What is wrong with my intuition of Modus Tollens?

Question

Abbreviate Modus Tollens to MT, Necessary Condition to NC, and Sufficient Condition to SC.
I pursue only intuition; please do not answer with formal proofs or Truth Tables. I already know of the fallacies of Affirming the Consequent and Denying the Antecedent.
_{Source: p 335, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley.
CAUTION: I changed the textbook's content marginally.}

4. ¬s ← ¬n    (I use ← instead of → to preserve the order of the letters)

I am not convinced by the above intuitive explanation. Are there better ones? Please ameliorate this post and tell me if you diagnose my anxiety (which I struggle to describe), but here is my try:

SCs may not be NCs; so a chasm exists between necessity and sufficiency.
How does this chasm fail to defeat MT?

It is counterintuitive that n is a NC in 1, but that ¬n is a SC in 4. How does negation and MT throw a NC (n) over the NC/SC chasm and then convert ¬NC (¬n) into a SC?

Answer 1

In the passage from s → n to ¬s ← ¬n the term n does not jump over the chasm between necessity and sufficiency, ¬n does. [This preceding paragraph answers an older version of the OP.]

It is the negation that does the trick. Think of a "condition" as a restriction on the class of things that satisfy it, the stronger the restriction the narrower the class. Normally necessary conditions are weaker than sufficient conditions, but negation always reverses the order of strength: the negation of a weaker condition is always stronger than the negation of a stronger condition. For example, being a solid is a weaker restriction than being a crystal, but being not a solid is a stronger restriction than being not a crystal (eg: 'wood' is not a crystal, but wood is a solid.) And this is the essence of MT: accepting MT amounts to getting directly from 1 to 4. MT then follows by applying modus ponens to 4.

I don't know if this will help or confuse you further, but modus tollens is accepted even by intuitionists, who have stricter demands on validity of arguments. They interpret logic in terms of proofs and reason as follows: s → n means that any proof of s can be converted into a proof of n, ¬n means that we are given a proof of ¬n. Thus, if we were given s we would acquire proofs of both n and ¬n, and we can not have that. It must be ¬s. If this resonates you may look at Contraposition in intuitionistic logic? on Math SE, if not disregard.

Answer 2

Here is an alternative wording which may help.

If we know that something is false, then anything that implies its truth cannot itself be true.

Schematically :

¬n ( if we know that n is false )

s ⇒ n ( and if we know that something implies its truth - i.e., s )

---

¬s ( we know s cannot be true )

It's not a million miles from the given description. It is obtained by changing the order of the hypotheses.

As jobermark points out in his comment to your question, the additional statement (statement 4) is logically equivalent to statement 1 and does not require deduction.

EDIT

Rereading your question this morning, I realise that I have not addressed the second part of your question.

Note that n is not an SC in statement 4. It is ¬n that is an SC in statement 4. So when n crosses the chasm from NC to SC, it is subject to change by negation.

Answer 3

Maybe the text you quote is a bit misleading in that it uses counterfactuals ("if we had f...") which we could be tempted to interpret in terms of possibility and necessity as you do (since you talk about necessary conditions). However the right interpretation to have of the text is not in terms of possibilities but of hypothesis: "if we had f" means "let us make the hypothesis that f is true to show a contradiction".

I can't be sure but I suspect that you're not convinced by the explanation because you interpret it in terms of necessity and possibility, not as a demonstration by contradiction.

I don't think one should talk about necessary and sufficient conditions here because first order logic is only about what is actually the case or not. For necessity statements, one should use modal logic, not first order logic. "f->w" does not say that w is necessarily the case when f is true, it only says that it's not actually the case that f is true and w is false. Talking about necessary or sufficient conditions only brings confusion.

Modus tollens can be explained intuitively as follows: if it's not the case that f is true and w is false, yet w is false, then it's not the case that f is true.

Having said that, it is true that an analog of modus tollens transforms necessary conditions into sufficient conditions.

"w is a necessary condition for f" means that w is required for f to occur although it might not be sufficient.

As an example, imagine you need a passport to enter a country. It is necessary but not sufficient (you must not be blacklisted, you must not carry food, etc...).

If w is necessary for f, then it suffices that w is false for f to be false. In the example: not having a passport is a sufficient condition to be rejected at the frontier.

So something like a modus tollens indeed transforms the negation of necessary conditions into sufficient conditions and thus this is not the right way to flesh out your intuition about what could be wrong with the text.

Answer 4

You reach the compact expression of Modus Tollens:

(s -> n) => (~s <- ~n)

In a modern sense, it is just the application de Morgan's laws to the 'long form' of implication (ie Material Implication: p → q  ≡  ¬p ∨ q). So you don't officially need a separate modus for negative deductions. Tradition just has a name for this form.

In terms of sufficiency and necessity. The opposite of a sufficient condition is a necessary one, and vice versus. If S suffices to prove N then the absence of S is required for N to be false. If S is required for N the absence of S suffices to disprove N.

In modus tollens, we are given 1) s suffices to indicate n. So we can say the absence of s is necessary for n to be false. We are also given 2) that n is false, so 3) s needs to be false as well.

(We don't bridge this chasm, we are given a negated premise, so we start from the opposite cliff.)

Answer 5

1. p --> q
2. ~q
   Ergo,
3. ~p

p is sufficent for q or ... conversely, q is necessary for p. Ergo, if ~q, we can, for certain, rule out p, which is to say, ~p.

Causally (the difference is subtle), p is always followed by q. So, ~q implies ~p.

Answer 6

SCs may not be NCs; so a chasm exists between necessity and sufficiency. How does this chasm fail to defeat MT?

It is counterintuitive that n is a NC in 1, but that ¬n is a SC in 4. How does negation and MT throw a NC (n) over the NC/SC chasm and then convert ¬NC (¬n) into a SC?

The notions of sufficient condition and necessary condition are relative to the position of the term in the conditional (or the implication). In the implication A → B, A is the sufficient condition by virtue of being the first term of the implication, and it is sufficient relatively to the truth of B. Thus, if the implication A → B is true, then A is sufficient just because it is sufficient that A be true for B to be true. Similarly, B is the necessary condition because it is the second term of the implication A → B, and necessary relatively to A; B is said to be the necessary condition because it is necessarily true that B is true if A is to be true (or, if B is false, A is not true).

Nota: This only applies to the logical implication A → B and to the conditional "If A, then B". It does not apply to the horseshoe A ⊃ B, which is just ¬A ∨ B, and therefore not A → B.