Is the law of excluded middle a law of logic? Introduction I will conclude that the intuitionist s attempt to rule out the law of excluded middle as a law of logic fails. They do so by appealing to harmony as a constraint on the deductive rules we have in our logic, but I shall show that such an appeal throws the baby out with the bathwater. This is because an appeal to harmony precludes us from incorporating identity into our logic. Additionally, a failure to give a plausible and precise definition of harmony suggests that harmony-type approaches to logic fail. Conceptual Clarification The law of excluded middle is the syntactic law that: P P In other words, it is provable from no premises that; it is a theorem that P P. A syntactic law, unlike a semantic law, introduces a claim that is provable from the deductive rules alone no premises are needed. A semantic law is a claim that is true on all possible interpretations. This is important because the syntactic law of excluded middle is sometimes conflated with the semantic principle of bivalence, x(px (Tx Fx) Which operates on the universal domain, and where Px: x is a proposition, Tx: x is true, and Fx: x is false. In English, it claims that every proposition is either true or false. They are not the same law; I shall focus exclusively on the syntactic law of excluded middle LEM. A logic is constituted of a language, deductive system and semantics. The language is the vocabulary of primitive symbols (,, ) and grammar (a definition of a sentence). The deductive system is the rules that tell us what follows from what, and thus which particular theorems are possible in that logic. The semantics is a specification of what, in the world, the symbols refer to. Implications of losing LEM The law of excluded middle is a law of logic, if and only if, our logic is (1) correct and (2) contains a deductive system such that LEM is a theorem. To deny (2), our deductive system shouldn t be able to prove LEM from no premises. Then, our logic would have to exclude all those rules that enable LEM to be a theorem. This includes tertium non datur (TND), which states 1 : 1 I have directly copied these proofs from O Griffiths Part IB lecture handouts on Intuitionism.
Since TND can prove LEM: We d also have to ditch double negation elimination (DNE), since it, too, can prove LEM: As well as Pierce s law, which states that (A B) A) A: Then, we d also have to get rid of any derived rule, which is provable only from DNE. This includes one of the CQ rules that takes us from xfx to x Fx:
And we d also have to get rid of any derived rule, which is provable only from TND. This includes a De Morgan rule that takes us from (A B) to A B: So in summary, ditching LEM actually has wider implications. We also have to get rid of TND, DNE, Pierce s Law, a CQ rule and a DeM rule. Keeping all other deductive rules of classical logic, we have created a system equivalent to the deductive system of Intuitionistic logic. Intuitionistic Logic Intuitionistic logic has the same language as classical logic. It deviates from classical logic insofar as it gets rid of the above deductive rules. Having lost TND, intuitionists use contradiction introduction as a surrogate for a negation elimination rule. Intuitionistic semantics also deviates from classical logic; the standard semantics is called the BHK interpretation, after Brouwer, Heyting and Kolmogorov, but my focus is on the syntactical deductive laws. Intuitionistic logic is not a three-valued logic, nor does it claim (P P). It merely doesn t assert LEM as a theorem. It s important that it doesn t claim (P P), because then we could, via De Morgan, derive P P, which is a contradiction that even intuitionistic logicians recognise 2. Having articulated a logic which denies that the law of excluded middle is a law of logic, I shall examine the motivations for embracing Intuitionistic logic in the first place. Tonk and Harmony Which deductive rules should we have in our logic? There seem to be some deductive rules which we definitely shouldn t have. The classic example is of tonk 3. The introduction and elimination rules for tonk are: Introduction: If you have A on line m, you can show A tonk B on some further line, citing the rule TI m. Elimination: If you have A tonk B on line m, you can show A on some further line, or (inclusive or) you can show B on some other further line, citing the rule TE m. 2 O Griffiths, Lecture Series 3 A N Prior, The Runabout Inference-Ticket
As such, tonk-introduction is similar to disjunction introduction, and tonk-elimination is similar to conjunction elimination. However, tonk is problematic, insofar as it lets us move via deductive rules from any proposition to any proposition, e.g. (for any A and B): 1 A 2 A tonk B TI 1 3 B TE 2 And indeed, it can even result in contradiction from any premise A: 1 A 2 A tonk A TI 1 3 A TE 2 4 A A I 1, 3 5 Contradiction Contradiction introduction Tonk is problematic because it is clearly a terrible connective - in some way, its introduction and elimination rules are faulty. One can conceive of the broader issue as a demarcation problem: how do we demarcate the good deductive rules from the bad? What makes tonk rubbish, and the proper rules good? One answer is the quality of harmony 4. Roughly, an elimination rule and introduction rule are said to be in harmony with one another iff they tell us no more and no less than they ought. For example, tonk is not harmonious, because we can move from any premise, to any conclusion, so tonk can tell us far more than it ought. There is another motivation for implementing harmony as a constraint on the possible deductive rules we should accept. Assuming we already accept the following principle, harmony is a good constraint because it fulfills the principle of innocence 5, which states: it should not be possible, solely by engaging in deductive logical reasoning, to discover hitherto unknown (atomic) truths that we would have been incapable of discovering independently of logic Harmony fulfills the principle of innocence because it guarantees that our deductive rules won t tell us too much. (I am being deliberately vague about the notion of harmony because - as I shall argue - I doubt whether there is any way to precisely define harmony without the concept losing its intended force) Now that we have our answer to the demarcation problem, we can implement it. We can get rid of all the rules that are not harmonious, keeping only those that are harmonious. Application of Harmony to Classical Negation Harmony definitely excludes the tonk-rules from our deductive system. But some have even argued that it excludes a classical elimination rule. In particular, it excludes DNE. This is because we can use DNE to arrive at a conclusion that we couldn t have otherwise established, as in the following proof: 4 N Belnap, Tonk, Plonk and Plink 5 F Steinberger, What Harmony Could and Could Not Be
