax George owes equals g.

Conclusion: d = g

You may assume these premises are true, e.g., that d and g are dollar numbers given in the respective “married filing separately columns”. Never mind deductions or the like.

[1] Credit goes to George!

[2] An inference can refer to the entire argument or the conclusion drawn from the premises. I will always indicate which is meant.

[3] 1. For all x, y If M*(x) and M*(y) ithen $(x) = $(y)

2. M*(duck) (i.e., the duck order has property M*)

3. $(duck) = $29.99

Therefore, $(mahi mahi) = $29.99.