When two or more alternative data are presented, of which only one is valid or relevant to a proposed argument, but we know not at first which the valid datum is, we have the _dilemma_ (_trilemma_, _tetralemma_, &c.) of logicians. In such conditions we have a double process to go through; we must first settle by observation or by an auxiliary argument which of the alternative data to select, and then work out the principal argument in the regular dogmatic form. Suppose we have to determine dialectically the specific gravity of a piece of metal, but do not know whether it is gold or gun-metal. It is evident we must first somehow make up our mind as to its identity, and then proceed on the usual method of argumentation. The 'making up our mind' is probably itself an argument, and might be of this character--'A piece of yellow metal stamped with what appears to be a hall-mark, is more likely to be gold than gun-metal; this piece of metal has traces of such a stamp; so I conclude _it is gold_.' Then we proceed to the principal question--'The specific gravity of gold is 19·26; I have concluded that this object is gold; I conclude further that _it has a specific gravity of_ 19·26.' We may work out all the alternative conclusions first and fix on a datum afterwards, as in deciding how we shall invest our money. 'If I put my money in Consols I shall have a small return with good security; if I buy Patagonian bonds I may have a large interest for a time, but the security is bad.' The next thing to settle is whether in our experience or on accepted principles small profit with good security is, or is not, to be preferred to large profit and bad security: having decided in favour of the former alternative, we now choose our investment dogmatically--'A good security with small profit is to be preferred; Consols are of this character; _they are a suitable investment for me_.' We may be unable to decide for any of the alternative data, but we work out all the possible arguments as hypotheses, and so are prepared in a degree for all the possible events. A person is seen approaching our residence, but we cannot discern whether it is A. B., who is a bore, or C. D., who is an entertaining companion. We argue rapidly--'If it is A. B. _I shall have a bad half-hour_, for he always wearies me; if it is C. D. _I shall have an agreeable distraction_, for he is very amusing.' According to the syllogists, the dilemmatic premises are a statement of alternative data and the choice of one of them, and the inference is the rejection of the remainder: or the rejection may be given as matter of fact and the selection as conclusion. In neither case have we argument.