Conditionals and Arguments

Conditionals & Arguments
Necessary and Sufficient Conditions Conditionals, typically (though not always!) expressed in English as sentences of the form "If .................. , (then) .................. , state necessary and sufficient conditions. In a conditional so expressed, the clause that follows the "if" is the antecedent; the other clause is the consequent. The state of affairs described in the antecedent is asserted to be a sufficient condition on the circumstance described in the consequent. To say that it is sufficient is just what it sounds like: it is to say that it is enough, nothing more is required to guarantee that the state of affairs described in the other clause obtains. Consider, for example (1) If someone is a mother then they're female. If you know that someone is a mother (not just a parent) that is enough to show that the person is female therefore being a mother is a sufficient condition on being female. It is not, however, a necessary condition on being female since being a mother is not a requirement for being female: you can be female without being a mother. On the other hand, being female is necessary for being a mother: if someone is not female they can't possibly be a mother. Thus (1) says that being a mother is a sufficient condition on being female and being female is a necessary condition on being a mother. In general, for any conditional whatsoever, the antecedent is a sufficient condition on the consequent and the consequent is a necessary condition on the antecedent. This may not always be obvious, for consider the following: (2) If you study then you'll pass. (2) clearly says that studying is sufficient for passing. It's not so clear that it says passing is necessary for studying and, in fact, it sounds peculiar because passing is something that happens after you study and it seems odd to suggest that the occurrence of a later event is necessary for the occurrence of an earlier event. How can it be that my ability to study now depends upon something happening in the future, i.e. my passing? The oddity arises because conditionals in ordinary English, like (2), may express something more than necessary and sufficient conditions: they may express causation and, hence, temporal succession. Let us however ignore this feature of some conditionals (we'll talk about it later when we "go formal") and consider them only insofar as they are expressions of necessary and sufficient conditions. If we strip away the causal and temporal connotations of (2) it becomes more plausible to see passing as a necessary condition on studying, especially if you recognize that once you discount tense (2) is equivalent to (3) If you didn't pass then you couldn't have studied. When one thing is necessary for another thing that means that if you haven't got the first then you haven't got the second. Water, for example, is necessary for plant life: no water, no plants. Similarly, (3) says no pass conclusively shows no study, hence that passing is necessary for studying. In general, for any sentence of the form, “If P then Q,” its contrapositive, “If not-Q then not-P,” is logically equivalent to it. Two sentences are logically equivalent if they necessarily have the same truth value. (3) is the contrapositive of (2), in idiomatic English. Now there is a difference between necessary and sufficient conditions: in (1), for example, being a mother is sufficient, but not necessary, for being female while being female is necessary but not sufficient for being a mother. This isn't always so: some times one thing is both necessary and sufficient for something else. Consider, for example, the following true proposition: (4) For any integers x and y , xy is odd if and only if both x and y are odd. (4) says that the oddness of xy is both necessary and sufficient for the oddness of both x and y . You can think of statements of necessary and sufficient conditions like (4) as, in effect, two way conditionals: each of the conditions is necessary and sufficient for the other. In fact it is extremely useful to think of them this way because in proving propositions like (4) the standard strategy is to prove that the first condition is sufficient for the second and then that the second is sufficient for the first. Conditionals, Arguments and Inferences Like arguments, conditionals may express inferences. Nevertheless, a conditional by itself is not an argument. The difference is that when you put forth an argument you commit yourself to the truth of all its parts--even if "only for the sake of the argument." When you assert a conditional, however, you do not commit yourself to the truth of either its antecedent or its consequent. Indeed, the whole conditional can be true even if both its parts are false. Compare the conditional and argument below: (5) If Michelle Bachmann is elected then I'll eat my hat. (6) Michelle Bachmann will be elected, therefore I will eat my hat. Indeed, someone who asserts a conditional like (5) is convinced that neither the antecedent nor the consequent is true--he is betting against bin Laden’s surrender! Conditionals, however, can figure as parts of arguments--as premises, conclusions or both. The following are arguments, which contain conditionals: (7) If you study then you'll pass. If you pass then you'll graduate. Therefore if you study you'll graduate. (8) If a number is even then it's divisible by 2 without a remainder. 4 is divisible by 2 without a remainder. Therefore, 4 is even. Corresponding Conditional For any given argument, the conditional that is formed by taking the conjunction (the "and-ing") of its premises as the antecedent and the conclusion of the argument as its consequent is the corresponding conditional to that argument. For example, (9) below is an argument and (10) is its corresponding conditional: (9) All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (10) If all men are mortal and Socrates is a man then Socrates is mortal. Notice that (9) is, intuitively, a valid argument: the premises really "force" the conclusion in the sense that if they are true then the conclusion must be true. Notice also that (10) is necessarily true. In general, an argument is valid if and only if its corresponding conditional is necessarily true.

