Conditional Arguments

I am confused between the VALID and INVALID conditional arguments. The book 'Being Logical' states that affirming the antecedent or denying the consequent yield valid arguments, while denying the antecedent or affirming the consequent yield invalid arguments. Consider the following example:

-If the weather is nice tomorrow, we will go on a picnic.

1- By denying the antecedent (the good weather), we can still decide to go on a picnic. Doesn't this yield a possible conclusion and a valid argument? 2- By affirming the consequent (going on a picnic), the good weather can really be the reason behind the picnic; hence, it is a possible conclusion. How could it be invalid?

Is the invalid argument only addressing the conclusions thay 'should' follow necessarily? such as: If Louise is running, then she is moving. Louise is not running. Therefore, she is not moving.

• `while denying the antecedent or affirming the consequent yield invalid arguments` ... are you sure about that wording? – virmaior Nov 23 '17 at 15:32

See page 48 of Being Logical:

The simplest argument is one composed of two statements, a supporting statement or premise and a supported statement or conclusion. Usually, the context of the argument will allow you to tell which is which, but we attach what are called "logical indicators" to statements in order to mark them clearly as either premises or conclusions. [...] Common logical indicators for conclusions are "therefore"...

Thus, the exposition of Conditional Argument (page 63) is a little bit sloppy: "if the weather is nice tomorrow, we will go on a picnic" is not an argument but a compound statement.

The complete argument will be:

"if the weather is nice tomorrow, we will go on a picnic; the weather is nice tomorrow. Therefore, we will go on a picnic."

It is an instance of the following "argument schema":

"if A, then B; A. Therefore, B".

Regarding validity, see page 60: validty regards "form" (the structure of the argument) and not "matter" (its content):

An argument is valid if its structure is sound, which means that its structure is such that true premises will ensure a true conclusion.

Thus, in order to assess validity of an argument, we have to consider the corresponding "argument schema", i.e. the formula obtained replacing statements with (sentential) variables:

An argument is valid when every "instance" (obtained substituting real sentences for the variables) of it has the nice property that:

if all the premises are true, so is the conclusion.

Please, note the clause "if all the premises are true"; in the above case, we have to assume that both premises are true.

But if we assume that the first premise: "if the weather is nice tomorrow, then we will go on a picnic" is true, then we cannot say "tomorrow there will not be good weather, but we can still decide to go on a picnic", because this is not the original premise.

The "argument schema":

"if P, then Q, and P; therefore Q",

is valid, as well as: "if P, then Q, and not-Q; therefore not-P".

The "argument schema": "if P, then Q, and Q; therefore P", instead, is not valid (see Affirming the consequent).

The validity of: "if P, then Q, and P; therefore Q" as well as that of: "if P, then Q, and not-Q; therefore not-P" are licensed by the truth table for the conditional propositional connective: "if..., then...".

For the first one, we assume that the first premise is true: thus rules out the 2nd line of the truth table (P true and Q false).

But we assume also P true, and this in turn rules out 3rd and 4th lines.

In conclusion, the assumption of both premises gives us only one possible state of affairs: P true and Q true: thus, we are licensed to safely conclude, from the two assumptions above, with the truth of Q.

Same approach to the second argument schema: the assumption of the first premise rules out the 2nd line of the truth table (P true and Q false).

And the assumption of not-Q as true (i.e. of Q as false) in turn rules out 1st and 3rd lines.

Again, we are left with only one line: the 4th one, where P is false, i.e. not-P is true.

Thus, we are licensed to safely conclude, from the two assumptions, with the truth of not-P.

