# Order of Premises in Logic

Does anyone have any examples of order-sensitive arguments (that is, arguments in which different conclusions can arguably be drawn depending upon the order that the premises are presented)?

In classical logic, and indeed in most logics, the order of the premises does not matter. This is encoded in the fact that classical logic, as do most logics, contain commutativity as a structural rule (which basically says order doesn't matter). So if you're looking for an argument in classical logic where order matters to the validity of the argument, you won't find any. There might be a psychological impact on presenting premises in some order or another (for instance, putting premises in different orders might make you implicitly assume something you weren't assuming before), but that's a long way from being able to draw different conclusions from those premises.

With that said, there is, in fact, a logic where order of premises do matter. It's known as "ticket entailment." It was (possibly) suggested by Ryle in 1949, and later developed by Anderson and Belnap in more depth (see Anderson and Belnap's Entailment, Vol. 1, section 6 for more details). Roughly, the idea is to treat conditionals ("law-statements") as "inference tickets" which allow you to get from the antecedent to the consequent. That is, when using (say) modus ponens, the conditional must always be the major premise, and never the minor premise. What this comes down to is that you must have established your conditional statement before you've established the antecedent in order to use modus ponens. This may sound weird, but there are some philosophical reasons to think that this (somewhat minimalist) system is interesting.

Sure, there is. Linear Logic is one example I know of. In classical logic, true values are always true. In linear logic, "truth" is an expensive resource. So, given a set ( or "multimap" in linear logic) of valid statements `Γ={A→B, A→C}`, and given another hypothesis `A`, in classical logic, you can derive `B⋀C`. In linear logic, given `Γ={A⊸B, A⊸C}` and `A`, either you can derive `B` or you can derive `C`, but not both.

Think of this as having just one apple, and two hungry horses (B and C). Either B is fed or C is fed. You don't have an apple after you feed one of them.

The application domain for linear logic is very different from that of classical logic. It is used by many formalisms in linguistics. Here, words have properties that are 'consumed'.

• What would be the philosophical motivation to this semantics?
– Lawl
Apr 29, 2013 at 16:31
• @Lawl: I don't know. I have only studied the aspect of it that's been used in linguistics. But the original paper that introduced it is here: iml.univ-mrs.fr/~girard/linear.pdf Apr 29, 2013 at 18:58

Any such example will probably be controversial as it would have implications for open debates on logic. But in general it will probably turn on the same principle by which "and" can express different things based on order. "He went to get beers and drove home," is different from "He drove home and went to get beers" because it suggests temporal ordering. If a murder happened at his home, the argument that he couldn't have been there could be stronger or weaker depending on which of these you use since one might suggest that he was there at the time and the other might not. If you want a more purely logical argument, then you could say, "Premise 1: he went to get beers; premise 2: he drove home; conclusion: He went to get beers before driving home," would apply. If you don't like this way of appealing to order then you're probably not going to like any examples of arguments that appeal to order.