A sign in the supermarket says, ‘Express lane – ten items or less’. An antisocial git at the front of the queue with a trolley full of trophies is locked in a battle of wills with the cashier while telling everyone else to mind their business and get their logic right. ‘Express lane – ten items or less’ is not a law. Is it logic?
The sign can be phrased in terms of a conditional: You can get in the express lane only if you have ten items or less in your basket or trolley. On the first order logic reading of the sentence, there are two issues at play here: the necessary versus sufficient condition, and the truth value of the conditional. To see how they interact, it is best, perhaps, to turn the sentence into a simple material conditional:
(1) If you are in an express lane, you have ten items or less.
By all accounts, having ten items or less is a necessary condition for getting in an express lane, while being in an express lane is only a sufficient condition for having ten items or less. You could after all go to a regular check-out with ten items or less and nobody would mind. They might even thank you.
The standard theory turns on just this symmetry: the consequent of a conditional is the necessary condition for the antecedent, while the antecedent is a sufficient condition for the consequent.
On the truth-value reading of conditionals, a conditional sentence is false when the consequent is false while the antecedent is true. If it is indeed the case that you are in an express lane but have more than ten items in the trolley, (1) is false as a reflection of reality. It is true under all other circumstances, including when you are not in an express lane but have ten items or less, which is just as it should be.
What would you make of (1) if antecedent and consequent were flipped around:
(2) If you have ten items or less, you are in an express lane.
Assuming the facts have not changed: you have more than ten items and are in an express lane, the antecedent in (2) is now false while the consequent is true. On the truth-value reading of conditionals, (2) is true.
The truth of (2) may appear somewhat counterintuitive at first but it is less so once you realize what it means to assert the truth of: If you have ten items or less, you are in an express lane. Nothing in this sentence implies that having more than ten items while being in an express lane is not true. It may be, and, from the logical point of view, is true, too.
Interpreting the sign in terms of (2) will please the antisocial git, as he can shrug his shoulders and point out that (2), as it stands, is true of reality – him having more items and being in an express lane.
But in what sense is being in an express lane a necessary condition for having ten items or less in the trolley?
You could win the argument with the antisocial git if you got him to admit that his interpretation of the sign is: You can have ten items or less in your trolley only if you get in the express lane. I rate your chances of a successful argument though along with the chances of the antisocial git holding the chair of the faculty of ethics in town.
The sign can be phrased in terms of a conditional: You can get in the express lane only if you have ten items or less in your basket or trolley. On the first order logic reading of the sentence, there are two issues at play here: the necessary versus sufficient condition, and the truth value of the conditional. To see how they interact, it is best, perhaps, to turn the sentence into a simple material conditional:
(1) If you are in an express lane, you have ten items or less.
By all accounts, having ten items or less is a necessary condition for getting in an express lane, while being in an express lane is only a sufficient condition for having ten items or less. You could after all go to a regular check-out with ten items or less and nobody would mind. They might even thank you.
The standard theory turns on just this symmetry: the consequent of a conditional is the necessary condition for the antecedent, while the antecedent is a sufficient condition for the consequent.
On the truth-value reading of conditionals, a conditional sentence is false when the consequent is false while the antecedent is true. If it is indeed the case that you are in an express lane but have more than ten items in the trolley, (1) is false as a reflection of reality. It is true under all other circumstances, including when you are not in an express lane but have ten items or less, which is just as it should be.
What would you make of (1) if antecedent and consequent were flipped around:
(2) If you have ten items or less, you are in an express lane.
Assuming the facts have not changed: you have more than ten items and are in an express lane, the antecedent in (2) is now false while the consequent is true. On the truth-value reading of conditionals, (2) is true.
The truth of (2) may appear somewhat counterintuitive at first but it is less so once you realize what it means to assert the truth of: If you have ten items or less, you are in an express lane. Nothing in this sentence implies that having more than ten items while being in an express lane is not true. It may be, and, from the logical point of view, is true, too.
Interpreting the sign in terms of (2) will please the antisocial git, as he can shrug his shoulders and point out that (2), as it stands, is true of reality – him having more items and being in an express lane.
But in what sense is being in an express lane a necessary condition for having ten items or less in the trolley?
You could win the argument with the antisocial git if you got him to admit that his interpretation of the sign is: You can have ten items or less in your trolley only if you get in the express lane. I rate your chances of a successful argument though along with the chances of the antisocial git holding the chair of the faculty of ethics in town.
It should read: 10 items or FEWER...by the way
ReplyDeleteThis should read: 10 items or FEWER.
ReplyDelete