Shantila's Inside Logic #17 Consistency; and the Law of Non-contradiction A set of sentences is consistent just in case it would be possible for all the sentences in the set to be true together. If so, a contradiction cannot be derived from that set of sentences. So a set of sentences is not consistent (that is, it is inconsistent) just in case a contradiction can be derived from that set. Consider, for example, this set of four sentences. Although it may not be obvious given a quick reading of these four sentences, it is not possible for all four of these sentences to be true. This set is inconsistent.
We can represent these sentences as follows, using the following symbols. J: Joe is happy. H: You are happy. L: You are sharing your lunch. G: You are giving some of your lunch to Joe.
Is this set of four sentences consistent? The answer is No! We can show the set is not consistent by deriving a contradiction from these four sentences, as follows.
What this derivation shows is that the four sentences (the premises here) are inconsistent (that is, not consistent). Line 10 says that given the four premises, it follows that J&~J, which is a contradiction: this set of four sentences entails a contradiction. --And this means that at least one of the sentences must be false. It is impossible for all four sentences to be true. Given any of the three as premises in this example, we could use RA to derive the negation of the other premise. On line 10 we have a proof of a contradiction based on lines 1,2,3, and 4. So, for example, we can add line 11 (using RA, of course) to show that the first 3 premises entail ~~G, and so also then line 12, G.
But equally we could have negated line 1 based on the other three sentences. So a different step from line 10 would be as follows:
Logic by itself will of course not tell use which of the four sentences to reject -- but it does tell us that certainly at least one of the four sentences must be false. * Here's another example, from current cosmology and physics. Many cosmologists currently believe the following. (1) Most of the matter in the universe is "dark" (i.e. cannot be detected from the light which it emits). Dark matter is "stuff" which cannot be seen directly (at least, we have no means of "seeing" it at this point.) So what makes us think that it exists at all? Its presence is inferred indirectly from the motions of astronomical objects (stars, galaxies and galaxy cluster/superclusters). These motions cannot be explained without assuming there is great deal more matter in the universe than we understand. The term "dark" matter is used to refer to this stuff that needs to be posited in order to explain the motions that are observed. (This is actual science, not science fiction!) But some theorists have argued against this view, claiming (2) If most of the matter in the universe is dark, it would have prevented the existence of everything we know in our cosmos. According to this alternative theory, the existence of dark matter would undermine the very existence of the things with which we are familiar. Common sense kicks in to say (3) If dark matter prevented the existence of everything we know in our cosmos, then nothing we know about exists now; and yet (4) It is not true that nothing we know about exists now! These four sentences can be represented as follows: (1) D (2) D>P (3) P>N (4) ~N It is easy for us to show that (1)-(4) here comprise an inconsistent set of sentences.
We can see (and everyone will agree) that (1)-(4) is an inconsistent set. At least one of them is false. Which one? It is an interesting dispute. The debate will center on whether (1) or (2) should be rejected since (3) and (4) are obviously true. --Of course it might turn out that, based on further research, both (1) and (2) should be rejected. * The basic logical notions we have been examining so far (if, and, or, not) are all quite simple. We have understood these notions in their basic forms (as represented in our Rules) since we were children -- as we have said several times before. Yet our minds are complex. We each have many diverse beliefs and values. And it is quite possible that there are inconsistencies in our minds. One potential value of logic is to alert us to this possiblity. There is no red flag that shows us automatically that the sentences about Joe and lunch (or the sentences about dark matter) are inconsistent. To see that they are inconsistent we have to "put two and two together." Now when we look closely, we easily can do that-- we can see that they are inconsistent! Indeed it is as simple as 2+2=4 once we understand our rules! But prior to looking closely, we might not realize that there is a problem with the inconsistent sets of sentences. We might have had the intuition they do not cohere together very well but we might not have been able to say why not. Given the tools we now have in hand, we can spell out what is wrong: the sets are inconsistent. At least one of the sentences in each set must be false. If a set of sentences is inconsistent, we can know on the basis of logic alone that at least one of the sentences must be false. Of course, logic by itself is not going to tell us which of the sentences is false. * The Law of Non-contradiction Since the rule RA reduces our assumptions by one (just as does CP), RA also can be used to prove logical truths even when CP is not used. For example, consider the proof of the following logical truth: 108 } ~(P&~P)
This might look like a trick, but it is not a trick. It is just a very simple proof. Recall that the rule RA says that if you make an assumption and can prove a contradiction, you can negate the assumption, where the premises of the new line are all those of the contradiction except the number of line negated. We have used RA correctly here. The unusual feature is that we simply assumed a contradiction on line 1, so given our rule RA, the derivation could not be simpler: the line where we make the assumption (line 1) is exactly the same as the line where we get the contradiction! This logical truth ~(P&~P) is known traditionally as the "Law of Non-contradiction." Notice that the proof of this law is simple, given our rule RA, because accepting RA as a rule means pretty much the same thing as accepting ~(P&~P) as a logical truth. There is nothing mysterious or tricky going on here. Intuitively it makes sense that we should be able to prove this sentence as a logical truth, since it says simply that no sentence P is such that both P and ~P are true. No sentence is both true and false. There is a passage in the writings of the Greek philosopher Aristotle, where in discussing this "law" of logic, he says that you can't really have a reasonable discussion with a person who does not accept it. For those "intransigent enough to deny even the principle of non-contradiction," the Aristotelian philosopher Avicenna advises "putting them to the fire or beating them, until they admit that pain is different from nonpain." I think he probably was joking but I'm not sure.
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