In the above schematic proof, we arrive at A, which we couldn t have otherwise arrived at. Therefore, DNE isn t harmonious. Therefore, DNE shouldn t be one of our deductive rules. Therefore, we should accept a logic that doesn t include DNE as a deductive rule. One candidate (albeit not the only possible candidate) for this is intuitionistic logic. So, in the absence of any feasible alternatives, we should embrace intuitionistic logic. Intuitionistic logic doesn t include LEM as a law of logic. So LEM is not a law of logic. Criticism 1: Harmony excludes Identity I shall focus on two criticisms of the above argument for intuitionistic logic (and thus indirectly for the LEM). First, harmony is too strict a constraint on our deductive rules, because it excludes the deductive rules around identity, and so throws the baby out with the bathwater. If we embrace harmony, we have to get rid of the identity rules. We need the identity rules. So we shouldn t embrace harmony. Identity elimination is plausibly not a harmonious rule, because we can start from the premises a=b and Fa, to arrive at Fb, which we could not have otherwise arrived at: But we need some kind of identity elimination rule. So harmony is too strong. Although it successfully gets rid of tonk, in doing so, it gets rid of too much. So we should reject harmony as a constraint. We thus lose any reason to reject DNE as a deductive rule, and so too lose any motivation for intuitionistic logic. Read s harmonisation of identity Read 6 attempts to defend harmony, by giving a new identity elimination rule that is harmonious. The rule, he suggests, could operate: If (1) F does not appear in any undischarged assumptions other than i 6 S Read, Identity and Harmony
(2) On some line i we assume Fa, and within the same subproof conclude Fb we can conclude a=b on some further line, citing the rule =I However, even if Read s new rule is harmonious, it must be false. This is because, as Griffiths 7 points out, Read s identity rule assumes the identity of indiscernibles. Read s rule, in English, can be paraphrased as For all of a s properties, if b has those same properties, then a is identical to b. This is equivalent to the identity of indiscernibles, which states that: F x y((fx Fy) x=y) However, even if two objects have exactly the same properties, this is insufficient to show that they are one and the same object. For example, consider a universe in which the only existing objects are two black spheres of equal size, orbiting around each other. They share the exact same properties. And yet they are not identical. So no matter how many properties we show two objects to share, this is insufficient to show that they are identical. So Read s new rule must be wrong even if it is harmonious. So his defence of harmony fails, and so too does his indirect defence of intuitionistic logic. Criticism 2: Putting pressure on the very concept of harmony Just what is harmony anyway? My rough definition of harmony claimed that harmony obtains iff the introduction and elimination rules of a connective don t tell us anything new. What constitutes something new? It can t mean not obvious, because obviousness is a psychological quality, and harmony is a logical relation as Frege repeatedly maintained in Begriffsschrift, logic is not psychology. Also, if new meant not obvious, then, given that different claims have different degrees of obviousness for different people, we d be forced to say that different rules are harmonious for different people, and so different people should accept different logics dependent on what it obvious to them and not obvious to them. This seems wrong. We can redefine it to be a logical relationship. For example we can define harmony as that relation that obtains iff the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from logical laws. But this, too, is problematic. Given that all logical laws are proved from, or are identical to, the introduction and elimination rules of the connectives, we can rephrase the definition, as the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from the introduction and elimination rules of the connectives. And if we embrace this definition of harmony, then all connectives pass the test of harmony. All connectives will pass the test of harmony, because they will always tell you something you can deduce by applying that very same connective. For example, even tonk passes this test of harmony, because tonk will never tell you something that you can t deduce from the connectives (including tonk itself!). In other words, tonk will never surprise you, if you already accept tonk. So this definition is too liberal it lets in too many connectives. So perhaps we can restrict the latter clause of the above definition to only include other connectives than the one we re currently testing. On this definition, the tonk rules can t be used to show that tonk is harmonious. Our definition is now: Harmony is that relation that obtains iff the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from the introduction and elimination rules of the other connectives. 7 O Griffiths, Harmonious Rules for Identity
However, this, too, is problematic, because it doesn t determine what the other connectives are. The upshot is that this definition seems to let in too many rules as a test for the particular connective we re testing. For example, if we re testing tonk for harmony on this definition, we just check and see if tonk can tell us anything that, for example,,, and any other non-tonk connective can t tell us. This isn t so problematic. But then what happens when we test for harmony? We just check and see if can tell us anything that, for example,,, tonk can t tell us. But it seems wrong to include the tonk-rules (or any other bizarre tonk-type rules) as a test to see if is harmonious. In other words, the phrase the other connectives in the above definition is still too indeterminate, because it lets us use tonk-typerules to assess other connectives. Alternatively, we could just stipulate that only the connectives we already accept should be included in the domain of other connectives. This excludes tonk-type-rules, because presumably nobody actually accepts tonk-type rules. The problem with this approach is that it makes harmony relative to the deductive rules that one already accepts, which seems wrong. On this account of harmony, different rules may be harmonious for different people. So it runs into the same sort of trouble that defining harmony in psychological terms ran into. All this trouble in defining harmony suggests that there is no coherent formulation of the concept. As such, we d be better ditching the concept. As before argued, this leaves us with no motivation to get rid of DNE, and so, no motivation to embrace intuitionistic logic, and so, no motivation to exclude the law of excluded middle from our logic. Conclusion I have shown that one attempt to argue for intuitionistic logic, which excludes the law of excluded middle, fails. If we use harmony as a constraint on our deductive rules, we throw the baby out with the bathwater by removing identity also. But an appeal to harmony has even deeper problems: there is doubt as to whether harmony can even be precisely defined in the first place. As such, one attempt to show that the law of excluded middle is not a law of logic, fails.