• So are you saying that the example of the weather is not a valid argument because the truth of its first premise is not an absolute one since it is predictive? – Yaman H Nov 22 '17 at 7:54
• @YamanH - not clear... "If the weather is nice tomorrow, we will go on a picnic" is not an argument but a sentence. It can be used as a premise of an argument; adding a second premise and a conclusion, we get an argument. The result can be valid or not according to the "logical form" it has. – Mauro ALLEGRANZA Nov 22 '17 at 7:57
• Then let's consider the argument you provided: "if the weather is nice tomorrow, then we will go on a picnic, and the weather is nice tomorrow; therefore we will go on a picnic". Is it valid even though the fact that the weather is nice tomorrow is a predicitve one (we cannot be certain about the weather tomorrow). In other words, how do we ensure the truth of the second premise while its content is not fulfilled yet? – Yaman H Nov 22 '17 at 8:17
• Validity doesn’t care whether the premises are true or false. A valid argument is one where the conclusion must be true IF the premises are. If all you know is that a given argument is valid, it could be that: (a) the premises are true and the conclusion is true; (b) the premises are false and conclusion is false; (c) the premises are false and the conclusion is true. The only thing you can rule out is that the premises are true and the conclusion false. Note that this corresponds exactly to the truth-conditions of ‘if…then…’. A valid argument with true premises is called sound. – MarkOxford Nov 22 '17 at 9:15
• My apologies. My previous comment was addressed to @Yaman H, and was supposed to help with the question of whether the above argument is valid “even though the fact that the weather is nice tomorrow is a predicitve one”. – MarkOxford Nov 22 '17 at 10:12

In logic, there isn’t really such a thing as a ‘possible conclusion’: either the conclusion follows from the premises, or it doesn’t. If it does not follow, it might still be consistent with the premises; but that’s not enough to render the argument valid. To elaborate this a little, let’s consider the following:

1: If P, then Q

2: P

3: Q

4: not-Q

5: not-P

Together, (1) and (2) yield (3), i.e. the argument that has (1) and (2) as premises and (3) as conclusion is valid. The argument pattern is called modus ponens. Similarly, the argument that has (1) and (4) as premises and (5) as conclusion, is also valid. This pattern is called modus tollens and it is the ‘opposite’ of modus ponens. (I think both inference patterns are quite intuitive, but I’ll provide an explanation if you’re not sure.)

Now suppose you have (1) and (5). This does not imply (4), i.e. the argument with (1) and (5) as premises and (4) as conclusion, is not valid. However, (1), (4), and (5) are consistent, i.e. they can all be true at the same time / under the same interpretation. The reason is that (1) is silent on what happens to Q if P is false. The conditional only covers the case where P is true. (At least, that’s so if you understand ‘if…then…’ the way logicians do.) In other words, (1) only tells you that you can’t have P without Q, but still allows you to have Q without P.

Turning to the combination of (1) and (3), much the same reasoning applies – except that (3) actually implies (1), i.e. the argument with (3) as premise and (1) as conclusion is valid. This may seem a bit counter-intuitive, but think of it this way: if Q is true, it’s true no matter what; i.e., it’s true regardless of whether P is true or false; i.e., it is true if P and if also if not-P; i.e., Q it is true if P.

Finally, remember that the logical ‘if…then…’ never asserts that one thing caused another: it only says that the one implies the other – and for that to be the case, the two propositions don’t have to be related.

Bonus question (if I may): suppose you take (1) again as premise and add (6) as a second premise. What does that imply?

6: If P, then not-Q

• Does it imply (5)? It is a tricky one even though I understood your explanation well. If so, would you elaborate on why it implies (5)? – Yaman H Nov 21 '17 at 20:37
• Together, (1) and (6) indeed imply (5); here’s how: Suppose P; then (1) yields Q and (6) yields not-Q – a contradiction! Thus, P cannot be true, whence not-P = (5) must be true. – MarkOxford Nov 21 '17 at 22:59

The all important phrase here is if and only if, which would make the conditional bidirectional, where you could affirm/deny the antecedent/consequent and thereby confirm/falsify the consequent/antecedent.

In unidirectional conditionals, such as the one in the example, there is uncertainty on its converse, hence denying the antecedent or affirming the consequent is invalid.

So, given that : "If weather is good, we will go on a picnic", saying "if the weather isn't good..." or "We went on a picnic.." isn't useful as it will always be false, if followed by the consequent/antecedent or their converse.