Conditional arguments

Conditional arguments are when we use a premise in an argument that says if one proposition is true, then another proposition must also be true. Here is an example of conditional reasoning:

• 1. If there is a fire, then the fire alarm will sound
• 2. There is a fire
• Therefore,
• 3. The fire alarm will sound

This argument is valid, because if it is true that if there is a fire, then the fire alarm will sound, and there is a fire, then the fire alarm will sound. Arguments in this form are always valid. In philosophy, the term given to this argument is Modus Ponens. The Modus Ponens logical form is this:

• If p then q
• P
• Therefore,
• Q

Now lets consider the next argument regarding fire alarms:

• If there is a fire, then the fire alarm will sound
• The fire alarm has sound
• Therefore,
• There is a fire

This argument is invalid, because even though if both premises are true, the conclusion could be false. The fire alarm may have sounded due to a malfunction, or someone was cooking too close to the alarm, and so on. The logical form of these arguments are always invalid, which is called affirming the consequent. The logical form of affirming the consequent is this:

• If p then q
• Q
• Therefore,
• P

Now onto to the next style of the argument:

• If there is a fire, the fire alarm will sound
• The fire alarm has not sound
• Therefore,
• There is not a fire

This argument is valid, because if it is true that if there is a fire, then the fire alarm will sound, and the fire alarm has not sound, then there is not a fire. This is due to that the conditional premise asserts that the fire alarm sounding must occur if there is a fire. These arguments are always valid, which are called Modus Tollens. Modus Tollens arguments in logical form look like this:

• If p then q
• Not q
• Therefore,
• Not p

Our final conditional argument is as follows:

• If there is a fire, then the fire alarm will sound
• There is not a fire
• Therefore,
• The fire alarm will not sound

This argument is invalid. This is because when we use a conditional premise, the latter proposition is a consequence of the former, but not the other way around. Even though there is not a fire, once again, malfunctioning or cooking may have set it off. These arguments are always invalid, which are called denying the antecedent. Denying the antecedent in logical form looks like this:

• If p then q
• Not p
• Therefore,
• Not q

Of course, to get knowledge of the conclusion, we need to know if all the premises are true. Consider the conditional premise ‘if there is a fire, then the fire alarm will sound. For the premise to be true relies on the rules very similar to testing validity, because for this to be true, there cannot be any circumstance where there is a fire but the alarm does not sound. So, we can prove a conditional premise false by providing a counterexample. And there is such an counterexample: the batteries have gone flat. So, the premise is potentially false (I say potentially, because you may be able to rule out these counterexamples, but that would require further argument, which will be discussedin later tutorials. Therefore, all of the above arguments are unsound, even both of the valid versions, since we need to know that the premise is true, and we cannot.

Test yourself on the four conditonal arguments below and see whether they are modus ponens, modus tollens, affirming the consequent or denying the antecedent. Also, check to see if they are sound:

1. If this animal is a whale, then it is a mammal. It is a whale. So it must be a mammal
2. If you a gambler, then you will be broke. You’re broke. Thus, you must be a gambler.
3. If I have at least one million dollars, then I’m a millionaire. I don’t have at least one million dollars. Hence, I am not a millionaire.
4. If I vote today, then I’ll be making a difference. I didn’t vote today. Therefore, I’m not making a difference.

If you wish to ask any questions, seek clarification, raise some objections, or check how you went on the test questions, please write them in the comments section and I will try respond as soon as I can.

I highly recommend purchasing the book ‘Understanding arguments’ by Walter Sinnott-Armstrong and Robert Fogelin, which is available for purchase in the link below. If you do purchase the book via this link, you are helping support this webpage. Thank you.