Falsity implies anything?!?!?

@JeffD2,

That's actually a pretty good point, especially considering that when you typically look at the basic truth table for a conditional, you tend to take it at face value and completely bypass the issue of why the value is what it is.

First, you may mean "why is a conditional proposition false if the antecedent is true and the consequent is false. This is the base truth table value for a conditional;

But basically, the topic deals with the issue of these values: T -->F |- F. I suppose you could say that the statement If T, then F is always false because the antecedent is a precondition for the consequent. What if I said to you, "If I post on philosophy forum, then I will be banned." Sounds logical, right? But what if I post and I am not banned. The antecedent is true but the consequent is false, which is the possibility that is accounted for in this instance, namely T -->F |- F.

But with your examples, especially in truth functional logic, the moon could be purple and unicorns could roam the earth devouring small kittens and the statements would still be true (or false)... the only thing that matters is the truth value we assign to them.

@JeffD2,

If you look into common sense, you'd know that it isn't. Two couldn't possibly be even due to our logic of arithmetic, just as jupiter is the first planet from the sun.

@JeffD2,

And yet this is not common sense...this is truth functional logic. But really... what contributes to common sense? irrationality? No. Common sense is an amalgamation between inductive and deductive logic. It may be the case that we take the formal systems we use everyday for granted.

In the grander scheme of logic, jupiter could be the sun for all intents and purposes, which is why logic is a closed system. Practical use of logic just requires... well... common sense.

@VideCorSpoon,

It is common sense it the sense that it is well known, i guess it's just simple logic then; not common sense.

@JeffD2,

first, i am loving the ambiguity of the conversation with "it"s and what not which can be mistaken to iterate different things. In terms of a formal logic system (or even a regular argument), that is not a really fantastic thing to do. but anyway...

Common sense in the sense of the term is what it means to be well known. This is a definitional fallacy, like saying Socrates has a snub-nose nose. The one implies the other. Within the ordered system of logic, it is again truth functional, although this is entirely dependent on the system created... or pre-existing. Within the closed world of logic, unicorns could roam the earth devouring small kittens. Like wise, the world we live in could be considered a closed system. So within this close-system world we live in, we can rely on common sense, and I do not disagree with this. But that common sense is reliant on a system of logic in one way or another, whether it be inductive, deductive, or whatever.

Why you separate common sense from logic is puzzling, but I suppose you have a good reason to suppose so, so I am eager to hear your thoughts on it.

@VideCorSpoon,

Common sense is something well known, logic in my opinion is basic reasoning which can be proven. You can have the false logic of "unicorns are real" but common sense tells you they are not real, and your eyes can see otherwise, though not everything you see is not 100% valid.

@JeffD2,

well, yes... proven to be true. This is a fundamental premise behind truth functional logic. But also, it seems as though you wouldn't call basic reasoning a factor in common sense? Wouldn't that be common sense?

On the the unicorns, common sense telling you they are not real, and the fact that your eyes can see other wise... think about it like this;

I have never been to Antarctica. Chances are you have never been to Antarctica. But does the fact that we have never been to Antarctica negate the possibility of it actually being real... of it actually existing? You could apply this to many different parts of the world, even to specific parts of the town you are in right now which you have never seen before. Your eyes, your empirical sensory perceptions only reveal that which you immediately perceive. False logic is actually what you imply, by inferring on common sense rather than the possibility of something not existing becuase you have not seen it... that is a fundamental flaw in inductive logic.

Logic is complex in the respect that it accounts for, to borrow from Leibniz's monadology, all possible worlds as well as the best of all possible worlds. Many possibilities as well as the apparent one, what you would call what you see what your own two eyes. Which is probably why we use truth functional logic to distinguish the constants in our particular universe.

This phrase you end with in post #7 is curious though.

"though not everything you see is not 100% valid"

common sense in logic would say that this is a self negated sentence. I would not think under your rubric of "fundamental" common sense that one would use negated sentential operators. Is this not an application of logic in "common sense?" This is in a sense a double negated sentence, roughly the same as saying "everything you see is 100% valid."

@Paggos,

Paggos;67011 wrote:

Common sense is something well known, logic in my opinion is basic reasoning which can be proven. You can have the false logic of "unicorns are real" but common sense tells you they are not real, and your eyes can see otherwise, though not everything you see is not 100% valid.

