Post by Eugene 2.0 on Dec 1, 2021 15:44:23 GMT
• An implication
can be translated into "If ..., then ...". For example, "If a coffee is cold, it tastes badly".
• Any implication has two sides: a head one (before the arrow) named "antecedent", and a tail one (after the arrow) named "consequent".
• Usually an implication means the next:
• Quite weirdly to have any precious and necessary forms of it, except for math and science. And even there the implication isn't so clear mostly. Let's exemplify it:
• In he first example of (a), we cannot claim this for certain: that a person should eat frogs, if the person appeared to be a Frenchman. The same cannot be said rigorously about (b), because it might happen, during the rain a single drop hasn't reached the ground (i.e. if the ground was covered with a carpet, etc).
• This isn't so good about next two examples of (c) and (d). We might be wrong about a certain electron (no scientists can check all the electrons), and even further, this cannot be said about the math.
• Indeed, what if those "twoes" could happen to be identical. In this case, the addition would be equal to two, not four. The same is fare if only one element among those twoes is shared by both; this time the sum will be equal to three.
• So, by the weak implication I propose the next one taken from the modal logic:
• We can change (i) to the form of (ii) finally having got:
• If our knowledge is so limited, and we want to be sure, we should prefer (iii) to (i).
• Also, considering a shared in any modal logic axiom M2
we can continue to modify the (iii) further into:
• Then, assuming (iii) is being necessary, using modus ponens for (iv), we've got:
• By definitions of modalities the (v) can be transformed into:
• And via conversion, and deleting two successive negations, it results in:
• Finally changing q to ~p, and p to ~q, while using the double negations removing:
• Let's look at the formula (viii). If modal rule M2 works (most logicians would agree), then (iii) can have ◇□p into its antecedent. This means, then (iii) is being proved by the laws of modal logic. And indeed, if we take a certain implication proposition, then there's no certainty in it, and we have to doubt whether or not something can be implied with necessity from it.
p → q (i)
can be translated into "If ..., then ...". For example, "If a coffee is cold, it tastes badly".
• Any implication has two sides: a head one (before the arrow) named "antecedent", and a tail one (after the arrow) named "consequent".
• Usually an implication means the next:
The implication proposition is true, if and only if its antecedent is false, or its consequent is true
a) if he's a Frenchman, he eats frogs
b) if it's raining, the ground is wet
c) if an electron has been charged positively, it achieves a piece of energy
c) if two adds two, then it equals to four
• In he first example of (a), we cannot claim this for certain: that a person should eat frogs, if the person appeared to be a Frenchman. The same cannot be said rigorously about (b), because it might happen, during the rain a single drop hasn't reached the ground (i.e. if the ground was covered with a carpet, etc).
• This isn't so good about next two examples of (c) and (d). We might be wrong about a certain electron (no scientists can check all the electrons), and even further, this cannot be said about the math.
• Indeed, what if those "twoes" could happen to be identical. In this case, the addition would be equal to two, not four. The same is fare if only one element among those twoes is shared by both; this time the sum will be equal to three.
• So, by the weak implication I propose the next one taken from the modal logic:
p → ◇p (ii)
• We can change (i) to the form of (ii) finally having got:
p → ◇q (iii)
• If our knowledge is so limited, and we want to be sure, we should prefer (iii) to (i).
• Also, considering a shared in any modal logic axiom M2
□(p → q) → (□p → □q) (iv)
we can continue to modify the (iii) further into:
□(p → ◇q) → (□p → □◇q) (iv)
• Then, assuming (iii) is being necessary, using modus ponens for (iv), we've got:
□p → □◇q (v)
• By definitions of modalities the (v) can be transformed into:
~◇~p → ~◇~~□~q (vi)
• And via conversion, and deleting two successive negations, it results in:
◇□~q → ◇~p (vii)
• Finally changing q to ~p, and p to ~q, while using the double negations removing:
◇□p → ◇q (viii)
• Let's look at the formula (viii). If modal rule M2 works (most logicians would agree), then (iii) can have ◇□p into its antecedent. This means, then (iii) is being proved by the laws of modal logic. And indeed, if we take a certain implication proposition, then there's no certainty in it, and we have to doubt whether or not something can be implied with necessity from it.