Whay are trying to do ? Are you checking if the arguments are valid ?

P2 is about classes or sets: the set of Tomatoes is not equal to the set of Fruits, or about an individual and a set: Bob does not belong to the set of Fruits ?

Is the second one stated correctly? It does not make sense. You cannot deduce anything from (not A => B and A). And as noted above there is either a category error or an abuse of notation in the premises. Either X is a property or Z is not an object (outside really frilly computing languages).

I am trying to check if the arguments are valid, yes. So would both X and Z need to be properties or sets, or individuals or objects for this to work? I have amended the second one so Z is a singular tomato, if that helps.

By using 'can' does this mean I am able to not necessarily include Z in Y? In other words, not-X is potentially Y and Z included in X, can we therefore state Z is potentially Y?

With elementary logic we cannot translate "can*: either Z is Y or Z is not Y.

@questioningthis I rewrote your question a bit to use more precise language, especially since "can be eaten" invites misinterpretations. Is is better to express the premises in terms of that the objects "is" and "is not" rather than what actions someone else can perform on the objects.

@MichaelK thanks very much for doing that. I was unaware that would not make sense but the edit is a much better framing. EDIT: I have just added to the second example (as a tomato is a fruit, or Z is X) that a tomato is not edible (Z is not-Y). Hope that is fine.

@MauroALLEGRANZA does this now make sense with the edit or is it still not possible to deduce Z included in Y? As a layman, it seems to follow but I am not sure.

@questioningthis As to the question itself: the first conclusion makes sense. The second does not. We do not know if the tomato is edible or not simply from the conclusion that it is a fruit. With P1, you have only made a statement about the edibility of things that are "not X". You have not said anything about the edibility of things that are X. So the second conclusion may be valid, or it may not be valid. But the second conclusion it is not supported by P1 and P2.

@MichaelK Thanks for the comment and I see what you mean, yes. Would it then be valid to add another premise? Stating effectively everything that is X is edible. I have edited to reflect this.

@questioningthis Yes, you would need to add P3: "Everything that is X is not Y" in order to support the second conclusion, or change P1 to be "All things, and only all things, that are not X are Y".

@MichaelK is there a material difference between adding P3 or modifying P1? I ask just in case it affects the logic of the argument. Sorry, I realise I am veering a bit into discussion and am happy to continue this in chat if that is preferable.

@questioningthis I cannot say for sure... formal logic is something I only have a "common sense" understanding of, I do not know the logical operations very good.

@questioningthis...I though though that no matter which one you choose you have not only made it possible to support both conclusions, but you have also made it possible do the inverse: "Is Bob edible?", "Yes, Bob is edible", "Then Bob is not a fruit". You did however introduce a possible problem: Is "Not Edible" and "Inedible" really the same thing? :-D

@MichaelK I think this is the case, yes, but I would like to make my question slightly less ambiguous as per Mauro's suggestions.

@MichaelK Edited to remove 'not edible' and replaced with 'inedible' in all cases; thanks for the catch :)

@questioningthis Actually... I would suggest that you do the opposite, and replace "inedible" with "Not edible". Because your premises deal with whether things are "edible" or "not edible". Only is "not edible" is completely equal to "inedible" can you use *"inedible" as a replacement for "not edible".

@MichaelK Sorry, yes you are right, as I am dealing with X, not-X, Y, not-Y, it would be better to use that syntax. I think in this context I am using not edible for not permissible to eat as opposed not being able to physically eat an object.

I suppose you also must be told that validity is not the same as truth. Because you have a valid argument does not mean the argument works in reality. You seem more interested in math the way you wrote your premises. You could have used standard English. Everything is not can be rewritten. Everything that is not a fruit is edible is equivalent to all non fruits are edible. Everything that is a fruit is not edible is equivalent to some fruits are not eligible. The rewrites are definitely easier to read. You cannot have an odd number of premises. Every two premises force a conclusion.

Hi @Logikal, yes. Is another way of saying this that validity is not the same as soundness? This is how I have conceptualised it previously. As regarding an odd number of premises, could you tell me why this is not permissible? A cursory glance on the matter yields this paper which seems to argue the opposite: scholar.uwindsor.ca/cgi/…