Saturday, 25 May 2019

Logical Pluralism

In this post I will raise an issue for the logical pluralism of Beall & Restall (hereafter 'B&R') - a much-discussed, topic-revivifying view in the philosophy of logic. My study of their view was prompted by Mark Colyvan, whose course on Philosophy of Logic at Sydney Uni I'm helping to teach this year. Thanks to Mark for encouragement and discussion. 

I'll start off in this post by looking at B&R's central thesis, and arguing that it fails to capture an interesting, controversial position in the philosophy of logic. The problem is not that the view is false, but that it's "too easily" true.

I take their 2000 paper ‘Logical Pluralism’ as the starting point, but where more detail is needed, I draw on their 2005 book Logical Pluralism.

Their claim concerns the following schema:
Generalised Tarski Thesis (GTT): An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion. (2005, p. 29.)
(This is more precise than the corresponding schema, (V) (for 'validity'), from their 2000 as it makes clear what is allowed to vary.)

They write:
Logical pluralism is the claim that at least two different instances of GTT provide admissible precisifications of logical consequence. (2005, p. 29.)
Being an existential, numerical claim (‘There are at least two…’), there are many ways the view could be true. Later in this series of posts I'll look at the ways they imagine it coming out true - B&R hold that 'cases' may be taken to be worlds, Tarski-style models, or situations (in the sense of Barwise & Perry's situation theory, at least in the first instance). But here I want to highlight presumably unintended ways in which it comes out true, or at least appears to me to do so. If I'm right about this, these unintended ways threaten to rob their view of its apparent bite.

According to B&R, ‘cases’ may be models ‘Tarskian style’ (2000, p. 480) or ‘along Tarskian lines’ (2005, p. 29).

But this permits differences over exactly what a model is (even for a given language L - to fix ideas, let's consider the language of first-order logic).

For instances, do models provide assignments to variables, or just to names? That depends on how you like to treat quantification when defining 'true on a model'. (Another option, taken by Tarski, eschews assignments to variables in favour of a trick involving sequences of objects.)

Some think which option you take here is philosophically significant - see for instance Smith's 'Truth via Satisfaction?'. But few I think would want to say that not all of these options lead to Tarskian style models (in a broad sense).

But this doesn't actually matter, since there other differences, which seem definitely trivial, over what exactly a model is taken to be in various presentations of first-order logic.

Is a model a tuple of the form <D, I> (where D is the domain and I contains semantic information about all non-logical terms)? Or is it a tuple of the form <D, P, N>, where predicates’ extensions are given separately from names’ referents? Or do we bundle these ingredients informally, as is often done in introductory texts? (That is, do we think that a model is just: a domain, extensions for predicates etc., without thinking of the model as a mathematical object in its own right?)

The point is that these differences, while pretty unimportant, do lead to real differences
over which objects are the ‘cases’. And so to different ‘precisifications’ of the notion of logical consequence.

And so it seems that, if you believe that there is more than one slightly different way of doing broadly Tarksi-style model theory, then you should be a logical pluralist in Beall & Restall's sense. But that seems like the wrong outcome! And in some sense, not what they mean. B&R wanted to carve out and develop a distinctive philosophical view, one which would for instance conflict with the views of someone who thinks that nothing that is not some form of classical logic counts as 'logic'.


Beall, Jc & Restall, Greg (2000). Logical pluralism. Australasian Journal of Philosophy 78 (4):475 – 493.
Beall, Jc & Restall, Greg (2005). Logical Pluralism. Oxford University Press.
Smith, Nicholas J. J. (2017). Truth via Satisfaction? In Pavel Arazim & Tomas Lavicka (eds.), The Logica Yearbook 2016. London: College Publications. pp. 273-287.

