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# Material Conditional (logic)

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1. I'm studying a module on elementary logic atm (it's propositional and first order logic) and have just run into the material conditional, which completely threw me. Googling it I've learned that it's considered "logic's first surprise".

It's not that I don't understand it (it's just a case of memorising the truth functions) it's that i completely reject it's foundational semantics. Most of the controversy seems to come from the last two rows of it's truth table, but i even reject the first row.

For those who don't know, material conditional is an "if then" statement captured through truth tables like this:

A B /A --> B
T T/ T
T F/ F
F T/ T
F F/ T

with A and B just being any variables, the arrow being the conditional, T and F being true or false, and the second column defining the truth of the conditional from the truth of a and b.

I do not accept that A -->B (if..then) can be defined by ¬(A n ¬B) (not the case that A and not B). Clearly it is defined that way so as to be able to give it truth a truth value from the truth values of the propositional variables, however this strikes me as the wrong way to go about assigning a truth value to it. When you say "If I go to the shops it will be to get chicken", your criteria for that statement's truth is not whether you went to the shop and got some chicken (after all, getting chicken from the shops is not a formal proof that it always follows that when you go to the shop whenever the conditional is true, you must get chicken). Clearly, the the truth value of the conditional which these semantics (syntactics?) are trying to symbolise are not captured by the truth of the irrelevant propositions either side of it. Right...?

So it seems clear to me that the truth values of conditionals cannot be dependant on the truth values of propositions.

Another problem I have is that the assignments assume truth unless proven false (as it is on line 2). How can this be the case? How can if A then B be proven to be true by the fact that both are false? The justification i can think of is that if both ¬A and ¬B, then it is true that it is not the case that (A n ¬B). That is intuitive when we're thinking of (A-->B) as ¬(A n ¬B) but as previously explained I don't accept this as a proper symbolisation of what is required for a conditional argument to be true.

Can anyone help me accept this piece of logic? Or explain a reason why it needs to be truth table-able? Can't we find a better way of incorporating if then statements into logic then assigning it a truth table?
2. I posted this thread in the Philosophy help forum but that's pretty inactive and only really used by AS religious philosophy students so I doubt it will get a response there.

http://www.thestudentroom.co.uk/show...1#post63834361

It's not necessarily maths but it is maths related (propositional logic). I hope some of you have studied this and can make it intuitive for me. Cheers.
3. (Original post by banterboy)
I'm studying a module on elementary logic atm (it's propositional and first order logic) and have just run into the material conditional, which completely threw me. Googling it I've learned that it's considered "logic's first surprise".

It's not that I don't understand it (it's just a case of memorising the truth functions) it's that i completely reject it's foundational semantics. Most of the controversy seems to come from the last two rows of it's truth table, but i even reject the first row.

For those who don't know, material conditional is an "if then" statement captured through truth tables like this:

A B /A --> B
T T/ T
T F/ F
F T/ T
F F/ T

with A and B just being any variables, the arrow being the conditional, T and F being true or false, and the second column defining the truth of the conditional from the truth of a and b.

I do not accept that A -->B (if..then) can be defined by ¬(A n ¬B) (not the case that A and not B). Clearly it is defined that way so as to be able to give it truth a truth value from the truth values of the propositional variables, however this strikes me as the wrong way to go about assigning a truth value to it. When you say "If I go to the shops it will be to get chicken", your criteria for that statement's truth is not whether you went to the shop and got some chicken (after all, getting chicken from the shops is not a formal proof that it always follows that when you go to the shop whenever the conditional is true, you must get chicken). Clearly, the the truth value of the conditional which these semantics (syntactics?) are trying to symbolise are not captured by the truth of the irrelevant propositions either side of it. Right...?

So it seems clear to me that the truth values of conditionals cannot be dependant on the truth values of propositions.

Another problem I have is that the assignments assume truth unless proven false (as it is on line 2). How can this be the case? How can if A then B be proven to be true by the fact that both are false? The justification i can think of is that if both ¬A and ¬B, then it is true that it is not the case that (A n ¬B). That is intuitive when we're thinking of (A-->B) as ¬(A n ¬B) but as previously explained I don't accept this as a proper symbolisation of what is required for a conditional argument to be true.

Can anyone help me accept this piece of logic? Or explain a reason why it needs to be truth table-able? Can't we find a better way of incorporating if then statements into logic then assigning it a truth table?
If this is what I think it is (kinda stopped reading half way through), the reason False False is true is that the truth of both statements is equal, it matters not whether they are true or not. In the morning I will see if I brought my copy of Principia Mathematica Vol. 1 home and quote what it says on the matter (it's really early on so probably the right thing), but I might also be thinking of something slightly different.

I'll try to think of an example to demonstrate.
4. (Original post by Jammy Duel)
If this is what I think it is (kinda stopped reading half way through), the reason False False is true is that the truth of both statements is equal, it matters not whether they are true or not. In the morning I will see if I brought my copy of Principia Mathematica Vol. 1 home and quote what it says on the matter (it's really early on so probably the right thing), but I might also be thinking of something slightly different.

I'll try to think of an example to demonstrate.
The bottom line is what apparently most students don't like but I equally reject the top line.
5. (Original post by banterboy)
The bottom line is what apparently most students don't like but I equally reject the top line.
For the bit you state you reject below the table, do a bit of reading on De Morgan's Laws. Perhaps the easiest way to consider, and note that it's been a bit since I did the relevant stuff so I may not be perfect on this, what you are having an issue with is the following statement "'if A then B' is logically equivalent to 'not A, or B'". Let us focus on the second half of the equivalence and break it down into the two options, namely "not A" and "B". Trivially, A being false trivially leads to the "not A" option, however, if A is true then B is true, hence "B". Not quite sure how to write it better than that and Principia is at uni and I am not going to try finding the passage I want in the dodgy archive copy I have found online.

Having just checked something I think I might be thinking of the definition of logical equivalence, so I will get back to you tomorrow if the above didn't help so I have time to read and think about material conditional, although reading the Wikipedia page on material conditional it sounds like I am thinking of the right thing. If I am it's just one of those funny things you just need to accept as a seemingly illogical thing due to the semantics.
6. (Original post by banterboy)
...
I've merged both your threads into one and whilst I don't have time to waffle on about why the implies connective makes quite a bit of sense at the very moment, I'll refer you to this which is what I used to learn about the connective the first time around, he explains it and give motivation for each truth table value (even though it is quite long-winded, and you'll need to settle in for a long read) -- it's well worth it.

Hope that helps.

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