I understand how, according to the accepted rules of logic, false things imply everything. What I do not understand is why the rules of logic are formulated in such a way as to permit this.
I am particularly puzzled as to why necessarily false statements (i.e. contradictions) are allowed to imply anything. Is not a contradiction completely meaningless, and therefore tantamount to gibberish? How, therefore, can it have any logical function?
Consider the following sentences:
1. If it is raining and not raining, then the moon is made of cheese.
2. If rain thinks hot memory squares, then the moon is made of cheese.
The first sentence is said to be true. Is the second sentence true also? If not, why not? 'It is raining and not raining' is no more meaningful than 'rain thinks hot memory squares'; both phrases are nonsense. (And if the second sentence is true, will any random sequence of words be OK?)
Why not just deny contradictions a truth-value? If they are regarded not as false but simply nonsensical, they cannot form part of any meaningful implication.
The sentence is true, the logical structure of the sentence has absolutely nothing to do with the meaning of it. Logic is a system similar to mathematics, in fact; it's more similar to mathematics than to natural language.
To better understand the If, then statement, understand that it can be built from or(v) and negation (-) thusly: p->q <=> -p v q since an 'if, then' is false only when the antecedent (p) is true, and the consequent (q) is false, so it is thought of from the standpoint of a promise which is kept in all other circumstances and not kept if you do what is asked of you and you don't get a reward.
So here denote A=go get me a chair; B=I'll give you a dollar. To set up the implication we have:
A -> B which in its English translation says If you get me a chair, then I will give you a dollar. Clearly this promise is not broken if you don't give him a chair, but he still gives you a dollar,which is the False implies True situation. The promise is ONLY broken if you give him a chair and he does not give you a dollar, so A ^ - B (A and Not B).
Remember that all of the logical laws (at least at the basic level we are talking about here) can be derived from disjunction (OR) and Negation (NOT). In actuality the reason that the english language has so little bearing on the Formal Language of logic, is that they have as much to do with each other as Mathematics and Shakespear. There is a whole separate branch of the study of Natural Language which is entirely different. The main applications of logic are in Computer Science and mathematics. Lingustics is not in the domain of logic, though there is some intersection.