Why/How does the definition of implication in mathematics work?

[jaja1990]

Basically, we don't know what the truth values for FT an FF (in implication) should be. We chose them to be true, but is there any basis for this choice?

[DrRocket]

Basically, we don't know what the truth values for FT an FF (in implication) should be. We chose them to be true, but is there any basis for this choice?

 

 

 

A B A-->B

 

T T T

T F F

F F T

F T T

[Shadow]

Try thinking about it this way. If you start with a true statement and deduce something from it, then you can only get another true statement (or, to put it another way, it is possible to start with a true premise and arrive at a true consequence, ie. the implication is true). On the other hand, if you start with a true statement, you cannot deduce something untrue from it (or to again put it another way, it is not possible to start with a true premise and arrive at a false consequence, ie. the implication is false). It is now obvious why FT and FF are both true; it is perfectly possible to start with something untrue and arrive at something that is also untrue, ie. "Gravity repels massive objects which means that apples fall up", and it is equally possible to start with a false premise and arrive at a true consequence, ie. "The derivative of |x| evaluated at the point zero is equal to zero and thus zero is a local extreme of |x|".

Edited February 24, 2012 by Shadow

[DrRocket]

Try thinking about it this way. If you start with a true statement and deduce something from it, then you can only get another true statement (or, to put it another way, it is possible to start with a true premise and arrive at a true consequence, ie. the implication is true). On the other hand, if you start with a true statement, you cannot deduce something untrue from it (or to again put it another way, it is not possible to start with a true premise and arrive at a false consequence, ie. the implication is false).

 

You have stated no "on the other hand". The second (compound)statement is a restatement of first statement followed by a statement of the contrapositive.

[MathIsCool]

[math](A\implies B):\iff (\neg A\vee B)[/math].

 

Implication [math]A\implies B[/math] is defined to be true iff A is false or B is true.

[Shadow]

You have stated no "on the other hand". The second (compound)statement is a restatement of first statement followed by a statement of the contrapositive.

 

I'm not sure I understand. What I did, or at least attempted to do, was write out all the situations that could arise when evaluating an implication and what it evaluates to, ie. T->T (=T), T->F (=F), F->T (=T), F->F (=T) and tried to give an intuitive explanation of why it is evaluated in this way. The second statement could be viewed as a restatement of the first, but that wasn't the point. The same goes for the contrapositive.

[DrRocket]

I'm not sure I understand. What I did, or at least attempted to do, was write out all the situations that could arise when evaluating an implication and what it evaluates to, ie. T->T (=T), T->F (=F), F->T (=T), F->F (=T) and tried to give an intuitive explanation of why it is evaluated in this way. The second statement could be viewed as a restatement of the first, but that wasn't the point. The same goes for the contrapositive.

 

I don't follow you, but all possible situations are summarized in a tuth table. The truth table for A--->B was my first post.

 

The entries for the truth table are intuitively clear if you read A--->B as "If A then B" withe perhaps the value when A is false being a bit mysterious until one realizes that a false hypothesis can imply literally anything.

 

The fact that a false hypothesis implies anything is important mainly in the implications for an inconsistent set of axioms. Given an inconsistent set of axioms, literally any sentence formulated using those axioms is both true and false, hence inconsistent sets of axioms are both uninteresting and to be avoided.

 

Thus "If pigs could fly then I woudl be rich" is both a true implication and an information-free statement.

 

I'm not sure I understand. What I did, or at least attempted to do, was write out all the situations that could arise when evaluating an implication and what it evaluates to, ie. T->T (=T), T->F (=F), F->T (=T), F->F (=T) and tried to give an intuitive explanation of why it is evaluated in this way. The second statement could be viewed as a restatement of the first, but that wasn't the point. The same goes for the contrapositive.

 

I don't follow you, but all possible situations are summarized in a tuth table. The truth table for A--->B was my first post.

 

The entries for the truth table are intuitively clear if you read A--->B as "If A then B" withe perhaps the value when A is false being a bit mysterious until one realizes that a false hypothesis can imply literally anything.

 

The fact that a false hypothesis implies anything is important mainly in the implications for an inconsistent set of axioms. Given an inconsistent set of axioms, literally any sentence formulated using those axioms is both true and false, hence inconsistent sets of axioms are both uninteresting and to be avoided.

 

Thus "If pigs could fly then I woudl be rich" is both a true implication and an information-free statement.