Abstract
Owen Griffiths has recently argued that Etchemendy’s account of logical consequence faces a dilemma. Etchemendy claims that we can be sure that his account does not overgenerate, but that we should expect it to undergenerate. Griffiths argues that if we define the relationship between formal and natural language as being dependent on logical consequence, then Etchemendy’s claims are not true; and if we define the relationship as being independent of logical consequence, then we cannot assess the truth of the claims without further information. I argue that Griffiths misconstrues Etchemendy’s theory and overstates the first horn of the dilemma: Etchemendy does see the relationship as being dependent on logical consequence, but that does not mean that his claims are not true.
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Notes
Griffiths also takes his dilemma as applying to the concept of logical consequence given by Shapiro (1998, 2005), but in a slightly different manner. Since I will only be concerned with how Etchemendy should respond to Griffiths’ charge, I will leave it to the reader to go back to Griffiths’ text if his comments on Shapiro are of interest.
At one point Griffiths suggest that (2’) and (NO’) are both false when formalisation is understood as being independent of logical consequence (2014: 12), though at other points he suggests that we cannot really tell without further explanation from Griffiths (2014: 15-16).
One might wonder as to the status of (NO’) once it is shown that (2’) is not trivially true. One could of course accept Etchemendy’s argument (1990: 147) for the truth of (NO’). But as Griffiths (2014: note 6) correctly notes, it does not seem like Etchemendy himself should accept it. Etchemendy (1990: 111) discusses formal sentences such as \(\exists x \exists y (x \neq y)\), \(\exists x \exists y \exists z (x \neq y) \wedge (x \neq z) \wedge (y \neq z)\) and the like. These are not model-theoretically valid and neither are their readings, on Etchemendy’s view, informal logical truths, because they make claims about the cardinality of the world. However, Etchemendy ought to have considered \(\exists x (x =x)\), as Griffiths notes. This is a model-theoretic validity, yet by the same reasoning Etchemendy would presumably claim that its reading is not an informal logical truth. Thus here is a counterexample to (NO’), the claim that there is no overgeneration. I take this point to be true, but if anything it shows that (NO’) also is not trivially true, as Griffith claims that it is.
This is not to say that the informal notion of consequence only applies to natural language. In fact, Etchemendy quite clearly states that any language gives rise to a logical consequence relation (2008: 283). The informal notion is therefore not limited to natural languages. We will return to discuss these issues in the next section.
The qualifier ‘with the terms in \({\Gamma }\) and \({\Phi }\) retaining their actual interpretation’ is important when the statements in \({\Gamma } \cup \{{\Phi }\}\) are mentioned rather than used. It serves to make sure that we are checking possible ways the world could have been relative to what the statements in question actually express. There are possible ways the world could be where ‘or’ means what and actually means. In such circumstances an utterance of ‘It is raining or it is not raining’ would be false (at least if we assume that the other terms do not change their meaning). But ‘It is raining or it is not raining’ is still a logical truth in Etchemendy’s informal sense because what ‘It is raining or it is not raining’ actually expresses could not fail to be the case, even in such circumstances.
Logical constant here means no more, and no less, than being given the same treatment as that which we standardly call the logical constants.
I am speaking about logical truth rather than logical consequence here. But the point extends to logical consequence by noting that a logical truth is a consequence—both in the model-theoretic and Etchemendy’s informal sense—of the empty set of premises. The same point applies to what follows.
Actually it is enough when we note the following: if Griffiths’ argument that (2’) is true by default is sound in the case of first order languages, then an analogous argument should be equally sound in the case of propositional languages. And the latter is clearly not the case as the counterexample shows.
Etchemendy’s example consists of two premises and a conclusion rather than being just one sentence.
The same, of course, could be said of the other counterexample as well: while it is true that ‘b is a triangle’ does indeed follow from ‘a is a triangle’ and ‘a and b have the same shape’, the notion of “following from” is not that of logical consequence, rather it is that of analytic consequence.
While it is true that there is no mention of any relationship between formal and natural language in Etchemendy’s own discussion of the squeezing argument (Etchemendy 1990), we have to be careful what consequences to draw from this. Griffiths’ dilemma is meant to apply to Etchemendy’s own notion of logical consequence and the underlying model theory, whereas the discussion of the squeezing argument which appears in Etchemendy (1990) concerns the standard Tarskian definition of model-theoretic consequence. As Tarski’s conception of model-theoretic consequence is not necessarily the same as the model-theoretic conception which underlies Etchemendy’s own notion of consequence, we should be careful not to conclude too much on this basis. Etchemendy discusses his own views of the informal notion of consequence and the role played by the underlying model theory in Etchemendy (2008).
I would like to thank an anonymous referee for drawing my attention to this.
In this case there would only be one argument involved in stating the relevant version of (U’):
There is at least one argument which is valid in Etchemendy’s informal sense, but which is itself not valid in the model-theoretic sense.
Is this version of (U’) then true or false? Again, I think the categorematic treatment of some terms makes it false. Consider the formal sentence \(\forall x (Bx \to Ux)\), but this time think of it as itself expressing that all bachelors are unmarried, and not by some relation it has with any English sentence. Clearly, if it does express this, then it is valid in Etchemendy’s informal sense. There simply are no bachelors that are married and there could not be such bachelors. But it is not valid in the model theory, since the model-theoretic treatment of the predicates B and U ignores their exact semantics. And while this is no longer a semantic feature that these predicates possess through a relationship with some English predicates, it nevertheless is a semantic feature that is ignored by the model theory. For this reason, the adapted version of (U’) is false.
This does not mean, of course, that there is only one language to which the informal notion applies, as Etchemendy quite clearly states that any language has a consequence relation in this sense. It only means that studying the consequence relation in a particular language never involves a relationship between two different languages.
On this view there is no such thing as the correct formalisation of a given natural language sentence. This, on Etchemendy’s view, is as it should be. It will always be relative to a choice of which terms and phrases to treat as the “logical constants” and which to treat categorematically.
I thank an anonymous referee for pointing this out.
We might say that Etchemendy tries to adapt one half of Kreisel’s squeezing argument, but this is the less controversial half, consisting of the completeness theorem and an argument for (1’). In fact, in his original presentation of the squeezing argument, Kreisel argues that his version of (1’) is true, so it does not seem that Etchemendy gets Kreisel wrong here either.
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Acknowledgements
Thank you to Edwin Mares and Max Cresswell for interesting and helpful discussions on how to understand Etchemendy’s writings. In addition I would like to thank two anonymous referees for providing valuable comments and suggestions.
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Christensen, K.H. Etchemendy on Squeezing Arguments and Logical Consequence: a Reply to Griffiths. Philosophia 46, 803–816 (2018). https://doi.org/10.1007/s11406-018-9955-z
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DOI: https://doi.org/10.1007/s11406-018-9955-z