When you say common sense, what I think of is inductive reasoning. When you say "unicorns are not real", you are basing that off of numerous observations and experiences. You can't deductively prove that. Its just an observation.

In my opinion though, common sense can be deductive reasoning or inductive reasoning.

Deductive Reasoning: Prove that the sky is blue. People would see that as common sense and would have no idea how to prove that other than by numerous observations, but they are wrong. In order to deductively prove that, one would first have to understand what blue is, or understand the definition of blue. Then that person would look up to the sky. Finally that person would compare the color of the sky to his/her understanding of the meaning of blue. (I know that the sky isnt always blue, but you get the point)

Inductive Reasoning: All human observations and experiences have never involved a unicorn. There are numerous human observations and experiences. Therefore, unicorns don't exist.

---------- Post added at 03:02 PM ---------- Previous post was at 02:44 PM ----------

Back to my original post though. Why is a conditional proposition always true if the antecedent is false? To me, it seems like it is just a definition.

In my opinion, the conditional proposition shouldnt be true or false. How can one start with an illogical assumption and end with a logical argument and a logical conclusion. Propositions with false antecedents seem illogical and meaningless to me.

Can you prove to me the following: If two is odd, then two is even. Absolutely not!!! How can you say this is true if you dont even understand why it is true. The only way is by defining it to be true.

It is not common sense that a conditional proposition is always true if the antecedent is false. Common sense is something that the majority of the population knows and understands. Only logicians, philosophers, mathematicians, etc. know of this "illogical logic".

@JeffD2,

I guess you're valid then, the two do intermix.

@JeffD2,

JeffD2;65179 wrote:

Why is a conditional proposition always true if the antecedent is false? For example, all of the following conditional propositions are true:

If 1=2, then 3=4
If 1=2, then 3=3
If 2 is odd, then 2 is even
If Jupiters is the 1st planet from the sun, then Jupiter is the 8th planet from the sun

Why?!?!?

You are talking about material conditionap, not what is ordinarily meant by the conditional.
Why is the truth table of the material conditional such that when the either the antecedent is false, or the consequent is true, the entire conditional is true, so that FT and FF both give us T? The answer is that it is important that the conditional be false when the antecedent is true and the consequent is false. That is essential for any conditional that makes logical sense. Given that essential, it is clear we want TT F; what about FT, and FF? Well, it is true that in some cases FT gives us F. For example, if Paris is in Germany, then Paris is in Europe. That conditional is true, even if the antecedent is false. We have to allow for that possibility. How about; FF? If Paris is in China, then Paris is in Asia. And we have to allow for that possibility too. So, as long as TF is always false, and as long as FT and FF are sometimes true, we have the truth values for the material conditional we have. And, it works.

Hope this helps.

@JeffD2,

JeffD2;65179 wrote:

Why is a conditional proposition always true if the antecedent is false? ...
Why?!?!?

Conditional statements "A -> B" are usually translated by "If A then B". This translation may mislead you, as it seems to imply some sort of causal link between A and B.

If you translate "A -> B" by "A is not truer than B", the principle "ex falso quodlibet"(*) you wonder about becomes quite obvious: The falsum is never "truer" than anything else. So your examples would be

"1=2" is not truer than "3=4" (correct, as both are false)

"1=2" is not truer than "3=3" (correct, as the second one is truer)

"2 is odd" is not truer than "2 is even" (correct, as the second one is truer)

"Jupiters is the 1st planet" is not truer than "Jupiter is the 8th planet" (correct, as both are false)

_____________________
(*) From the false, everything [can be concluded].

@JeffD2,

The principle of explosion ( {a, ?a} ⇒ q ) is rejected in paraconsistent logics, because the proof for them relies on at least one inference rule that isn't accepted. Most proofs go like this:

1. P (Premise)
2. P∨Q (Add.)
3. ?P (Premise)
4. Q (DS)

DS is Disjunctive Syllogism: {P∨Q, P} ⇒ Q.

Most paraconsistent logics reject DS, because it entails the principle of explosion. They also usually reject reductio ad absurdum and Double Negation Elimination, two other inference rules. The wikipedia page is surprisingly adequate.

