Logic - negation (1 Viewer)

mathsbrain

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Just wonering if we want to find the negation of an implication, is it wrong to add in the word "IF", and if so why?
For example, the negation of "If A, then B" should be "A and not B", but why can't we say "IF A and not B" or are these both correct?
 

Drongoski

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Truth Table

p q p => q Not(p => q)

T T T F

T F F T

F T T F

F F T F
 

fan96

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Just wonering if we want to find the negation of an implication, is it wrong to add in the word "IF", and if so why?
For example, the negation of "If A, then B" should be "A and not B", but why can't we say "IF A and not B" or are these both correct?
The reason that the negation of "if A then B" is "A and not B" is because of the following.

"If A then B" is logically equivalent to "not A or B", i.e. "either A is not true or B is true".
(You can prove this with a truth table or just reason it out.)

In particular, the latter form is not an implication, it is another logical statement that is equivalent to the implication. Basically we're converting the implication into another form (that's easier to work with) in order to be able to negate it.

Anyway, taking the negation of this leads to "A and (not B)", i.e. "A is true and B is not true".
So that's why this is a correct negation.

"If A and not B" is an incomplete statement. The word "if" means we're talking about an implication. An implication is of the form "if ( ... ) then ( ... )". So you would need to complete this somehow. "If (A and not B) then ( ... )."
It's not that it's "wrong to add the word if", it's that you need to find some way to do it that actually gives a correct negation. Essentially, you want to express the negation of an implication as another implication. And this isn't really possible, so we just stick with what we had before.
 

Drongoski

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You put up the post. I was surprised. I did not think it is in 4U. Have to check the new syllabus again.
 

Drongoski

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So do the new Maths Ext 2 textbooks cover Logic concepts like? - proposition, T and F, disjunction, conjunction, negation, exclusive disjunction, Truth Table, logical equivalence, implication, equivalence, converse, inverse, contrapositive, tautology, contradiction? If not, it does not include a study of Logic.

You don't need to study Logic to apply the ideas of implication, converse, negation and contrapositive in proofs. In fact many complete their Maths degrees without studying Formal Logic.

But, interestingly, Logic is part of the lowest level IB Diploma maths subject: Mathematical Studies (SL), but not in the higher level subjects Mathematics (SL) and Mathematics (HL)!
 
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lolzdj

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So do the new Maths Ext 2 textbooks cover Logic concepts like? - proposition, T and F, disjunction, conjunction, negation, exclusive disjunction, Truth Table, logical equivalence, implication, equivalence, converse, inverse, contrapositive, totaulogy, contradiction? If not, it does not include a study of Logic.

You don't need to study Logic to apply the ideas of implication, converse, negation and contrapositive in proofs. In fact many complete their Maths degrees without studying Formal Logic.

But, interestingly, Logic is part of the lowest level IB Diploma maths subject: Mathematical Studies (SL), but not in the higher level subjects Mathematics (SL) and Mathematics (HL)!
Terry Lee does not cover negation and the like. Both Fitzpatrick and Cambridge do - although Cambridge does delve into set theory and venn diagrams and stuff. Not sure about the others.
 

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