Thursday, 25 April 2019

Williamson's Metaphysical Modal Epistemology and Vacuism about Counterpossibles

Timothy Williamson has argued that our capacity for metaphysical modal judgement comes along with our capacity for counterfactual judgement. This passage gives a flavour of his view:
Humans evolved under no pressure to do philosophy. Presumably, survival and reproduction in the Stone Age depended little on philosophical prowess, dialectical skill being no more effective then than now as a seduction technique and in any case dependent on a hearer already equipped to recognize it. Any cognitive capacity we have for philosophy is a more or less accidental byproduct of other developments. Nor are psychological dispositions that are non-cognitive outside philosophy likely suddenly to become cognitive within it. We should expect cognitive capacities used in philosophy to be cases of general cognitive capacities used in ordinary life, perhaps trained, developed, and systematically applied in various special ways, just as the cognitive capacities that we use in mathematics and natural science are rooted in more primitive cognitive capacities to perceive, imagine, correlate, reason, discuss… In particular, a plausible non-skeptical epistemology of metaphysical modality should subsume our capacity to discriminate metaphysical possibilities from metaphysical impossibilities under more general cognitive capacities used in general life. I will argue that the ordinary cognitive capacity to handle counterfactual carries with it the cognitive capacity to handle metaphysical modality. (Williamson, The Philosophy of Philosophy (2007), p. 136. Found in Section 3 of the SEP article 'The Epistemology of Modality'.)
In order to argue for this, Williamson takes a schematic semantic story about counterfactual conditionals:
Where “A>B” express “If it were that A, it would be that B”, (CC) gives the truth conditions for subjunctive conditionals: A subjunctive conditional “A>C” is true at a possible world w just in case either (i) A is true at no possible world or (ii) some possible world at which both A and C are true is more similar to w than any possible world at which both A and ¬C are true.(Formulation from Sec 3 of the SEP article.)
On the basis of this, he proves the following equivalences:

(NEC) □A if and only if (¬A>⊥)
It is necessary that A if and only if were ¬A true, a contradiction would follow.

(POS) ◊A if and only if ¬(A>⊥)
It is possible that A if and only if it is not the case that were A true, a contradiction would follow.

(Renderings and spellings out from the SEP article. The box and diamond are metaphysical modal operators, and the falsum - which looks like an upside-down 'T' - represents a contradiction.)

But this only works because the schematic theory of counterfactuals Williamson adopts is understood as working against a background of a notion possible worlds, where Williamson understands this as the notion of metaphysically possible worlds. This theory deems true all counterfactuals with metaphysically impossible antecedents, and this is crucial to his demonstation of the equivalences on the basis of his assumed schematic theory.

There are lots of reasons not to adopt a theory of counterfactuals which uses the notion of metaphysical possibility in this way. One reason to move away from a theory like this is if you think that there are counterpossibles - counterfactual conditionals with metaphysically impossible antecedents - which have their truth-values non-vacuously. But you might be agnostic about that. For instance, you might be happy with metaphysical modal distinctions but doubt that there are any clear cases where countepossibles have truth-values non-vacuously. In that case you might see no good reason for the backdrop of worlds or scenarios in a theory of counterfactuals to be exactly the metaphysically possible ones. Or, you might be skeptical about the very distinction between metaphysical possibility and impossibility, in which case you won't want to understand the backdrop in a way that involves that distinction. And it seems that you can get most, or even all, of the theoretical benefits of a Stalnaker-Lewis approach to counterfactuals without using that distinction. It doesn't really seem to play a starring role in the theories. 

If a semantic theory for counterfactuals which does not draw on any bright line between metaphysically possible and metaphysically impossible scenarios is as good or better than the theory that Williamson uses to prove his equivalences, that seriously undermines the equivalences, and in turn Williamson's story about how we get metaphysical modal knowledge.

(And note that this turns on Williamson's understanding his chosen theory so that 'possible world' means 'metaphysically possible world' - even accepting an identically worded theory, but where 'possible world' is understood in a way which does not involve the distinction between metaphysical possibility and impossibility, would make Williamson's equivalences unavailable.)

Sunday, 7 April 2019

Contradictory Premises and the Notion of Validity

When evaluating arguments in philosophy, it can be tempting to call an argument 'invalid' if you determine that it has contradictory premises. For example, in an introductory philosophy course at the University of Sydney, students are taught that a particular argument for the existence of God - called the Argument from Causation - is invalid because two of its premises contradict each other. It is tempting to call such an argument invalid because we can determine a priori that it is not sound, i.e. that it isn't both valid and such that its premises are true. But on a classical conception of validity, any argument with contradictory premises counts as valid, since it is impossible for all the premises of an argument with contradictory premises to be true, and so a fortiori impossible for the argument to have true premises and false conclusion.

