2

I came across this argument when looking at categorical syllogisms.

All hounds are creatures that howl when they see strangers.
The hound at the mansion didn't howl when he saw the thief
Therefore, the thief was not a stranger

What's the best/ simplest way of formalizing this as a syllogism or in another formal method?

Thanks!

  • This clearly is Modus Tollens. You can look for it's formal notion. – rus9384 May 17 '18 at 1:01
  • How would the symbolization look like? My best attempt, though very much an amateur one, is: H > (S > W), H ^ ~W, / ~S. It just seems awkward to symbolize – anonymous May 17 '18 at 1:19
  • I think it would be 1) (H ^ S) > W | H ^ ~W => 2) ~S. But actually at discrete math I was kinda bad in predicate logic (but good in boolean), so can't say. – rus9384 May 17 '18 at 1:24
  • Syllogism is a sub-system of predicate logic. – Mauro ALLEGRANZA May 17 '18 at 6:34
2

We can drop the words 'creatures' and 'mansion' since they are not really contributing to the logic, and we might take the 'seeing' as read. In first order predicate logic your premises can be represented as:

  1. (∀x)(∀y)((Hx ∧ Sy) → Howl(x,y))
  2. (∃!x)(∃!y)(Hx ∧ Ty ∧ ¬Howl(x,y))

A rough proof goes as follows:

From 1, we have by contraposition and an application of de Morgan's rule:

  1. (∀x)(∀y)(¬Howl(x,y) → (¬Hx ∨ ¬Sy))

From 2, by two applications of existential instantiation we get:

  1. Ha ∧ Tb ∧ ¬Howl(a,b)

In effect, all we have done here is name the hound 'a' and name the thief 'b'. From 4 we have by simplification

  1. ¬Howl(a,b)

which combined with 3 gives by modus ponens

  1. ¬Ha ∨ ¬Sb

From 4 by simplification we have

  1. Ha

and from 6 and 7 by disjunctive syllogism

  1. ¬Sb

From 4 by simplification

  1. Tb

and hence by conjunction from 8 and 9

  1. Tb ∧ ¬Sb

Which proves that the thief, b, is not a stranger. If we wanted to be really rigorous, we would translate out the unique existentials (∃!x) into their full explicit form, but the proof remains valid.

1

Major premise. All hounds are creatures that howl when they see strangers. (All M are P)

Minor premise. The hound at the mansion didn't howl when he saw the thief. (No M are S)

Conclusion. Therefore, the thief was not a stranger. (No S are P)

The syllogism is AEE in the third figure. This syllogism is invalid. If the major term (stranger) is distributed in the conclusion (it is), then it must be distributed in the premises (it is not).

The reasoning certainly sounds valid, but it does not write that way.


Added edit. These statements describe the line of reasoning:

Premise: If a hound sees a stranger, the hound howls.

Hypothetical assumption: The thief was a stranger and the hound saw him.

Hypothetical conclusion: Therefore, the hound howled. Modus ponens

Premise: But the hound did not howl when he saw whomever he saw.

Final conclusion: Therefore, the hound did not see a stranger. Modus tollens

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