# How to work with a premise containing multiple 'if' s?

This was alleged erroneous; so please explain as though I were 10 years old. See the bolded in the following. 3 and 4 confuse me because they are premises, but they themselves are conditional sentences. So what do I do? How do I make sense of them?

Source: 6 mins 33 seconds juncture, Lecture 8-3 (transcription), ... How to Reason and Argue, by Prof W Sinnott-Armstrong    [A screenshot of the original; I simplified his diction]

5. If ALL of the following conditions are met:

1. We have not found any case where X present and Y absent.

2. We have tested a wide variety of cases,
including cases where X is present and cases where Y is absent.

3. If there are any other features [call hese F] that are always absent where Y is absent,
then we have tested cases where those F are absent but X is present.

4. We have tested enough cases of various kinds that are likely to include a case where X
is present and Y is absent, IF there is any such case.

6. [Then] we have good reason to believe X IS a sufficient condition of Y

(1) states that we haven't found an obvious counterexample that disproves (6) immediately.

(2) states that we haven't been restricting our test cases to cases that obviously couldn't be counterexamples (a case where X is absent cannot be a counterexample, and neither can a case where Y is present be a counterexample, we had examples outside these two cases). That means that to some degree we haven't avoided counterexamples in our test cases.

(3) states that cases where X and F are both present cannot be counterexamples, but we haven't restricted the test cases to those cases either. That means again we haven't avoided counterexamples in our test cases.

(4) states explicitly that if there was a counterexample, then we would have been likely to find one, based on the number of different tests.

All of these together do not prove that X is a sufficient condition for Y, but give some indication according to our tests (especially 4) that counterexamples will be rare. However, it is always possible that rare counterexamples exist, and it is even possible that a huge number of counterexamples exist, that for some reason we have not considered.

An example: If you pull the plug in a filled bath tub, the water always swirls in the same direction. You won't find counterexamples. Until you get on a plane to the other side of the equator. All your test cases would not have noticed this.

If you translate an "if ... then ..." construction into propositional logic, you get something of the form P → Q. Recall the truth table of material implication:

`````` P | Q | P -> Q
---+---+--------
0 | 0 | 1
0 | 1 | 1
1 | 0 | 0
1 | 1 | 1
``````

This also shows that P → Q ≡ ¬ P ∨ Q ≡ ¬ (P ∧ ¬ Q), which may help you understand the nature of the implication. In natural language, these three are thus also all equivalent:

• If P is true then Q is true
• P is false and/or Q is true
• It is not the case that P true is but Q false

The nested expression (P → Q is nested in a large implication)

If C1, C2, ..., Cn are all true and P → Q is true, then X is true.

translates into propositional logic to (C1 ∧ C2 ∧ ... ∧ Cn ∧ (P → Q)) → X. This can be evaluated as usually with the truth table above, but this may be easier in natural language. Per the above, the quoted expression is equivalent with

If C1, C2, ..., Cn are all true and "it is not the case the Q but not P", then X is true.

Both your bolded expressions are simple implications (i.e. of the form P → Q where P and Q are atomic expressions) and can thus be rewritten into any of the three forms above. I find especially the last one very helpful when trying to evaluate an expression in natural language.

If every attempt to understand fails, a truth table can still help.