The Two Ways We Argue

two ways we argue
Each of us in our daily lives hears a lot of arguments, and all of them are different. But, in one respect, there are only two basic ways to argue. We might call these two kinds of argument “arguing forward” and “arguing backward.”

The First Way: Modus Ponens

Arguing forward involves beginning with a principle or a universal truth and inferring a conclusion that is its consequence. This kind of argumentation is characteristic of mathematics and philosophy.

Every square is a four-sided figure
A chess board is a square
Therefore, a chess board is a four-sided figure

The universal truth here is the fact that every square is a four-sided figure, and the conclusion is the specific statement that a chess board is a four-sided figure.

But this kind of argument can be used in politics and other areas as well. Abraham Lincoln is a good example of someone who argued forward. On the issue of slavery, Lincoln began with the assumption that no human being should be enslaved. That was his broad, general principle. He then employed the specific assertion that African Americans were human beings, from which it followed that no African-American person should be enslaved.

No human being should be enslaved
Every African-American person is a human being
Therefore, no African-American person should be enslaved


If no human being should be enslaved, then African Americans should not be enslaved
No human being should be enslaved
Therefore, African Americans should not be enslaved

This argument affirms that “no human being should be enslaved,” and concludes that African Americans should not be enslaved. This kind of argumentation is what is used in geometrical proofs. In fact, along with the King James Bible and the plays of Shakespeare, Lincoln carried around with him a copy of the Elements, Euclid’s ancient treatise on geometry. He even appealed to Euclid’s principle of geometrical equality to bolster the idea of human equality in his case against slavery.

Arguing forward involves starting with a broad, universal truth and deducing a specific conclusion. This kind of argumentation can take the form of deduction if it is in the form of a categorical syllogism (like the square example and the first slavery example above):

All M is P
All S is M
Therefore, all S is P

The movement forward in the argument can also take the form of what is called Modus Ponens in hypothetical reasoning (like the second slavery example above):

If P, then Q
Therefore, Q

It starts by affirming the antecedent (P) and ends affirming the consequent (Q). It affirms forward.

The Second Way: Modus Tollens

Arguing backward also involves deduction, but employs it in a different way. In the process of arguing backward, we make an inference, and then, rather than arguing forward to a positive conclusion, we argue backward to a negative conclusion:

Every animal that is kosher chews the cud and is cloven-hoofed
A pig does not chew the cud nor is it cloven-hoofed
Therefore, a pig is not kosher


If the material is copper, then it will conduct electricity
This material does not conduct electricity
Therefore, this material cannot be copper

This way of arguing starts by affirming the consequent (Q) and ends denying the antecedent (P). This kind of argumentation is particularly suited to a consequentialist argument; that is, an argument against some idea or policy on the basis of the bad consequences that are likely to ensue.

For example, at the time of this article’s publication, the debate on health care is raging. The argument made against the Affordable Care Act (“Obamacare”) by many conservatives is that if we don’t repeal and replace it, premiums will continue to skyrocket and insurers will continue to leave the market. And since we want neither of these things, we should be opposed to Obamacare.

But supporters also argue backward when they argue that, if we repeal the ACA, then more people will be uninsured. And since we don’t want people uninsured, we should not repeal the ACA.

Both of these example arguments take the same form, which is called Modus Tollens:

If P, then Q
Not Q
Therefore, not P

Modus Ponens argues positively forward, while Modus Tollens argues negatively backward. Both are legitimate and effective ways to argue.

logic sequence

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