An invalid argument with one or more false premises

Question

This is an exercise from A Modern Formal Logic Primer (Teller):

4-1. Give examples, using sentences in English, of arguments of each of the following kind. Use examples in which it is easy to tell whether the premises and the conclusion are in fact (in real life) true or false.

e) An invalid argument with one or more false premises

Validity of an argument has been defined as follows:

The argument "$X$. Therefore $Y$" is valid just in case the sentence $\neg (X \land \neg Y)$ is a logical truth.

Now my confusion is twofold:

1. An invalid argument with one or more false premises appears to require that at least one premise and the conclusion are contradictions. Otherwise there are cases in which all premises are true. But for the argument to be invalid, there must be at least one case in which the conjunction of the premises is true while the conclusion is false. How is this possible?

2. I am not sure how to interpret "logical truth" in terms of English sentences with invariable truth values. Here is the solution given in the book:

Anyone who loves logic is bald.
Robert Redford is bald.

Therefore Redford loves logic.

As far as I can tell, both of the premises are false (is it correct to interpret invariably false natural language sentences as contradictions?). Which makes $\neg (X \land \neg Y)$ true. Now since the truth value of the premises is invariable, there is no case in which the conjunction of the premises is true while the conclusion is false. So how is this an invalid argument?

Answer 1

An invalid argument does not require any contradictions at all. Example: 1. Unicorns exist. 2. It rains everywhere on the Earth at all times. Therefore, 3. I am flying to San Francisco.

Your second example commits the fallacy of the undistributed middle. You haven't talked about all bald people, so it can't be the middle term in an Aristotelian syllogism.

Answer 2

As far as I can tell there are two different interpretations here. Clearly if a conclusion $C$ is independent of a premise $P$, then $P \not\vDash C$, but just a clearly $P, \neg P \vDash C$.

The Law of Non-Contradiction establishes that no proposition may be both true and false in exactly the same context; so, in the context in which we assume some false premise $P$, we’re allowed to infer anything, e.g. $C$. In order to syntactically capture the notion that a contingently false premise doesn’t validate just anything, something like modal logic seems required. For example, $(\Diamond P \land \neg P) \to Q$ is invalid in most modal logics, and captures the notion that the falsehood of a contingent formula $P$ does not contradict that $P$ could be true. $\Diamond (P \land \neg P) \to Q$ validates the notion that if a contradiction holds in any accessible context/scenario/possible world, then anything follows.

Answer 3

is it correct to interpret invariably false natural language sentences as contradictions?

While the sentence "unicorns exist" is invariably false in our world, it is satisfiable in another world, so is not a contradiction.

The argument "$X$. Therefore $Y$" is valid just in case the sentence $$\neg (X \land \neg Y)$$ is a logical truth.

Anyone who loves logic is bald.
Robert Redford is bald.

Therefore Redford loves logic.

Now since the truth value of the premises is invariable, there is no case in which the conjunction of the premises is true while the conclusion is false. So how is this an invalid argument?

The given argument is $$\lnot\Big(\big(\forall x(Lx\to Bx)\land Br\big)\land\lnot Lr\Big).$$ It is not a logical truth, as it is false for the following case:

• universe = $\{7\}$

  $Lx\quad:\Leftrightarrow\quad x$ is less than $5$

  $Bx\quad:\Leftrightarrow\quad x$ is bigger than $5.$

The point is, assessing the validity of an argument considers its logical structure, not its particular truth value in our world.