Propositional logic Natural Deduction

The Rules of Implication of Addition says:

p Therefore: p or q

This rule doesn't make sense, and my book doesn't explain it. Rather, it just expects me to accept it and to continue to the exercises,

Is there any explanation why it is so?

• Comments are not for extended discussion; this conversation has been moved to chat. Jul 9 at 8:19
• @causative what do you mean by 'grant him' as i understand it just mean to start from a false proposition, i don't think there is a special meaning to the expression grant him in this manner. But although the conclusion is valid it is an unsound because propositional logic is a truth preserving system as i understand from bumble answer, but i have found a way arounds it, we can prove that the sun is cold by using quantum mechanics and the electron position. Jul 9 at 8:20

The rule of addition is correct in classical logic, which is presumably what you are learning. Classical logic is usually understood as the logic that is truth-preserving, i.e. for a valid argument it requires that if the premises of an argument are true then the conclusion follows by necessity, or that it is impossible for the premises to be true and the conclusion false. In practice, this can be quite a weak criterion. It does not require that the premises are relevant to the conclusion at all, and it allows for the principle of explosion, which is that from a contradiction anything follows, and also the principle of implosion, which is that a logical truth follows from any premises whatever.

In simple cases this need not seem odd. If I have a coin in my left pocket, it follows that I have a coin in my left pocket or a coin in my right pocket. But it can definitely seem odd in cases where the introduced disjunction is completely irrelevant. It's just a feature of classical logic. The important thing about it is that it is a truth-preserving move. It will never be the case that A is true and "A or B" is false.

There are other logics that impose different conditions on what qualifies as a valid argument, or which can be understood as preserving something other than just truth. There is a family of logics that go by the name 'relevance logic' that either lack the addition rule, or impose restrictions on it, precisely to block the move from A to "A or B", and thence to "if not A then B". Some of these distinguish two kinds of disjunction, an extensional one that allows the move from A to "A or B", and a separate intensional one that is required to go from "A or B" to "if not A then B".

• My last question, as far as you know is it ok to apply the addition rule to a false proposition because this example confused me a bit, (in my opinion it shouldn't be allowed) cse.buffalo.edu/~rapaport/191/addition.html Jul 9 at 3:11
• I mean to this part: " 2. (1+1=3) ∨ "I, Bertrand Russell, am the Pope " Jul 9 at 3:14
• The addition rule simply allows you to go from A to "A or B". It is not concerned with the whether A, B are true or false. The whole point of it being a rule is that it works any which way. The example quoted in your referenced page is an example of the principle of explosion. It takes as a premise "1+1=3", which is false, and also uses ¬(1+1=3), which contradicts it, and from this anything follows. Jul 9 at 3:18
• Ok, that mean i need to learn about those principles. Thank you,frome here i can manage on my own. Jul 9 at 3:22
• @bumble I learnt (from another comment of yours?) the term "hyperintensionality". The problem with classical logic is that it breaks off relevance as a separable concern. Since relevance is a desideratum in informal reasoning this produces the most bizarre effects in baby-examples! Assuming extension intention and hyperintension are relevant to the issue of relevance and so to this q, it would be great if you could put in something about that in your answer! Jul 9 at 7:47

In formal logic, OR is an inclusive or -- it is true when either of the propositions is true.

This is logic, not sense. Normal discussions do not contain statements such as are routinely used in logic, sentences having no connection except formal truth value.