UC Berkeley, Philosophy 142, Spring 2016

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1 Logical Consequence UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Intuitive characterizations of consequence Modal: It is necessary (or apriori) that, if the premises are true, the conclusion is true. Provability: It is possible to prove the conclusion from the premises. No formal counterexamples: It is impossible to find a formal counterexample (a substitution instance, preserving the argument s form, where the premises are true and the conclusion false). Tarski s account develops the no formal counterexamples idea. 2 From Tarski s account to the modern model-theoretic account Remove intermediate step of variables; directly interpret nonlogical constants. Variable domain. 3 Interpretational and representational semantics Distinction due to John Etchemendy [3]: Representational models represent different situations, different ways the world could be [holding meanings fixed]. Interpretational models represent different meanings the expressions could have had [holding the world fixed]. 4 Prawitz s criticism of the Tarskian account What do we want in a valid argument, besides truth-preservation? that the truth preservation be due to logical form that the truth preservation be necessary Prawitz [4] criticizes Tarski s account for not capturing the modal or epistemic element of consequence: It is said that with the help of valid inferences, we justify our beliefs and acquire knowledge. The modal character of a valid inference is essential here, and is commonly articulated by saying that a valid inference guarantees the March 10,

2 5. Standard problems with proof-based accounts truth of the conclusion, given the truth of the premises. It is because of this guarantee that a belief in the truth of the conclusion becomes justified when it has been inferred by the use of a valid inference from premises known to be true. But if the validity of an inference is equated with (1) [counterexamplefree] (or its variants), then in order to know that the inference is valid, we must already know, it seems, that the conclusion is true in case the premises are true. After all, according to this analysis, the validity of the inference just means that the conclusion is true in case the premises are, and that the same relation holds for all inferences of the same logical form as the given one. Hence, on this view, we cannot really say that we infer the truth of the conclusion by the use of a valid inference. It is, rather, the other way around: we can conclude that the inference is valid after having established for all inferences of the same form that the conclusion is true in all cases where the premises are. [4, p. 675] How persuasive is this argument? (Compare a very similar argument in Etchemendy [3, p. 93].) 5 Standard problems with proof-based accounts 1. Proof is system-relative, but it seems arbitrary to link the definition of consequence to any particular formal system. 2. If we define consequence in terms of provability in some system, we face the problem that some systems are not sound. If we say in some sound system, our definition is circular, since sound just means that you can t derive anything in the system that isn t a logical consequence. 3. Not all proof systems are complete. Gödel s incompleteness theorem implies that, at least in the case of higher-order logics, there are always going to be logical consequences that can t be proved in a given system. 6 Prawitz s strategy Give a purely proof-theoretic account of what it is for a derivation to be a valid argument: Certain arguments (called canonical arguments ) count as valid because they are self-justifying. These will be applications of the introduction rules for the constants. Introduction rules are self-justifying because they determine the meanings of the expressions they introduce. Other arguments count as valid if there is a procedure for converting valid arguments for their premises into canonical proofs of their conclusions. March 10,

