Logic

[Daecon]

Hi I hope someone can help me.

 

Are there any good websites for learning logic?

 

I've just been trying to figure out antecedents and consequents and the two valid ways and the two invalid ways of arguing them.

 

I'm confused by the following sets of propositions:

 

If I buy the correct ticket then I'll win the Lottery.

I'll win the Lottery only if I buy the correct ticket.

 

The consequents and antecedents are reversed from one the the other, yet one of them can be a true statement and the other not?

[cosine]

Hm interesting question.

The first statement is

T -> L,

the second statement says that if we know you won the Lottery, then we know you bought a ticket. So it is

L -> T.

If you want to pursue this you can conclude that T <-> L

 

Does this help?

 

Also for mathematical topics I always reccoment Mathworld and Wikipedia, mathworld for more technical overviews, and wikipedia for everything else. So in this case I would reccoment searching "logic" on Wikipedia.

[Sisyphus]

An equivalent example which ought to make it clear how one could be true and the other not:

 

If I am in Boston, then I am in the United States.

 

vs.

 

If I am in the United States, then I am in Boston.

[Tom Mattson]

Hi I hope someone can help me.

 

Are there any good websites for learning logic?

 

There are free textbooks and lecture notes all over the place:

 

Mathematical Logic - An Introduction

Mathematical Logic

A Problem Course in Mathematical Logic

[Daecon]

[nevermind]

[Daecon]

I'm officially confused.

 

I cannot work out how a situation of an IF, THEN statement can return a value of true when the Antecedent is false and the Consequent is true.

 

Surely it's an invalid argument which ever way around you look at it, so how can it be deemed true?

 

[EDIT:] Nevermind.

 

I just realised that if the consequent is true then the statement is true regardless of the truth of the antecedent.

 

I suppose the consequent is equivalent to the conclusion of an argument, validity isn't the same as soundness.

 

Well, at least I feel like I'm finally learning something... this is good.

[Tom Mattson]

I cannot work out how a situation of an IF' date=' THEN statement can return a value of true when the Antecedent is false and the Consequent is true.

[/quote']

 

Here are 2 plausibility arguments to convince you. Hopefully one of them will make sense to you.

 

1.) When we have statements of the form [imath]A \rightarrow B[/imath] you have to keep in mind that both the antecedent and consequent could be any truth-functional statement, even a compound one. So let [imath]A \equiv P \wedge Q[/imath] and let [imath]C \equiv Q[/imath]. Then our conditional looks like [imath]P \wedge Q \rightarrow Q[/imath]. Because [imath]Q[/imath] appears on both sides of the arrow, this schema looks like it should be true regardless of the truth values of [imath]P[/imath] and [imath]Q[/imath]. So let [imath]P=F[/imath] and [imath]Q=T[/imath]. Then [imath](P

\wedge Q \rightarrow Q) = (F \rightarrow T) = T[/imath].

 

2.) Consider a statement such as, "If you live in Miami, then you live in Florida" (schema: [imath]M \rightarrow F[/imath]). Clearly this is equivalent to the statement "Either you live in Florida or you don't live in Miami" (schema: [imath]F \vee (\neg M)[/imath]). After thinking about it for a while you should be able to convince yourself that [imath]P \rightarrow Q \equiv Q \vee (\neg P)[/imath] under any possible interpretation. Then all you have to do is work out the truth table for [imath]Q \vee (\neg P)[/imath] and you're done.

 

Surely it's an invalid argument which ever way around you look at it, so how can it be deemed true?

 

A conditional statement is not an argument at all, invalid or otherwise. It is a truth-functional statement.

[Daecon]

Ah, yes - thanks. I'm confusing conditionals with syllogisms...

 

I'll get the hang of it eventually. I hope.