Conditionals & Arguments

Necessary and Sufficient Conditions

Conditionals, typically (though not always!) expressed in English as sentences of the form "If .................. , (then) .................. , state necessary and sufficient conditions. In a conditional so expressed, the clause that follows the "if" is the antecedent; the other clause is the consequent.
The state of affairs described in the antecedent is asserted to be a sufficient condition on the circumstance described in the consequent. To say that it is sufficient is just what it sounds like: it is to say that it is enough, nothing more is required to guarantee that the state of affairs described in the other clause obtains. Consider, for example

(1) If someone is a mother then they're female.

If you know that someone is a mother (not just a parent) that is enough to show that the person is female therefore being a mother is a sufficient condition on being female.
It is not, however, a necessary condition on being female since being a mother is not a requirement for being female: you can be female without being a mother. On the other hand, being female is necessary for being a mother: if someone is not female they can't possibly be a mother. Thus (1) says that being a mother is a sufficient condition on being female and being female is a necessary condition on being a mother.

In general, for any conditional whatsoever, the antecedent is a sufficient condition on the consequent and the consequent is a necessary condition on the antecedent. This may not always be obvious, for consider the following:

(2) If you study then you'll pass.

(2) clearly says that studying is sufficient for passing. It's not so clear that it says passing is necessary for studying and, in fact, it sounds peculiar because passing is something that happens after you study and it seems odd to suggest that the occurrence of a later event is necessary for the occurrence of an earlier event. How can it be that my ability to study now depends upon something happening in the future, i.e. my passing?

The oddity arises because conditionals in ordinary English, like (2), may express something more than necessary and sufficient conditions: they may express causation and, hence, temporal succession. Let us however ignore this feature of some conditionals (we'll talk about it later when we "go formal") and consider them only insofar as they are expressions of necessary and sufficient conditions.

If we strip away the causal and temporal connotations of (2) it becomes more plausible to see passing as a necessary condition on studying, especially if you recognize that once you discount tense (2) is equivalent to

(3) If you didn't pass then you couldn't have studied.

When one thing is necessary for another thing that means that if you haven't got the first then you haven't got the second. Water, for example, is necessary for plant life: no water, no plants. Similarly, (3) says no pass conclusively shows no study, hence that passing is necessary for studying. In general, for any sentence of the form, “If P then Q,” its contrapositive, “If not-Q then not-P,” is logically equivalent to it. Two sentences are logically equivalent if they necessarily have the same truth value. (3) is the contrapositive of (2), in idiomatic English.
Now there is a difference between necessary and sufficient conditions: in (1), for example, being a mother is sufficient, but not necessary, for being female while being female is necessary but not sufficient for being a mother. This isn't always so: some times one thing is both necessary and sufficient for something else. Consider, for example, the following true proposition:

(4) For any integers x and y , xy is odd if and only if both x and y are odd.
(4) says that the oddness of xy is both necessary and sufficient for the oddness of both x and y . You can think of statements of necessary and sufficient conditions like (4) as, in effect, two way conditionals: each of the conditions is necessary and sufficient for the other. In fact it is extremely useful to think of them this way because in proving propositions like (4) the standard strategy is to prove that the first condition is sufficient for the second and then that the second is sufficient for the first.

Conditionals, Arguments and Inferences
Like arguments, conditionals may express inferences. Nevertheless, a conditional by itself is not an argument. The difference is that when you put forth an argument you commit yourself to the truth of all its parts--even if "only for the sake of the argument." When you assert a conditional, however, you do not commit yourself to the truth of either its antecedent or its consequent. Indeed, the whole conditional can be true even if both its parts are false. Compare the conditional and argument below:

(5) If Michelle Bachmann is elected then I'll eat my hat.

(6) Michelle Bachmann will be elected, therefore I will eat my hat.
Indeed, someone who asserts a conditional like (5) is convinced that neither the antecedent nor the consequent is true--he is betting against bin Laden’s surrender!

Conditionals, however, can figure as parts of arguments--as premises, conclusions or both. The following are arguments, which contain conditionals:

(7) If you study then you'll pass. If you pass then you'll graduate. Therefore if you study you'll graduate.

(8) If a number is even then it's divisible by 2 without a remainder. 4 is divisible by 2 without a remainder. Therefore, 4 is even.

Corresponding Conditional

For any given argument, the conditional that is formed by taking the conjunction (the "and-ing") of its premises as the antecedent and the conclusion of the argument as its consequent is the corresponding conditional to that argument. For example, (9) below is an argument and (10) is its corresponding conditional:

(9) All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

(10) If all men are mortal and Socrates is a man then Socrates is mortal.
Notice that (9) is, intuitively, a valid argument: the premises really "force" the conclusion in the sense that if they are true then the conclusion must be true. Notice also that (10) is necessarily true. In general, an argument is valid if and only if its corresponding conditional is necessarily true.