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Am unsure (leaning more towards no): true P and true C is valid, false P and false C is valid, True P and false C is invalid, how about false P and true C?
P= Premises; C= Conclusion.
People who says.. Yes, it is valid, I would like to hear some answers for
How can we derive the truth out of falsehood?
If we can, then True P - False C is valid too (tell me why it is invalid?).
Yes, an argument with false premises and a true conclusion can be valid. For example:
All cats are human
Socrates is a cat
Therefore, Socrates is human
The argument has false premises and a true conclusion. But the argument is valid since it's impossible for the premises to be true and the conclusion false. In other words, if the premises are true the conclusion is guaranteed to be true, which is how validity is defined.
A valid argument guarantees that the conclusion shall be true whenever all premises are true.
This guarantee is broken only when the conclusion may be false when all premises are true.
So a valid argument does allow for a case where the conclusion is true while some (or all) of the premises are false. Its guarantee is not broken by that.
Could an argument with false Premises and a true Conclusion be logically valid?
Validity is assessed on form only. Whether the premises are actually true or actually false is irrelevant.
For example,
Donald Trump is a martian;
All martians are Presidents of the United States of America;
Therefore, Donald Trump is President of the United States of America.
Valid argument, false premises, true conclusion. QED.
Note 1 on logical validity
The truth of the conclusion is not derived from the truth of the premises since the premises are (presumably) false. And it is also clearly not derived from the falsehood of the premises.
The truth of the conclusion is derived from the form of the argument, and by assuming that the premises are true.
If you understand the argument, then you should be certain, once you assume the premises, that the conclusion is true.
There is nothing else to it.
Aristotle didn't provide any more details as to how we arrive at the certainty that the argument is valid. And, so far, nobody else did, even though many great thinkers since Aristotle pondered the issue.
Note 2 on logical validity
Many logicians accept as valid arguments which are not formally valid. For example:
Everyone is female.
So, any siblings are sisters.
This argument will be accepted on the semantic ground that, first, the definition of the word "sister" in English makes any sister female by definition and, second, the definition of the word "sibling" in English makes any sibling either male or female.
However, semantic is a murky issue and admitting validity on semantic grounds can only lead to endless debates about the meaning of the words used in the argument which are not logical terms (i.e. not "or", "imply", etc.).
Further, any definition accepted as relevant to justify validity on semantic ground, is de facto an assumption, i.e. an implicit premise.
Whenever an argument is admitted as valid on semantic ground, it should be possible to make it formally valid by making explicit all relevant definitions by incorporating them as additional premises of the argument.
Thus, the argument above could be made formally valid by making it "formal", as follows:
For all x, Brother(x) implies not Female(x);
For all y, Sibling(y) implies either Sister(y) or Brother(y);
For all z, Female(z);
Therefore, for any a, Sibling(a) implies Sister(a).
Here, we can ignore the semantic of the non-logical terms. The validity of the argument is now entirely a function of the form of the argument.
To qualify an informal argument as valid, without any qualification, is therefore seriously misleading. An informal argument is valid to you only because you admit, if only implicitly, all relevant definitions.
Try to get anyone who doesn't know the definition of the English words used in the argument to agree that the argument is valid! Good luck. And would you yourself sign a document written in any language you don't understand on being told that the document is valid?
All formally valid arguments are also informally valid. However, informally valid arguments are not necessarily formally valid.
Thus, it is never misleading to use the word "valid" to refer to formally valid arguments, but it is misleading to use it to refer to informal arguments. When talking about the validity of informal arguments, we should use the expression "informally valid".
I will answer your question first by talking about the definition of 'validity' (which I think is necessary to consider very precisely) and then explaining the reasoning behind this definition.
As already mentioned here, validity is a property not of a concrete (single) argument, but rather of the form of an argument.
The argument
(1) If it is raining, the street is wet. (2) It is raining. Conclusion: The street is wet.
has the following form: (1) If p, then q. (2) p. Conclusion: q.
Now the definition of validity says: An argument form is valid if and only if it is not possible that all premises are true and the conclusion is false.
That means, in order to prove that an argument form is valid, we have to prove that whenever we insert true propositions for its variables (here in the example: p and q), the conclusion must be guaranteed to be true.
I will show the general steps for this prove before considering your question regarding false premises. (Possibly you are already familiar with these steps.)
In the easiest case (like ours), a truth table can be made. A truth table shows all possible combinations of the truth values of the premises of our argument form.
The truth table for the above argument form would look as follows:
p | q | p -> q (1. premise) | p (2. premise) | q (conclusion)
-------------------------------------------------------------------------
F F T F F
F T T F T
T F F T F
T T T T T
(The '->' is the common symbol for 'if... then' in logic; an explanation of how the truth value for this logical operator is calculated can be found here.)