I have heard this anomaly explained away by appeal to the fact that, while an argument with contradictory premises may count as formally valid, we are looking at informal validity, and in an informal sense perhaps any argument with contradictory premises should count as invalid. But I don't think that's right. If 'formal' is meant to signal that we are not interested in the meanings of non-logical terms and are only interested in what can be shown on the basis of the form of the argument, then that is clearly a different issue: premises could be determined to be contradictory on the basis of form alone, or in part on the basis of the meanings of the non-logical terms. The issue of contradictory premises is similarly orthogonal to the issue of 'formality' if 'formal' is instead meant to signify something like 'in an artificial language' or 'in a precise mathematical sense'. 

In fact, it's arguable that the standard treatment of validity of arguments in classical formal logic should be supplemented, so that an argument counts as valid iff it has no countermodel and its premises are jointly satisfiable.

If we defined 'valid' that way in classical logic, then to test an argument for validity using the tree method, you might have to do two trees. First, one to see if the premises can all be true together. If the tree says No, the argument is invalid and we can stop, but if the tree says Yes, then we do another tree to see if the premises together with the negation of the conclusion can all be true together, and if the tree says No, the argument is valid.

Whether or not it's worth adopting in practise, it is worth noting that this augmented definition of 'valid' in classical logic seems to correspond more closely to the ordinary, informal notion of deductive validity than the usual definition. This even delivers at least one of the desiderata which motivate relevance logic.

However, note that while we seem pretty disposed to call an argument invalid if it has contradictory premises, there is no equally strong tendency to say corresponding things using 'follows from', 'is a consequence of', or 'implies'. This is interesting in itself. It looks like, when we're talking about implication, our focus is on the putative implier or impliers and what can be got out of them, whether or not they're true. By contrast, when we talk about arguments, we're often more focused on the conclusion and whether it is shown to be true by the argument in question, so that validity is treated as one of the things we need to verify along the way. If validity is playing that role, it makes sense to declare an argument invalid if we work out that its premises can't all be true.

Tuesday, 5 February 2019

Two Sources of Interest in Metaphysical Modality (and Kripke in Light of Them)

Source 1: We want to know the vocabulary and syntax of being, or as Rosen puts it in 'The Limits of Contingency', 'the combinatorial essence of the world'. What are the basic elements and how may they be combined?

Source 2: We want to know, as it were in advance, whether various kinds of statements that are not put in terms of the basic elements count as high-level descriptions of any possible world. 

Many of Kripke's arguments that certain kinds of statements are necessary furnish considerations which purport to show that whatever the possibilities are exactly, none of them is correctly described as one in which '...'. 

This mode of argumentation on Kripke's part can make it look like his modal notions can ultimately be explicated along conceptual or semantic lines. But, as Putnam came to appreciate in between 'The Meaning of "Meaning"' and 'Is Water Necessarily H2O?', this is not so.

In this connection it is notable that all of Kripke's distinctive modal theses are negative as regards possibility. (His claims, in the course of arguing against descriptivism about names, that well-known facts about Aristotle etc. could have been different, are an exception, admittedly - but for those arguments, I don't see that he needs these alternative possibilities to be real, metaphysical possibilities. The modality used in those arguments could be deflated to conceptual or semantic without affecting the arguments, which are after all for a semantic conclusion - that names aren't synonymous with descriptions.) As far as I know, Kripke never seriously argues that such and such really is possible, really is a way the world could have been. Rather, he just works on the assumption that there are many quite various ways things could have been, but then seeks to draw some limits in high-level vocabulary.

But these Kripkean results, if that's what they are, only satisfy interest in metaphysical modality that derives from the second source. How we might satisfy our interest that derives from the second source is largely left untackled, and this is one reason why many have found Kripke's work frustrating. He elicits epistemic hopes, someone might complain, without giving us even so much as a roadmap for how so satisfy them.

His suggestion that metaphysical possibility may coincide more closely with physical possibility than has often been supposed may however be suggestive. On the other hand, he is against physicalism, so this couldn't be the whole story from his point of view.

