Little Bit of Logic (5 mini problems for the reader)

Little bit of logic (5 little problems for you)[i]

Deductively valid arguments can readily have false conclusions! Yes, deductively valid arguments allow drawing their conclusions with 100% reliability but only if all their premises are true. For an argument to be deductively valid means simply that if the premises of the argument are all true, then the conclusion is true. For a valid argument to entail  the truth of its conclusion, all of its premises must be true.  In that case the argument is said to be (deductively) sound.

Equivalently, using the definition of deductive validity that I prefer: A deductively valid argument is one where, the truth of all its premises together with the falsity of its conclusion, leads to a logical contradiction (A & A).

Show that an argument with the form of disjunctive syllogism can have a false conclusion. Such an argument take the form (where A, B are statements):

(Either A or B), ~B, therefore A.

A better way to write this is in symbols:  (A V B), ~B |= A

Abbreviate it as D.S.  An example of D.S. with a false conclusion is:

Either you have been influenced by Zizek’s works or you’re not a philosopher.
Mayo hasn’t been influenced by Zizek’s works.
Therefore Mayo is not a philosopher.

The culprit, of course, is premise #1. It’s not true.

(a) Now you give me an example of D.S with a false conclusion. Put it in the comments.

(b) What do you know about the premises of a valid argument that has a false conclusion?

Deductively validity is easy as pie to obtain. Any argument to a conclusion C can readily be made deductively valid. One way is just to add C as a premise (often disguised so it’s not so obvious the conclusion is merely snuck into the premises. We call this a circular argument, but it’s valid. A second way is to add a contradiction to the premises. Yes, an argument with contradictory premises is deductively valid. That’s because it’s impossible to have all its premises true and its conclusion false (since its premises can never all be true)*.

Invalid Deductive Argument: One where it’s possible to have all true premises and a false conclusion. Example: affirming the consequent.

(H -> E), E |= H

I’m using “->” as a material conditional, an “if then” claim.

(c) Give an example of an argument following the form of affirming the consequent that has all true premises and a false conclusion. (See my slides for one. then supply one of your own in the comments.)

See a bit more logic below these slides.

*This is all semantics-relating to truth and meaning. We want a system that matches the semantics with proofs and derivations (part of its syntax). In particular, we want two things:

(1) If an argument:  P1, P2,…Pn |= C is deductively valid, then C is derivable from P1, P2,…Pn.

(2) If C is derivable from P1, P2,…Pn, then P1, P2,…Pn |= C is deductively valid.

(d) One of these is speaks to the consistency of the system, the second, to its completeness. Which is which?

(e) I said any deductive argument with contradictory premises A, ~A is valid. That follows the semantic definition of validity. On the syntactical or proof-theoretical side, show how to derive any arbitrary claim C from A and ~A.

[i] Earn up to 5 blog points.

You will find a brief discussion of logic in Excursion 2 Tour I of Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (Mayo 2018, CUP).