## What is the logical form of the definition of validity?

4

My text book gives the definition of validity as "An argument is valid IFF if the premises are true, then the conclusion must be true". Using a conditional elimination on the RHS of the IFF yields "An argument is valid IFF the premises are false or the conclusion is true".

Yet, this seems strange to me. Am I doing something incorrect with my manipulation or is this definition actually true? Surely the criteria for a valid argument goes beyond a false set of premises or a true conclusion.

4

"An argument is valid IFF the premises are false or the conclusion is true".

misses an important feature in the textbook's definition. Namely, you've lost the must, but the must is crucial.

The validity of an argument does not hinge on the truth or falisty of its premises or the truth of its conclusion. Instead, validity looks at the sum of all of the operations and rules of inference in an argument and evaluates it in light of every possible condition of the truth and falsity of every premise and the conclusion.

E.g., consider the following two arguments:

Argument 1
(1) If the moon is made of cheese, Kaguyahime lives there.
(2) The moon is made of cheese.
Therefore Kaguyahime lives there.


This argument is valid on your definition (at least one false premise). And valid on the must definition -- if the premises are true, then the conclusion must be true.

Argument 2
(1) The moon is smaller than the sun
(2) The moon is not made of cheese
Therefore, Apollo 11 went to the moon


This argument is valid according to your definition -- it has a true conclusion. But it is not a valid argument on the textbook definition. Why? Because there is an imaginable set of the variables where the premises are true and the conclusion is not (it is not the case that the conclusion must be true if the premises are true).

## WHY?

Argument 2 can be rewritten: (1) S (2) C (3) A

Per the fundamental rules of logic, S can be either T or F, C can be T or F, A can be T or F. This gives us 2^3 (8) possible arrangements of these variables. And one of these is this: S is true, C is true, and A is false. This breaks the must

Thus, your sentence is not an accurate articulation of validity because you've lost the modal consideration.

I like this answer. I think you should consider including "rules of inference" in your statement "validity looks at the sum of all of the operations in [the] argument and ...". Here, I read "operations" as "logical operations" which obviously do not include rules of inference. – Nick – 2014-10-10T23:41:38.533

@NickR good point and done. – virmaior – 2014-10-10T23:54:23.257

1

You point at the "counter-intuitive" aspect of the truth-functional definition of the conditional :

if A, then B

which is equivalent to :

either not A or B.

But it is also equivalent to :

not (A and not B).

In this case, the definition of logical consequence or valid argument is :

"An argument is valid iff it is not possible that the premisses are true and the conclusion is false

which is quite "sound".

1

An argument from premises P, P′, ... to C is valid just in case (P ∧ P′ ∧ ...) → C is a logical truth.

Since → is defined in terms of disjunction & negation, that will be the case just in case ¬(P ∧ P′ ∧ ...) ∨ C is a logical truth. Now that will be the case iff an arbitrary interpretation M either does not satisfy all of (P ∧ P′ ∧ ...) or satisfies C. This is a very strong definition, because establishing the validity of an argument requires that we establish the logical truth of the corresponding conditional, not simply its truth.

To do so, we would have to start by saying something like "let M be an arbitrary interpretation of" whatever the language we're working with "such that M satisfies P, P′, P′′, ...". The goal would then be to show that M also satisfies C. If we manage to show that M satisfies C, since we assumed nothing about the nature of M, it would mean that all interpretations that satisfy the premises satisfy C.

Contrast that with the strange one:

( ! ) An argument from premises P, P′, ... to C is valid just in case (P ∧ P′ ∧ ...) → C is true.

Had that been the definition, validity would be a very weak notion, because to show that (P ∧ P′ ∧ ...) → C is true, it would suffice to find some interpretation that either falsified one of the Ps or satisfied C!

By this definition validity of an argument is defined in terms of premises and conclusion only. What is colloquially understood by "argument" is what's in between. One is free to make definitions of course, but this one is idiosyncratic to a point of being useless for explanatory purposes. To illustrate: P)1=1; S) 1=2; C)2=2 is a "valid" argument under this definition. Even Wikipedia does better https://en.wikipedia.org/wiki/Validity#Validity_of_arguments

– Conifold – 2015-07-09T01:02:27.500