Thursday, 10 January 2019

Rigidity and General Terms: Two Different Analogues of the Singular Case

This post wrestles with and begins to settle on a view about the confusing issue of how Kripke's notion of rigidity may apply to general terms. 

One analogue of rigidity for general terms: how about we think of it as a rigid connection between properties (or alternatively, an additional connection between the "rigid" term and a further property).

So for example, 'water' in the first instance picks out the property of being water, which is tied rigidly to the property of being (mainly composed of) H20.

Or 'cat' is tied in the first instance to the property of being a cat, and that is tied rigidly to the property of being an animal.

But now there is another kind of thing which seems different, and comes up with sentences like 'John has the property we talked about yesterday'.

Suppose the property we talked about yesterday was the property of having the property that is discussed in Book A. And suppose Book A contains a discussion of the property of redness. Now 'has the property we talked about yesterday' rigidly designates the property of being a property we talked about yesterday, but it also non-rigidly designates the property of having the property that is discussed in Book A, as well as the property of redness.

This motivates the picture of, behind a predicate, a stack of properties, where the top one is designated rigidly, and the ones below not.

Problem: we might want to say that 'cat' rigidly designates a certain kind of animal. And I may then want to rephrase that as: 'cat' rigidly ascribes animality. 

But doesn't the 'rigid' bit here fall away? Take the phenomenological, underlying-nature-neutral counterpart of 'cat' - 'catty thing'. Now even if in our world all the catty things are cats, it doesn't sound right to say that 'catty thing' ascribes the property of being a cat at all - it's not that it ascribes that property, only non-rigidly. 

So now it is beginning to look like the distinction we are after here is between a term merely covering things with property P, vs. ascribing to them property P.

But then that seems wrong when we go back to 'John has the property we discussed yesterday', since if what we talked about yesterday was the property of redness, there is a sense in which that sentence ascribes redness to John. 

This is hell!

But this whole problem, occupying the last few paragraphs, perhaps only arises from mixing together two different analogues of rigidity that we get when we look at predicates.

It may be protested that  'John has the property ...' is not a property ascription syntactically at all, but rather a 2-place relational statement with a non rigid second term.

Be that as it may, we can still classify 'has the ...' as a predicate and can still talk about a rigid/non-rigid distinction. And so I think we need to recognise that there are at least two quite different things going on here - two different things which are a bit similar to rigidity/non-rigidity as applied to names.

One is the difference between 'has the property discussed earlier' and 'is red'. Another is the difference between 'is water' and 'is watery' (one brings being composed of H20 along with it in counterfactual scenario descriptions, and the other doesn't), or 'is a cat' and 'is a catty thing'. 

One reason the second analogue may be counterintuitive if presented as a kind of rigidity is that in the case of singular terms, rigidity is associated with simplicity (both syntactic and semantic), but in the case of predicates (let's look at 'is gold', 'is water', 'is a cat' and put aside 'has the property...') it's the opposite. The "non-rigid" predicates just don't take any further property along with them, but the rigid ones do. I.e. 'is a catty thing' or 'is catlike' just picks out one property, but 'is a cat' is tied to the further property of being an animal.

Actually, the 'any' in 'any further property' is probably wrong! Maybe all predicates rigidly take some further properties along with them. So this second sort of "rigidity" we can talk about in connection with general terms should be thought of as relative to whatever further property is in question. (For instance, 'is a pencil' is arguably counterfactually locked to 'is a physical object'. So 'is a pencil' rigidly picks out physical objects - we might want to say something like that.)

One thing that is emerging here is that the 'has the property discussed earlier' vs. 'is red' thing is one distinction which pattern-matches with Kripke's discussion of rigidity as applied to names, but there is also another thing going on - predicates dragging further properties along with them in counterfactual scenario descriptions - which actually corresponds better with Kripke's informal applications of the notion of rigidity to general terms.

Now it looks like the general-term-"rigidity" considerations in Naming and Necessity are actually closer to the "necessity of constitution" and similar considerations than they are to the "necessity of identity" considerations.

Background Reading:



- Kripke, Saul (1980). Naming and Necessity. Harvard University Press.

- Soames, Scott (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press

- Salmon, Nathan (2004). Are general terms rigid? Linguistics and Philosophy 28 (1):117 - 134.

