How can you rewrite without any conditionals 'If A then B; A; therefore B' ?

Question

_{Source: p. 150 Top. Informal Logic: A Pragmatic Approach (2 ed 2008) by Douglas Walton. I edited the argument forms.}

However, modus ponens is not the only form that the argument [in green, on the left] has. It also has this less specific form [in the red, on the right]:

Instead of representing [1.] as a conditional, we could also represent it as a simple proposition, A. Of course, representing it as a conditional would be more specific, but if we did represent it in the less specific form above, that would break no rule of logic we have, so far, required. And that form of argument is invalid. Even if both A and B are true, it is quite possible that C could be false, for all logic tells us.

The red argument purposes to rewrite green without any conditional statement, per the following:

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.

But how does the green argument logically equal the red? One problem is the arguments' difference in conclusions: 3 does not equal 6.

Answer 1

But this is not what the author says.

The example rewrites the original argument form, from modus ponens into:

the less specific form: A; B; therefore C.

Instead of representing the first premise as a conditional, we could also represent it as a simple proposition, A. [...] And that form of argument is invalid [emphasis added]. Even if both A and B are true, it is quite possible that C could be false, for all logic tells us.

And the conclusion is :

So we have to be careful here. Even if we know an argument has an invalid form, it need not automatically follow that the argument must be invalid.

The author is not asserting that: "A and B; therefore C" is "equal" (equivalent) to: "if A, then B, and A; therefore B".

He is asserting that an "incorrect" formalization of an argument can "hide" its validity.

Answer 2

The argument in red cannot be deduced validly from the argument in green. All they have in common is that they're both arguments with two premises.

To answer the question in the title, every conditional sentence (i.e. any hypothetical proposition) has a logically equivalent disjunction wherein the protasis is a negated disjunct and the apodosis is not a negated disjunct.

"If P then Q" means "Q or not-P". In conventional notation of symbolic logic, P→Q is a hypothetical proposition that is logically equivalent to the disjunction Q∨¬P. Therefore, modus ponens has the following form:

• P
• Q∨¬P
• ∴Q

Answer 3

note that your reference is about informal logic. which is arguably an oxymoron. as other answers have pointed out, the red "argument" is not the logical equivalent of the green one. the greenie is a syllogism; the reedie is an enthymeme, a rhetorical device that approximates a syllogism, but omits a premise. In the red argument, the missing premise is precisely implicational form that would justify the conclusion. in this case, that could be sth like "if A and B, then C". but it could also be "if A or B, then C".