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.

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  1. ¬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?

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    You're assuming that 1.1 is true, and at the same time you ask us to imagine that 1.1 is false. You are merely denying one of the premise. Although, involving necessity only brings confusion. If you want to reason on necessity you should rather use modal logic. First order logic is only about what is actually the case. Implication "->" does not assume any kind of relation (of necessity, causality...) between the antecedent and the consequent. It only says that either the antecedent is false or the consequent is true. – Quentin Ruyant Jan 4 '16 at 22:48
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    If "the pond is refilled with special fluid, different from water and created by scientists, in which F can exist" then you are directly denying that "Water is a Necessary Condition for Fish to exist" despite saying that you agree it is true. – Conifold Jan 5 '16 at 1:25
  • 4. is not a deduction here, right? Do you mean (f -> w) => (~f <- ~w). Because that is effectively the point. If you apply de Morgan's laws to implication, you don't officially need a separate modus for negative deductions. But tradition has a name for this form... – user9166 Jan 5 '16 at 4:42
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    @LePressentiment w did not jump from NC to SC. w is NC and its negation is SC. Example: you need a passport to enter the country (NC). If you don't have a passport you'll be rejected (SC). – Quentin Ruyant Jan 5 '16 at 10:29
  • w is the necessary condition, because if we assume f ⊃ w as true, when f is true also w must be (i.e. we cannot have f "without" w). – Mauro ALLEGRANZA Jan 5 '16 at 14:37

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.

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  • Thank you. 1. Can you pleas explain your use of 'condition' and 'restriction', in your 2nd paragraph? Are they synonyms or antonyms? 2. Are there clearer words than 'stronger' and 'weaker'? – Accounting Jan 5 '16 at 21:58
  • @LePressentiment Sorry, I added a sentence at the beginning of the paragraph to clarify. – Conifold Jan 6 '16 at 22:52

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.


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.

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  • +1. Thank you as always once again. I changed my OP thanks to your and others' correction that w is not an SC in statement 4. Please allow me to postpone acceptation to allow time for more (changes to) answers. – Accounting Jan 5 '16 at 21:46
  • Statement 4 is indeed logically equivalent but it's a different statement so yes, it does require a deduction... – Quentin Ruyant Jan 5 '16 at 22:40
  • @quen_tin It's hard to argue with that! It does indeed require deduction. I had intended to mean in the context of explaining modus tollens, though obviously it does not read that way. – Nick Jan 5 '16 at 23:49
  • @NickR oh I see sorry... – Quentin Ruyant Jan 5 '16 at 23:53

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.

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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.)

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  • Yes. I know that MT means 1 => 4, and not just 4 itself. I wrote 4 to combine 2 and 3 in one line . Please change my post if I failed to enunciate this. Does this answer your questions above? – Accounting Jan 5 '16 at 21:51
  • +1. Thanks. 1. You wrote 'de Morgan's laws', but need you this really? Did you mean only 'Material Implication' and commutativity: 1 ≡  ¬p ∨ q ≡ q ∨ ¬p ≡ 4 ? 2. What do you mean by 'start from the opposite cliff'? Sorry, I cannot behold your visual extension of my metaphor. – Accounting Jan 20 '16 at 19:04

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