## A Study On Modus Ponens

• 11.9k
In natural deduction, one valid argument form is known as modus ponens.

Modus Ponens
1. If P then Q
2. P
Ergo,
3. Q

1a. If it rains then the ground will be wet
2a. It rains
Ergo,
3a. The ground will be wet

If the premises 1 and 2 are true, it's impossible for the conclusion, 3, to be false.

Associated fallacies

Affirming the consequent fallacy
4. If P then Q
5. Q
Ergo ???
6. P

4a. If it rains then the ground will be wet
5a. The ground is wet
Ergo ???
6a. It rained (leaking pipes could make the ground wet)

The above argument form is considered invalid (a truth table will show you why). Anyway, the problem with this fallacy is that there could be more than one way Q could be true i.e. there's no necessity that P be true if Q is. So far so good.

However, notice something important here: If Q is true, that P is false doesn't follow i.e. P could be true. In other words, though affirming the consequent is considered a fallacy, P can be true if Q is true.

Affirming the consequent (valid version)

7. If P then Q
8. Q
Ergo,
9. It could be P

7a. If it rains then the ground will be wet
8a. The ground is wet
Ergo,
9a. It could have rained

Denying the antecedent fallacy

10. If P then Q
11. Not P
Ergo ???
12. Not Q

10a. If it rains then ground will be wet
11a. It didn't rain
Ergo ???
12a. The ground won't be wet (leaking pipes could wet the ground)

There's no point discussing the fallacy itself for it's common knowledge. However, pay attention to the fact that if Not P, though it doesn't follow that Not Q, it's possible that Not Q; after all one sufficient condition for Q, P is not the case.

Denying the antecedent (valid version)

13. If P then Q
14. Not P
Ergo,
15. It could be not Q

13a. If it rains the ground will be wet
14a. It didn't rain
Ergo,
15a. The ground could be dry (not wet)
• 984
All mortals are men. Felicia (Alice's sister) is mortal. Therefore, she 's a man.
• 944
Some Felicia is man
My cat is called Felicia
Therefore my cat could be a man.
• 539
A causes X
B causes X
C causes X...

X

Ergo

A and/or B and/or C,...
• 9k
Modus Ponens
1. If P then Q
2. P
Ergo,
3. Q

1a. If it rains then the ground will be wet
2a. It rains
Ergo,
3a. The ground will be wet

If the premises 1 and 2 are true, it's impossible for the conclusion, 3, to be false.

What makes it impossible that 3 is false? It's not one of the three fundamental laws, identity, non-contradiction, or excluded middle. And I don't believe it's a combination of the three, it's a completely different principle.

Can we call this a principle of logical priority? We say that Q is logically prior to P. The ground being wet is logically prior to it raining, as raining necessitates logically, the ground being wet, as a logical requirement. Notice that in this case there is a reversal of ontological, or temporal causation. If "P causes Q", is an ontological determination, through a necessary temporal relation which implies P as temporally prior to Q, then Q is logically prior to P, as the determination of cause and effect is what validates the inverse logical priority.

Cause and effect is just one example, as there are many other ontological principles which validate a logical priority.
• 11.9k
What makes it impossible that 3 is false?

1. If P then Q
2. P
Ergo,
3. Q

3 has to be true; no possible world exists where 1 and 2 are true with 4 false.

As for temporal aspects of sufficient and necessary conditions and causality, we can forgo discussion on them for they muddy the waters.
• 158
Validity means that the premises cannot be true and the consequent false. To test validity we assume all premises are true and see if there is a spot on the truth table where the conclusion is false.

p | q | conditional
—————————
T | T | T
T | F | F
F | T | T
F | F | T

Is the truth table for the conditional as traditionally defined. Note that the conditional can be true in three cases and false in only one.

The conditional is always true when the consequent (Q) is true, meaning that knowing that the conditional is true and Q is true tells you no information about P, i.e. p then q,q therefore (p or ~p). These are entries one and three on the table and correspond to the fallacy of affirming the consequent.

The conditional is always true when the antecedent (P) is false, meaning that knowing that conditional is true and P is false tells you no information about Q, i.e. p then q, ~p therefore (q or ~q). Conditions 3 and 4 therefore correspond to the fallacy of denying the antecedent.

The conditional can be true or false when the antecedent (P) is true. Condition one is the only case where the premise is true and the conditional is true. The only truth value for Q in this situation is T. This entry corresponds to the valid inference of Modus Ponens.