3 7. Carnap on meaning and inference 7 Carnap on meaning and inference Rudolf Carnap, Logical Syntax of Language: Up to now, in constructing a language, the procedure has usually been, first to assign a meaning to the fundamental mathematico-logical symbols, and then to consider what sentences and inferences are seen to be logically correct in accordance with this meaning. Since the assignment of the meaning is expressed in words, and is, in consequence, inexact, no conclusion arrived at in this way can very well be otherwise than inexact and ambiguous. The connection will only become clear when approached from the opposite direction: let any postulates and any rules of inference be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols. By this method, also, the conflict between the divergent points of view on the problem of the foundations of mathematics disappears. For language, in its mathematical form, can be constructed according to the preferences of any one of the points of view represented; so that no question of justification arises at all, but only the question of the syntactical consequences to which one or other of the choices leads, including the question of non-contradiction. [2, p. xv] Old way: means if... then. Now, what rules are valid given this meaning? New way: here are some rules for ; these determine a meaning for it. 8 Prior s Tonk Prior s Runabout Inference Ticket [6] is a tongue-in-cheek response to this Carnapian line of thought. Look first at how Prior describes the view he s criticizing: The meaning of and can be completely given by laying down the standard introduction and elimination rules. Anyone who has learnt to perform these inferences knows the meaning of and, for there is simply nothing more to knowing the meaning of and than being able to perform these inferences. In particular, you don t have to grasp some concept that goes beyond what you get just by learning these rules. Prior asks, How do we know that for any two statements P and Q there is a statement with the properties ascribed to P&Q, i.e. a statement from which P and Q can both be derived, and which follows from P and Q together?...on the view we are considering such a doubt is quite misplaced, once we have introduced a word, say the word and, precisely in order to form a statement R with these properties from any pair of statements P and Q. The March 10,

4 9. Responses to Prior doubt reflects the old superstitious view that an expression must have some independently determined meaning before we can discover whether inferences involving it are valid or invalid. With analytically valid inferences this just isn t so. Now, the critical part. Prior points out that if all this is right, we can introduce a connective tonk governed by these rules: tonk tonk-intro A A tonk B tonk-elim A tonk B B These are perfectly clear rules, and we stipulate that tonk is the connective governed by them. So, these inferences are valid just in virtue of what tonk means. The nifty thing is that in a language with tonk we can derive anything from anything! Which, you might think, is bad. 9 Responses to Prior 9.1 Stevenson s Response J. T. Stevenson [7] replied: this just shows that we need to take the semantic viewpoint. To introduce a connective, you need to give a semantics (e.g. a truth table). The rules are then justified in terms of the semantics. The idea that we can define a connective in terms of inference rules governing it is just misguided. So, the problem with tonk is that there doesn t exist any truth table that validates these rules. In addition to existence, Stevenson points out, there are issues of uniqueness. Suppose I define % as follows: %-intro A A % B I haven t said enough to pick out exactly which connective % is. Here s what we know about the truth table: A B A%B T T T T F T F T? F F? But the? s can be filled in any way we like and the rule will still hold. We could make them both F, so that A%B is equivalent to A, or we could make it like disjunction, or we could make them both T. Stevenson concludes that we have to introduce connectives semantically and justify the rules in relation to the semantics. March 10,

5 9.2 Belnap s Response 9.2 Belnap s Response Belnap [1] thinks this is an overreaction to Prior s article. He contrasts the synthetic mode of explaining parts in terms of wholes with the analytic mode of explaining wholes in terms of parts. Both are useful, he thinks, in logic as elsewhere. Prior doesn t show that we have to give up the synthetic mode; rather, that we need to understand its limitations. It would... be truly a shame to see the synthetic mode in logic pass away as a result of a severe attack of tonktitis. So, what, according to Belnap, goes wrong with tonk? The problem is that the rules for tonk are inconsistent with assumptions we ve already made. So tonk is bad in just the same way as Peano s?, which is defined as follows: a b? c d = d e f a + c c + d This definition seems clear enough, but it leads to bad results! 1 1?1 2 = 2 3 and 1 1?2 4 = 3 5 But we had already assumed that 1 2 = 2 4, and that equals can be substituted for equals yielding equals. Unless we give up these antecedent assumptions, we can now derive 2 3 = 3 5. What antecedent assumptions does tonk contradict? Well, we have assumed that the following rules exhaust the universally valid statements about deducibility that can be made without using any logical constants: Identity A A Weakening A 1,...,A n C A 1,...,A n B C Permutation A 1,...,A i, A i+1,...,an C A 1,...,A i+1, A i,...,a n C Contraction A 1,...,A n, A n C A 1,...,A n C Transitivity A 1,...,A m B&C 1,...,C n, B D A 1,...,A m, C 1,...,C n D Now, if we add tonk, we get another universally valid rule that we didn t have before: Anything Goes A B So the addition of tonk is not a conservative extension of our earlier system. The addition of a new connective C is a conservative extension of an old system iff all new statements of deducibility (i.e. all those that weren t implied by the old system) contain C. That is: nothing not involving the new vocabulary can be proved that wasn t provable before. conservative extension March 10,

