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This is a question from Course Notes of Introduction to Mathematical Proofs by Shay Fuchs. It asks:

In the following cartoon, the dog concludes that he is a cat. Find the flaw in his argument. Which connectives and quantifiers are used? Can you relate this to any of the standard truth tables?

The cartoon says,"All cats have four legs. I have four legs. Therefore, I am a cat."

EDIT I am stuck as to what are the propositions here, specially how to translate the first sentence_All cats have four legs. I can thankfully manage the rest.

Though it's evident the argument is false, but what can be the best way to convince myself by a sound logical argument?

My Attempt
Utilizing the comment of TheSilverDoe, I took:
A = "I am a cat"
B = "I have four legs"

The two premises here are:
A $\to$ B = "If I am a cat then I have four legs" = "All cats have four legs"
B = "I have four legs"

The Dog is infering:
A = "I am a cat"

We can construct the truth table here:

$$\begin{array}{|c|c|c|c|c|} \hline \\ \ A & B & A \to B & P = (A \to B).B & P \to A \\ \hline \ 0 & 0 & 1 & 0 & 1 \\ \hline \ 0 & 1 & 1 & 1 & 0 \\ \hline \ 1 & 0 & 0 & 0 & 1 \\ \hline \ 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$$

We do not get a tautology in the last column of the truth table which leads us to the fact that the conclusion is wrong by the principle of syllogism.

My Doubt:

Now here we can note the last column is identical to the truth table of B $\to$ A which in turn means: "If I have four legs then I am a cat" and that is the logical equivalent of what the Dog is saying and exactly the converse of A $\to$ B = "All cats have four legs". Now we know that the converse of a conditional statement may/may not be true. The dog is asserting that the converse is essentially true, which is the flaw as per my understanding. Is that correct?

Also here apparently no quantifiers were used although the question is asking for it. I speculate that there can be a better interpretation of the problem if we can use some quantification but I do not know how.

EDIT-1

I also got an explanation here.

Please suggest if my argument is okay or if there's a better approach. Thank you.

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    $\begingroup$ If $A=$"I am a cat" and $B=$"I have four legs", the dog is saying that $((A \Rightarrow B) \wedge B) \Rightarrow A$. $\endgroup$ Commented Aug 24, 2020 at 21:21
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    $\begingroup$ What do you think and why? $\endgroup$ Commented Aug 24, 2020 at 21:23
  • $\begingroup$ @TheSilverDoe Thanks! I got it! :) $\endgroup$
    – S.S
    Commented Aug 24, 2020 at 21:23
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    $\begingroup$ @ShatabdiSinha Maybe you could write a self answer? $\endgroup$
    – user400188
    Commented Aug 25, 2020 at 3:06
  • $\begingroup$ @user400188 I have added what I have understood so far and also specified my doubt. Kindly have a look if there can be some improvisation. $\endgroup$
    – S.S
    Commented Aug 25, 2020 at 9:38

1 Answer 1

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The meaning of phrases of the form "all X are Y" have had different interpretations in the history of logic. One of those interpretations is:

"all X are Y" means the same as "if X, then Y".

I think that logicians of the terminist school (in contrast to what can get called the "Fregian" school of logic) such as the Fred Sommers have had this interpretation recently.

Thus, the truth table/conditional approach you used isn't necessarily flawed. Under that interpretation, I find your analysis convincing. One might also classify the argument as having this form under that interpretation:

(B→A)
A
-----------------
B

Arguments of this form get classified as a fallacy known as "affirming the consequent", since the consequent of a conditional gets affirmed as if it's sufficient to infer the antecedent of a conditional. But, if B were false, A were true, then both (B $\rightarrow$ A) and A are true, while B is false. The dog is, yes, asserting the converse of (A$\rightarrow$B). Under this interpretation no quantifiers are used at all.

Under another interpretation of "all X are Y" it has the same meaning as "for all p, if p has property X, then p has property Y". So, "all cats have four legs" has the same meaning as "for all p, if p has the properties of a cat, then it has the properties of a four-legged animal." This could get translated as:

∀p(Cp→Lp)

where "C" means "has the properties of a cat" and "L" means has the properties of a four-legged animal.

"I have four legs" could get translated as:

Li

with 'i' being a constant instead of a variable.

And the conclusion of the argument would then translate as

Ci

Under this interpretation, one quantifier does get used, but the flaw in the argument is similar.

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  • $\begingroup$ Thank you for the elaboration. I feel more or less it's the same I had supposed it to be. The fallacy of converse is the flaw here. $\endgroup$
    – S.S
    Commented Aug 25, 2020 at 18:32
  • $\begingroup$ @ShatabdiSinha You're welcome. $\endgroup$ Commented Aug 25, 2020 at 19:13

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