The conditional can be true or false when the predecessor (Q) is false. Condition four is the only case where the conditional is true and the antecedent false. The only truth value for P in this situation is F. This entry corresponds to the valid inference of modus tollens,

Condition 2 is the only entry where the conditional is false. In a valid argument all premises must be true, so condition 2 cannot be the premise of a valid argument, i.e. P cannot be true and Q false in an argument where P then Q is a premise (true). This means that knowing that the conditional is true, it must be the case that either P is false (since P being true can lead to a false conditional) or Q must be true (which always leads to a true conditional), i.e. ~P or Q, which corresponds to the valid inference of material implication.

Validity and the rules of inference are established definitionally. There is no “logical” relationship between the parts other than what is defined.

This lack of relationship between the individual statements, e.g. P, and their presence in a valid argument leads to odd semantic outcomes and the formal rules of logic seem to stop functioning as useful tools of thought. The conditional in particular has been criticized for leading to absurd results do to its use in statements of cause and effect and identity and people have tried to rehabilitate it with ideas such as entailment and necessity.

Introducing the concept of “could” to replace the logically defined concept of “or” in valid proofs is a semantic shift that leads to confusion rather than clarity. If this concept results in some philosophical question for you, it strikes as an invention of your word choice. Perhaps you can expand a bit on what issue with the conditional/modus ponens you are having. The formal logic does not seem problematic so far as it goes.
• 158
Or maybe somewhat differently, do you want to talk about modal logic but used non-modal references for the sake of inclusivity?
• 9k
3 has to be true; no possible world exists where 1 and 2 are true with 4 false.

Well that's not telling me what makes it true. You are just repeating that it has to be true, necessarily. What makes 3 true, when 2 is stipulated as true, is the relationship between P and Q which is stated in 1. The truth or falsity of 2, in your example of rain, is a direct empirical determination The truth or falsity of 1 is a logical determination. So we can ask what type of logic supports such premises.

As for temporal aspects of sufficient and necessary conditions and causality, we can forgo discussion on them for they muddy the waters.

If you're just going to repeat over and over again, that 3 is necessarily true, if 1 and 2 are true, then there is no discussion to be had.
• 11.9k

All I can say is that we're discussing validity, about the form of an argument and inasmuch as that matters, the conclusion follows from the premises.
• 9k

The form of premise 1, as a conditional statement, is crucial to the validity of the conclusion, as what is used to determine the truth table. For example, if the premise was changed to a biconditional, the truth table would be different.
• 11.9k
The form of premise 1, as a conditional statement, is crucial to the validity of the conclusion, as what is used to determine the truth table. For example, if the premise was changed to a biconditional, the truth table would be different.

Agreed!
• 9k

I'm glad you agree. Now consider that there are a number of different forms of these crucial statements. I would say that they are definitional. So in the famous example from Aristotle: All men are mortal; Socrates is a man; Therefore Socrates is mortal, you can see clearly that "man" is defined as mortal, and "Socrates" is defined as a man. The way A explains it, the concept of "mortal" is within the concept of "man". I say "within" means a necessary part of the definition, "necessary" being relative to that specific logical proceeding.

The premise in your example is slightly different but similar. "Rain" is defined by "the ground will be wet" as necessary for the logical procedure. So you place the concept of "the ground will be wet" within the concept of "rain", as necessary. It is what I would call definitional.

I would say that the principal difference between the two examples above, is that the former defines a type of object (making that the subject) "men", through predication, "are mortal", while the latter defines an activity "rain" (making that the subject) through a causal relation "the ground will be wet". You could have stated something different as your definition, like "If it's raining there will be drops of water in the air". This is more like a simple predication. We replace the causal relation with a straight relation of predication, but the situation is the same, logically. Every time it is raining, there are drops of water in the air, but not necessarily every time there are drops of water in the air is it raining. The concept of "drops of water in the air" is placed within the concept of "rain" through this definitional act. And the definitional act enables the judgement of validity, relative to the conclusion.
• 158
I say "within" means a necessary part of the definition, "necessary" being relative to that specific logical proceeding.

For those following at home: Logical necessity/entailment. also a bit of relevance logic.

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?" — Wiki

. . .Relevance logicians have attempted to construct logics that reject theses and arguments that commit “fallacies of relevance”. Relevant logicians point out that what is wrong with some of the paradoxes (and fallacies) is that the antecedents and consequents (or premises and conclusions) are on completely different topics. . . . — SEP

And the ideas of semantic vs. syntactic consequence.

Syntactic consequence does not depend on any interpretation of the formal system.
. . .
A formula A is a semantic consequence of a group of premises G where the set of the interpretations that make all members of G true is a subset of the set of the interpretations that make A true.
— Wiki with some liberties
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