(2017-10-25, 06:40 PM)Chris Wrote: [ -> ]So if it's ["p implies q" --Laird] restated as "in all cases where p is true, q is also true", then that implies that all cases where p is not true are consistent with the rule - the rule doesn't mention them, so they can't contradict it.
And this is the heart of the quirk to which I've been alluding.
In everyday logic, if somebody said to us, "I am trying to decide (for some given 'p' and 'q') whether 'p' implies 'q', in other words, whether, in all cases where p is true, q is also true. How helpful are the cases where 'p' is false in making this decision?", we would probably respond: "Not at all. They can't help you to decide that the implication is true, and they can't help you to decide that it's false. They are simply irrelevant. Only the cases where 'p' is
true can help you with that decision".
Unfortunately, in propositional logic, we can't put a couple of "irrelevant"s into the truth table - we are restricted to one of "true" or "false". I think you explain well in your post why the best choice out of these for the two "not p" rows is "true". But this does cause a break with everyday logic, and some very counter-intuitive results.
Rather than putting it into my own words, I've found
a PDF online which explains this well. First off, on the issue just discussed - what the truth-table value should be for the material implication operator when the antecedent is false, given that we can't use a value like "irrelevant" - it puts it in an only slightly different way than you do, Chris:
Quote:Logicians have decided to take an “innocent until proven guilty” stance on this issue. An if—then statement is considered true until proven false. Since we cannot call the statement p implies q false when p is false, our only alternative is to call it true.
The paper goes on to describe some of the counter-intuitive results of this quirk:
Quote:We emphasize again the surprising fact that a false statement implies anything.
It provides several examples - the one I find most illustrative is this one:
Quote:(1) If elephants can fly, then the Cubs will win the World Series this year.
(2) If elephants can fly, then the Cubs will lose the World Series this year.
Both statements are true—assuming, of course, that elephants can’t fly.
It also provides an example that
doesn't flow from any case where "not p" is true - but rather follows from the fact that there is not necessarily a causal connection between "p" and "q" when both are true and thus "p implies q" is true, as we would require when saying in everyday logic that "p implies q":
Quote:(1) If hydrochloric acid (HCl) and sodium hydroxide (NaOH) are combined, then table salt (NaCl) will be produced.
(2) If March has 31 days, then dogs are mammals.
Both statements are true. The first statement is an example of cause and effect and reflects the chemical equation
HCl + NaOH = NaCl + H2O.
In the second statement, there is clearly no causal relation between days of the month and dogs being mammals. Both the premise and the conclusion of Statement 2 are true, so according to the chart, the implication is true. The point here is that the use of implies in logic is very different from its use in ever[y]day language to reflect causality.
Anyhow. Hopefully that goes some way to explaining what I've been going on about re the unreliability of the material implication operator as a reflection of everyday-logic implication. It allows for such supposed entailments as we have discussed - e.g. (¬Q→¬P)→(P→Q) - that we would never accept in everyday logic.