Etydhv

Is it nonsense to say B → [A → B]? I would have thought so, it seems to say that every fact deductively follows from every other. But I was looking at a recent closed question, on natural deduction, and my thinking about it (not as natural deduction) got be wondering.

It seems to me that B with the law of excluded middle and disjunction introduction and conjunction introduction ⇒ [[~A ∧ B] v [A ∧ B]] which with law of non contradiction and disjunctive syllogism ⇒ [A → [A ∧ B]] which with conjunction elimination ⇒ [A → B].

B → [A → B]

Sorry for being so naive, but, assuming I can be understood here, what have I worked out wrong?

When proving a conditional one assumes the antecedent, B. The goal is not to derive this, but from this assumption to derive the consequent which happens to be A > B.

But A > B is another conditional. Since it is a conditional one derives that in the same way. First assume the antecedent, A. Can one derive B? Yes, one can, because in this derivation we already assumed B in the first line. We can "reiterate" it in the next line as B.

Since from A one can derive B in a subproof, one can introduce a conditional (→I), A > B. This is the same as saying: if I have A, I can derive B. It is a shorthand for that subproof.

But if one has A > B one can introduce the conditional again and write B > [A > B].

Here's the result using the proof checker linked to below:

What this shows is that the argument is valid. The argument might be nonsense depending on what A and B are. Suppose A is Unicorns are white and B is The sky is blue. What this says is If the sky is blue then if unicorns are white then the sky is blue. Suppose the sky is blue. It doesn't matter whether unicorns are white or not. The sky is still blue.

For the reiteration rule (R) see section 16.1 of the forallx text below.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

The topic you want to research is 'paradoxes of material implication.' That is, you are right to think there's something odd about this formula. But it does not mean that every fact deductively follows from every other. Consider your target string:

B-->(A-->B)

Now note that the convention is to define the connectives such that the conditional 'P-->Q' is logically equivalent to '~PvQ'. Hence, replacing each instance of the conditional with the disjunctive equivalent, your target string is logically equivalent to:

~B v (~AvB).

i.e., Either not-B, or else not-A or B. On this formulation, it is clear that this formula does not mean what you suggest it means. Furthermore, this disjunction is a trivial logical truth . There are two ways the world could be with respect to B: either B is true or it is not true. If B is true, this formula is satisfied by the second disjunct. If B is not true, this formula is satisfied by the first disjunct. Hence, it is logically true.

This formula is one among many reasons why philosophers have thought that the material conditional of first-order logic does not capture any interesting epistemically-rich notion of implication. When formulated as a conditional, this leads to valid implications that are counterintuitive, pragmatically odd, or prehaps even false. That is, we usually expect implications to carry some kind of epistemic or pragmatic force, to deliver some information beyond the initial terms. So carrying out this kind of inference, though valid, holds no water in ordinary discourse. Depending on your views about the relationship between semantics and pragmatics, then, there is actually an argument to be made that such "implications" are a kind of nonsense.

Monotonicity is another feature of logic that you may want to look into. Entailment in first-order logic is a monotonic operation, which in this setting means, intuitively, that once something B is proven, the addition of new assumptions will never make B false. One way to think about your formula is through the monotonicity of entailment. Suppose we have some string of background assumptions T, such that T entails B:

T |- B

The monotonicity of entailment ensures that the addition of new assumptions will never prohibit concluding B. Hence, the extension of T by arbitrary new assumption A will not alter the result:

T, A |- B

Many systems are intuitively non-monotonic, such as when we reason in a default way or when we assume that implications should have some relevance to one another. If you're in the United States, then you do default reasoning whenever you pay your taxes. For example, by default you owe the standard amount of taxes for your income level. But the addition of other assumptions --- such as itemized deductions, being the owner of a business, the head of a household, of retirement age, capital gains, student loan interest payments, etc. --- can make those default truths false. We also usually demand that the terms of an inference be relevant to one another. This is, at least arguably, one point of departure between traditional Aristotelian logic and modern mathematical logic. Aristotle's logic exhibits features of a relevance logic, i.e., the assumption that the premises in an argument must have some relevance to one another by virtue of which the terms of the premises can be combined or separated. This is one of the many ways in whcih Aristotle's understanding of syllogisms and deduction differs from our modern notions of validity. How to pin down 'relevance' is an entirely other can of worms. In a Wittgensteinian vein, we might also complain that someone who asserted a conditional like yours in conversation would be failing to make a move in the language game.

First-order logic validates all kinds of at best pragmatically useless and---again, depending on your background philosophical views---perhaps false entailments.

Consider:
Priest, Graham (2001). Introduction to Non-Classical Logic. Cambridge University Press, pp.12-13 (on material implication)

https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093883397

https://en.wikipedia.org/wiki/Monotonicity_of_entailment

https://plato.stanford.edu/entries/logic-relevance/

https://plato.stanford.edu/entries/logic-nonmonotonic/

I take a statement to be ridiculous iff it contains or implies a contradiction. That being said, B --> (A-->B) is not ridiculous, but it is a tautology, which makes it vacuous.

I feel the previous answers over complicated things, but here are two short arguments showing that it is a tautology.

1. B --> (A --> B) Assumption 
2. B --> (~A v B) 1, Implication
3. ~B v (~A v B) 2, Implication
4. ~B v (B v ~A) 3, Commutation
5. (~B v B) v ~A 4, Association

You get the same solution using the rule of Exportation

1. B --> (A --> B) Assumption
2. (B ^ A) --> B 1, Exportation
3. ~(B ^ A) v B 2, Implication
4. (~B v ~A) v B 3, De Morgan
5. (~A v ~B) v B 4, Commutation
6. ~A v (~B v B) 5, Association 
7. (~B v B) v ~A 6, Commutation 

~B v B is logically true. It is trivial to show that (~B v ~B) v ~A is also logically true.

(@Frank Hubeny Thanks for keeping me honest and reminding me to include the Commutation steps in both proofs).

It is a tautology.

•If B is true, then A->B is true. Then B->(A->B) is B->T. B->T is T.

•If B is false, then B->(A->B) is F->(A->B). F-> Something is always True. That's why B->(A->B) is true.