[The following is summary. Bracketed commentary is my own. I highly recommend Agler’s book.]
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.1: Propositions, Arguments, and Logical Properties
Section 1.3: Deductively Valid Arguments
Section 1.4: Summary
An argument is deductively valid if it is impossible for the premises to be true and the conclusion false, and it is invalid otherwise. An argument is sound if the premises in fact are true and as well it is valid. It is unsound if either it is invalid or if any of the premises are false.
1.3 Deductively Valid Arguments
Elementary symbolic logic is concerned primarily with deductively valid arguments (14).
1.3.1 Deductive Validity Defined
Deductive validity applies only to arguments (15).
There are two ways to define deductively validity, both given below:
Validity: An argument is deductively valid if and only if, it is necessarily the case that if the premises are true, then the conclusion is true. That is, an argument is deductively valid if and only if it is logically impossible for its premises/assumptions to be true and its conclusion to be false.
We thus define deductive invalidity in the following way:
Invalidity: An argument is deductively invalid if and only if the argument is not valid.
We now recall the second formulation: “an argument is deductively valid if and only if it is logically impossible for its premises/assumptions to be true and its conclusion to be false” (16). We now need to define this impossibility. [Something is logically impossible if it involves a logical contradiction, and there is a contradiction among propositions if they are false no matter what the circumstances may be. Thus, if the premises in combination with the conclusion are false under any possible circumstance, then the inference leading to that conclusion is invalid.]
Impossibility: Something is logically impossible if and only if the state of affairs it proposes involves a logical contradiction.
A proposition is a logical contradiction if and only if, no matter how the world is, no matter what the facts, the proposition is always false.
Contradiction: A proposition is a contradiction that is always false under every circumstance.
Agler illustrates with these examples:
(1) John is 5'11, and John is not 5'11.
(2) Toronto is in Canada, and Toronto is not in Canada.
(3) Frank is the murderer, and Frank is not the murderer.
As we can see, sentences 1-3 are false, because no matter how we imagine the world, they cannot be true. “Since the state of affairs they propose involves a contradiction, each one of these is logically impossible. That is, | under no situation, circumstance, or way the world could be can John be two different heights, can Toronto be in Canada and not in Canada, or can Frank be the murderer and not be the murderer” (16).
And as we noted, if it is impossible [that is, if it leads to a contradiction] that the premisses are true and the conclusion is false, then the inference is valid: “we would be uttering something contradictory (always false) if we were to say that a deductive argument’s premises/assumptions were true and its conclusion was false” (17).
1.3.2 Testing for Deductive Validity
Given the above definition and explanation of deductive validity and deductive invalidity, in order to test an inference as to which it is, we can ask the following question:
Is it logically impossible for the premises to be true (condition 1) and the conclusion to be false (condition 2)?
Agler then provides a step-by-step procedure for testing validity in this way, called the Negative Test for Validity. The steps are depicted in the chart below:
So let us consider some possibilities to illustrate.
(1) All men are mortal.
(2) Barack Obama is a man.
(3) Therefore, Barack Obama is mortal.
Is it possible for all the premises to be true? Certainly (for in fact they are true). Assuming them to be true, is it also possible for the conclusion to be false? No it is not, because this leads to a contradiction. [If Barack Obama is a man, and all men are mortal, it cannot be that Barack Obama is not mortal.]
Consider now these propositions:
(1) Some horses are domesticated.
(2) All Clydesdales are horses.
(3) Therefore, all Clydesdales are domesticated.
Is it possible for all the premises to be true? Yes, it is. Assuming them to be true, is it also possible for the conclusion to be false? In fact, yes, it is. [If it said, “All horses are domesticated”, then the situation would be different. In that case, it would be impossible for both the premises to be true the conclusion to be false. But as it is written in the example above, it is possible for the conclusion to be false. For, Clydesdales could be a subgroup of horses that are not domesticated, since only some horses are.]
[Agler then given an example for when all the premises are false, but the conclusion is true:
(1) All humans are donkeys.
(2) James the donkey is human.
(3) James the donkey is a donkey.
But I am not sure that this is an instance that illustrates the other case we have not addressed, where it is impossible for the premises to be true. I can imagine that new evolutionary evidence is discovered that tells us that humans are a species of donkey. In other words, I do not see a logical contradiction among the premises. It might be helpful to consider an inference that Graham Priest offers in Ch.2 of his book Logic: A Very Short Introduction:
(1) The Queen is rich.
(2) The Queen is not rich.
(3) Therefore, pigs can fly.
Here it is not possible for the premises to be true, because that leads to a contradiction. Therefore, the inference is valid, even though the conclusion is false.
On of the valid inferences [which I skipped] was:
(1) All men are immortal.
(2) Barack Obama is a man.
(3) Therefore, Barack Obama is immortal.
What is notable here is that premise (1) is false. No man is immortal. Even so, the whole inference is valid. In this case, we would say that it is valid but unsound. Agler explains:
Deductive arguments can either be valid or invalid, and if they are valid, they can be sound or unsound. An argument is sound if and only if it is both valid and all of its premises are true.
Sound: An argument is sound if and only if it is valid and all of its premises are true.
One way of thinking about soundness is through the following formula:
Validity + all true premises = sound argument
An argument is not sound (or unsound) in either of two cases: (1) if an argument is invalid, then it is not sound; (2) if an argument is valid but has at least one false premise, then it is not sound.
Unsound: An argument is unsound if and only if it is either invalid or at least one premise is false.
Logic can determine whether or not an argument is valid. But whether each premise is true or false is something formal logic is mostly unable to do. So, “if a premise is contingent (one whose truth or falsity depends upon the facts of the world), then its logical form does not tell us whether the premise is true or false, and while the argument may be valid, we will not be able to determine whether or not it is sound without empirical investigation” (20).
Agler summarizes how we met our stated goals in this chapter, namely, we discussed propositions, arguments, and deductive validity (20).
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
Or if otherwise noted:
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.