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).

Categories: Error Statistics

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22 thoughts on “Little Bit of Logic (5 mini problems for the reader)

  1. On the Existence of Insistent Systems …

    Modus Dolens

    • Maybe they’re insisting on a reductio argument or inferring that one of the premises, presumably the first, isn’t true.

  2. Who can give a deduction of an arbitrary statement C from the premises A and ~A? I can do it in 4 lines.

  3. Here is one possible argument, using the elegant natural deduction system from Richard Kaye’s book, The Mathematics of Logic (chapter 6).

    1. ~C (assume)
    2. | A (premise)
    3. | ~A (premise)
    4. | false (contradiction rule with 2, 3)
    5. ~~C (RAA with 1, 4)
    6. C (~-elimination with 5).

    • Is that a syntactic rule? I mean obviously it can be derived, but does he give it as part of the proof system? OK, I accept this.

      • The step in 4 is syntactic. The rule says if p and ~p have been written down already, then you can write down “false”, which Kaye treats as a primitive sign (he actually has a special symbol for it, but keyboards these days…). Then his RAA says if “false” has been written, you can write down the negation of any given assumption.

    • I would start with the theorem
      A -> (~A -> C)
      and apply MP 2x

  4. Andrew J. Sutter

    D.S. also works if the 2nd premise is ~A:

    Either my cat likes asparagus, or she always votes for the ruling coalition.
    My cat abhors asparagus.
    ∴ My cat always votes for the ruling coalition.

    (In fact, the entire household is in the opposition camp.)

  5. Peirce’s spin on it, where he basically takes it as a definition of negation —

    Ex Falso Quodlibet

    • But what Peirce writes there speaks to a very idiosyncratic way to understand “x is false”, (which doesn’t mean X is inconsistent, see*)–one which holds only because of the peculiarities of the material conditional.

      Peirce: “One way of saying “not x” or “x is false” is to say “x implies α” where “α” is taken to mean “any proposition whatever”.”

      *A possible equivocation also should be noted:
      If X is false, then (X –> a) is true, because of the peculiarity of the material conditional, but you can’t derive an arbitrary a from an X that is merely false (e.g., Mayo can fly)..

      In the rest of this passage I see that Peirce gets to the question I posed (under the slides) about the consistency of a system.

  6. I think the underlying issue is whether we want our connectives to be truth-functional or whether we are seeking some sort of “relevance logic”. In the first case the supposed “content” of a proposition, e.g., “Mayo can fly” is irrelevant, only its truth-value enters into the truth-functional conditional. And the truth value of “Mayo can fly” is further irrelevant to the truth of “p ⇒ Mayo can fly” if p is false.

    • Sure, but then it’s not really a formal logic, is it. There are formal logics which might do better in capturing actual reasoning, but that’s beyond our discussion. I still think the quote from Peirce (one of my heroes) is a little odd, because that’s not how we view negation.

  7. Thomas Lumley

    1 is completeness: if it’s valid, it’s provable
    2 is consistency: if it’s provable, it’s valid

    • Minor pt: replace provable with derivable, C is derivable from the premises.

  8. What makes logic formal would be another good discussion.

    Peirce defines logic as “formal semiotic” —

    C.S. Peirce • On the Definition of Logic”>

    But he uses formal here in a sense roughly meaning the same thing as normative, as we can tell from statements he makes elsewhere.

  9. Golde Holtzman

    What does the vertical line “|” mean?

    • Tim was following the use in many books to indicate an assumed premise which then has to be “discharged”. Another is an -> arrow, with a straight line down from the tail until the assumption is discharged.

      On, a different topic, I’d like to inform you about our summer seminar

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