Since the definition of validity only talks about the case of true premises, all other lines of the truth table can be completely ignored. In this simple example, it is therefore only the last line that is relevant. Since the last line of the truth table yields a true conclusion, we know that this argument form is valid.
Before we have this proof, we can make no inference regarding validity: An argument with false premises could either be an instance of a valid argument form as well as of an invalid one.
One last illustration: If we again take our simple argument form above, we could really construct an instance with false premises and a true conclusion:
Let 'p' stand for 'You are a cat.', and q for 'You are a human.' (inspired by the answer by Eliran). The our concrete argument would look as follows: (1) If you are a cat, you are a human. (2) You are a cat. conclusion: You are a human.
Both premises are false, and the conclusion is true. From the reasoning above we know that the argument form of which this argument is an instance is valid.
The reasoning behind the concept of validity:
Maybe these illustrations can also make clear why 'validity' is defined precisely in this way. Validity can (metaphorically spoken) be seen as a quality criterion of argument forms. Argument forms can be seen as the logical construct that lies behind a concrete argument. If only we insert true premises into this construct, we are guaranteed to come up with a true conclusion. If this were not the case, we would have made a logical mistake. Correct logical reasoning guarantees that truth is preserved! If we however insert false premises... Well, since the logical construct is still the same, we have not made a logical mistake. We have rather made some other mistake (we have false beliefs regarding reality etc.) So our argument, from the logical point of view, still deserves this quality criterion. That's what validity is about.
Of course, validity is not everything. We indeed want to have true premises. That's why there is also the notion of 'soundness', as also already mentioned here. A sound argument is a valid argument + true premises. That is, a sound argument does not only involve correct logical reasoning but more: E. g. correct beliefs about our world.
Trivially, the argument could be:
Conclusion:
This argument is perfectly valid. The fact that both premises are wrong: umbrellas do not keep you dry on windy days, and it is not currently raining, does not change the fact that the valid argument led to a true conclusion.
The Material Conditional doesn’t always cleanly match our expectations of a relation of inference. This is well enough known that there is a Wikipedia article on Paradoxes of Material Implication.
One particular concern is the thought that assuming this as a blanket rule potentially allows not just valid arguments but definitionally Sound arguments of arbitrary statements as long as the conclusions are in fact true.
Intuitively, we may wish to demand that truth on its own is not quite enough on which to hang implication. This is a common theme in Relevance Logic, which is a topic that might interest you.
Yes.
The definition of validity is extremely narrow. From IEP:
A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
The definition does not mention what happens when the premise is logically false (that is, a contradiction). If the premise is logically false, the argument is trivially valid (more precisely, we call this "vacuously valid" but trivial works as well).
This is because if the premise is a contradiction, then it can never be true, and so it is impossible for the premise to be true while the conclusion is false.
This truth functional situation of false premises making a conditional true is my favorite example of how formal languages do track natural languages but with different aesthetics. It seems the aesthetics of all languages formal and natural in the end may be the same and thus imply there may be foundational knowledge there somewhere.
The concept of material implication is universally known in human reasoning: “if Socrates is human, then he is mortal” written logically as P⊃Q. Though it is universally used by all, as your question brings out, there are serious intuitive problems with such reasoning such as its truth conditions. If the antecedent P is false or if both antecedent and consequent are false then the compound implication in its entirety is true. “If life exists on the moon, then life exists on earth” and “If the moon is made of green cheese, then life exists on other planets” are both considered valid reasoning by most of modern logic because it is impossible for their truth conditions to be: the antecedent is true and the consequence is false. Seems absurd in our natural language but it makes perfect sense and the rationality is easier to see written aesthetically in natural language: “If congress passes serious immigration reform, then I am a monkey’s uncle”. This statement expresses what I consider to be a true proposition: “congress will not pass serious immigration reform”. Yet, I express it in the form of a false antecedent that by being false is taken even in natural language to prove anything; also, both antecedent and consequent are false but I use this conditional to state a truth.
Surprisedly, this is as common in mathematical reasoning as it is in natural language. The statement “if n is a perfect square, then n is not a prime number” is true throughout. But, “if 3 is a perfect square, then 3 is not prime” is also a true conditional though both its antecedent and consequent are false. So, as much as I hate to admit it, the nonsensical aesthetics of non-analytical or continental philosophy writers such as Sontag, Foucault, Derrida et al may actually contain something with which I agree but is simply not worth the effort of getting to it. Aesthetics trumps intuition and perhaps much more.