More relevant to the first post, material conditional (what corresponds to "If A, then B", or in logical symbols, A→B) is logically equivalent to ?B∨A, or Not B, or A". The antecedent and consequent, unlike their English counterparts, don't need to he related. For example:

[indent](1) If 3+3=6, then the sky is blue.[/indent]

(1) has the same form as (2):

[indent](2) If 3+3=6, then 3+3=6[/indent]

Logics, as formal systems, are not concerned with the meaning of the propositions they evaluate, only the form.

To prove that A→B ⇔ ?B∨A, you could use a truth table:

http://dl.getdropbox.com/u/107106/mathematica/material_conditional.png

@JeffD2,

Just a minor remark.

Leafy wrote:

material conditional [which] corresponds to "If A, then B"

It does only sort of. Most people find it very anti-intuitive that if both P and Q are false, then "P→Q" is true. We never ever use it like that in practice. Some logicians have tried to develop alternative system that agree with these intuitions.

A similar thing is that of relevance logics.

Or perhaps it is the same issue. I'm not entirely sure. I haven't read that much about it.

The symbols that I and Leafy use for formalization can be found here. (We're using "→" to mean material conditional and "⇒" to mean logical implication. Similarly with material bi-conditional and logical equivalence.

@Emil,

Emil;92536 wrote:

Just a minor remark.

It does only sort of. Most people find it very anti-intuitive that if both P and Q are false, then "P→Q" is true. We never ever use it like that in practice. Some logicians have tried to develop alternative system that agree with these intuitions.

A similar thing is that of relevance logics.

Or perhaps it is the same issue. I'm not entirely sure. I haven't read that much about it.

The symbols that I and Leafy use for formalization can be found here. (We're using "→" to mean material conditional and "⇒" to mean logical implication. Similarly with material bi-conditional and logical equivalence.

C.S. Pierce pointed out that it doesn't really matter if F>T, or if F>F, since if the argument depends on Modus Ponens, a premise will be false anyway, and the argument will be unsound even if valid.

@kennethamy,

kennethamy;93669 wrote:

C.S. Pierce pointed out that it doesn't really matter if F>T, or if F>F, since if the argument depends on Modus Ponens, a premise will be false anyway, and the argument will be unsound even if valid.

Yeah, I had noticed that too. FT and FF conditionals are, in a sense, useless.

@Emil,

Emil;93786 wrote:

Yeah, I had noticed that too. FT and FF conditionals are, in a sense, useless.

Well, redundant, because the salient line in the truth table is T >F, which is F. And then all we want is that the other lines not get us into trouble in arguments. After all, it is certainly possible for conditionals with false antecedents to be true, and, as long as it is impossible for conditionals with true antecedents, and false consequents to be true, we cannot get into trouble. After all, too, we can "define" the truth table for the conditional as ~p v q, in other words, the conditional is true whenever the antecedent is false, or the consequent is true.

@kennethamy,

If the antecedent is false and the consequent is completely nonsensical, is the proposition still true? For example:

"If Paris is in Germany, green questions walk coldly."

And what if the antecedent is false and the consequent presupposes a contradiction? Such as:

"If Paris is in Germany, both of the square circles are green."

@ACB,

ACB;93949 wrote:

If the antecedent is false and the consequent is completely nonsensical, is the proposition still true? For example:

"If Paris is in Germany, green questions walk coldly."

And what if the antecedent is false and the consequent presupposes a contradiction? Such as:

"If Paris is in Germany, both of the square circles are green."

Both the antecedent and the consequent must be propositions. Else, the conditional is nonsense.

@JeffD2,

JeffD2;65179 wrote:

Why is a conditional proposition always true if the antecedent is false? For example, all of the following conditional propositions are true:

If 1=2, then 3=4
If 1=2, then 3=3
If 2 is odd, then 2 is even
If Jupiters is the 1st planet from the sun, then Jupiter is the 8th planet from the sun

Why?!?!?

I would start by taking Aristotle, and using Venn diagrams to graphically depict what he was saying. It turns out, only a very small part of what Plato tried to teach him, but it is a start. Then I would ask, Since logic starts by abstracting from things, can we still continue to "rationalize" by writting gibberish? That not complying with the first principle of any logic system, a convention of names.

One can write gibberish all day long when the name being manipulated is a NOP.

I will caution that Plato was very good at knowing, using, and trying to teach the principles of predication, however, Aristotle may mention them correctly a few times, but on the whole generally got them wrong.

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Falsity implies anything?!?!?

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