6 10. Prawitz s Response Belnap appeals to the fact that adding tonk would give us a non-conservative extension of our original theory as a way of cashing out the idea that there is no such connective as tonk in proof-theoretic terms, without appealing to semantics, truth tables, etc. So, conservativeness is a proof-theoretic analogue to the existence of a truth-function... Belnap also supplies a proof-theoretic analogue of uniqueness: To say that plonk describes a unique connective is to say that if another connective plink is given the same introduction and elimination rules, then they are proof-theoretically equivalent: Aplonk B Aplink B and Aplink B Aplonk B. Check your understanding by convincing yourself that the rules for the connective %, described above, do not satisfy Belnap s uniqueness requirement. That is: if we had two connectives, % 1 and % 2, governed only by the introduction rule for %, we could not show that A% 1 B A% 2 B and A% 2 B A% 1 B. Then show that the standard introduction and elimination rules for conjunction do satisfy Belnap s uniqueness requirement. That is, if you have two connectives, and &, that satisfy these rules, then you can prove that A B A&B and A&B A B. (This is not a formal assignment to be turned in, but you should do it so you understand the point.) 10 Prawitz s Response Prawitz s approach [4] is a bit different from Belnap s. Belnap has given up, effectively, on the idea that introduction rules are self-justifying. On Belnap s view, a set of rules is justified only if it yields a conservative extension, and that s something that might require an external guarantee or proof. So, the rules are not self justifying. Prawitz wants to keep the idea that introduction rules for connectives are self-justifying. (For a list, see p. 685, and note how bottom and negation are handled.) In [5] he puts the point this way: if somebody asks why the rule for &-introduction... is a correct inference rule, one can answer only that this is just part of the meaning of conjunction: the meaning is determined partly by laying down that a conjunction is proved by proving both conjuncts, and partly by understanding that a proof of a conjunction could always be given in that way. And, in the more recent paper that you read: this amounts to making inferences by introduction valid valid by definition, so to say [4, p. 694]. How does he avoid Prior s criticism? By giving up the idea that elimination rules are self-justifying. March 10,

7 REFERENCES REFERENCES The view, rather, is this: you can stipulate that the connective is governed by whatever introduction rules you like. (There are formal constraints: the rule must introduce just one connective, and the premises must be subformulas of the introduced sentence, etc.) Since you can t get a non-conservative extension just by adding introduction rules, 1 we re okay so far. What about the elimination rules? We show that they are valid by showing that anything that can be proved using elimination rules could, in principle, be proved without them. (This process of eliminating elimination rules is called normalization.) The details are discussed in the handout on Prawitz s proof-theoretic account of consequence. normalization References [1] N. D. Belnap. Tonk, Plonk and Plink. In: Analysis 22 (1961), pp [2] R. Carnap. The Logical Syntax of Language. Trans. by A. Smeaton. Open Court classics. Open Court, [3] John Etchemendy. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press, [4] Dag Prawitz. Logical Consequence From a Constructivist Point of View. In: The Oxford Handbook of Philosophy of Mathematics and Logic. Ed. by Stewart Shapiro. Oxford: Oxford University Press, 2005, pp [5] Dag Prawitz. Remarks on Some Approaches to the Concept of Logical Consequence. In: Synthese 62 (1985), pp [6] A. N. Prior. The Runabout Inference-Ticket. In: Analysis 21 (1960), pp [7] J. T. Stevenson. Roundabout the Runabout Inference-ticket. In: Analysis 21 (1961). 1 Convince yourself of this! March 10,

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