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What is the Truth Table Analysis of the below argument?

If a football is more demanding than tennis and John is good football player, then John can become a good tennis player.But if football is not more demanding than tennis, that does not hold.Hence, John is a good player.

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    The logical conncetive is "If ___, then ...", i.e. the conditional: P → Q is false only when P is true and Q is false.
    – Mauro ALLEGRANZA
    Commented Oct 22, 2014 at 6:03
  • But your argument is ambiguous; what does it mean : "if football is not more demanding than tennis, that does not hold". What is "that" ? The argument is : (1) A and B → C ; (2) ¬A → ¬ ? From (1) and (2) C follows. If (2) is simply ¬A, the argument does not conclude, i.e. bot C and ¬C are "compatible" with the premise.
    – Mauro ALLEGRANZA
    Commented Oct 22, 2014 at 8:53

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Then is not really a logical connective at all. It is used in conditionals, but it is just an adverbial discourse marker. What it seems to show is that the speaker is prepared to assert what's coming next on the basis of whatever it is that then is referring back to being the case. It doesn't only occur in conditionals:

  • A: I'm leaving on Tuesday. B: Then you're going to miss the party.
  • You shouldn't do that, because you'll then be open to accusations of plagiarism.

Notice that there are no sentences in which the inclusion or omission of the adverb then makes any difference whatsoever to the truth conditions of the sentence. In the Original Poster's question the sentence has exactly the same truth conditions whether or not then is omitted:

If a football is more demanding than tennis and John is good football player, then John can become a good tennis player. But if football is not more demanding than tennis, that does not hold. Hence, John is a good player.

If a football is more demanding than tennis and John is good football player, then John can become a good tennis player. But if football is not more demanding than tennis, that does not hold. Hence, John is a good player.

The fact that adding or omitting then can never change our assessment of whether a conditional sentence was true, is evidence of the fact that it is not actually a logical connective. It merely adds a higher order comment about the speaker's attitude to a proposition.

Probably the best known work on the use of then in conditionals is by Sabine Iatridou On the contribution of conditional THEN. However, readers beware, the grammaticality judgements therein are, in my opinion, rather suspect in several places.

Iatridou pointed out that we are most likely to use then when we would not expect Q to happen unless P. There are many other interesting observations as well.

In short, however, then is not a logical connective at all.

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    Why was this deleted?
    – Joseph Weissman ♦
    Commented Oct 21, 2014 at 14:56
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Assuming the last sentence means "John is a good football player", and "that does not hold" applies to the entire previous statement, I got:

A: Football is more demanding then tennis 
B: John is a good football player 
C: John can become a good tennis player 

A and B -> C 

-A -> -(A and B -> C)

That's not a truth table, but is possibly helpful.

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