- Linsky, Bernard (2006). General Terms as Rigid Designators. Philosophical Studies 128 (3):655-667.

- Martí, Genoveva & Martínez-Fernández, José (2011). General terms, rigidity and the trivialization problem. Synthese 181 (2):277 - 293.

- Schwartz, Stephen P. (2002). Kinds, general terms, and rigidity: A reply to LaPorte. Philosophical Studies 109 (3):265 - 277.

- de Sa, Dan López (2007). Rigidity, General Terms, and Trivialization. Proceedings of the Aristotelian Society 107 (1pt1):117 - 123.

- Marti, Genoveva (2004). Rigidity and General Terms. Proceedings of the Aristotelian Society 104:131-148.

- Zouhar, Marián (2009). On the Notion of Rigidity for General Terms. Grazer Philosophische Studien 78 (1):207-229.

- Orlando, Eleonora (2014). General terms and rigidity: another solution to the trivialization problem. Manuscrito 37 (1):49-80.

- Gómez-Torrente, Mario (2004). Beyond Rigidity? Essentialist Predication and the Rigidity of General Terms. Critica 36 (108):37-54.

- Kosterec, Miloš (2018). Criteria for Nontrivial General Term Rigidity. Acta Analytica 33 (2):255-270.

Wednesday, 26 December 2018

Moral Concepts as Unsystematic on the Introduction Side

Here is a metaethical idea which I find appealing and under-explored. What if what makes metaethics and the cognitivism vs. non-cognitivism debate so difficult is that we have failed to consider a certain possibility. Namely, that expressions can count as expressing concepts if there is enough systematicity in what happens once an application is accepted, even if there is much less systematicity in what happens before an application is accepted. 

On the view I am interested in, the reason moral terms count as expressing concepts at all, and count as linguistically meaningful, is that they are fairly systematic with respect to what lies downstream from them. This is compatible with them being much less systematic with respect to what they lie downstream from - and it looks like they are much less systematic in this respect.

By analogy to natural deduction systems in logic, we might think of moral concepts as coming with elimination rules, but no definite, agreed-upon set of introduction rules. 

So, concepts like right and wrong have fairly systematic and shared relationships to, roughly, what follows from statements involving them - in terms of other statements following logically, but also more probabilistic effects, and motivations and actions flowing from their acceptance. This systematicity is what makes them concepts - and it is enough! When it comes to what licences their introduction in the first place, on the other hand, there just isn't much there in the way of tight, systematic connections. Or there may be quite a bit there, but still not enough to, for example, motivate the view that these concepts are co-extensive with any of what we'd happily call descriptive concepts. 

On this view, the famous Open Question Argument comes as no surprise. Maybe you can always, under any descriptive supposition, intelligibly still ask 'But is it right?' - and that is just because there simply are no tight systematic links of the right sort between uncontroversially descriptive concepts and moral concepts. 

(By the way, the view isn't that all such concepts - concepts with unsystematic introduction behaviour - are moral concepts. The moral concepts are a subset of these, and what distinguishes them as moral as opposed to, say, aesthetic, has to do with what lies downstream from them.)

I wonder if this sort of view has been anticipated, as it strikes me as having real potential.

Thursday, 6 December 2018

Different Attitudes about Meaningfulness in Philosophy, and Their Consequences

What makes a question or a statement meaningful as opposed to nonsensical? This is a hard question, and perhaps not a very fruitful one in philosophy since the falling out of favour of verificationism. But what if we instead ask: what difference does it make whether we think something makes sense?

I have sometimes been struck by a certain kind of self-conscious, almost defiant claim of meaningfulness that can be found in work by philosophers writing after the resurgence of metaphysics in the second half of the twentieth century. You may have noticed it yourself. Often in the course of making a point, a philosopher will say something like 'Well, the question is certainly meaningful...' or 'Whether or not it is true, there can be little doubt that this is a coherent position'. 

This liberal attitude to the question of whether something even makes sense contrasts starkly, and is often expressed in reaction to, the wary attitude of other philosophers, especially earlier in the twentieth century.

Often, the liberal attitude is supported with the idea that the wary attitude depended on a discredited view - verificationism - about what it takes to be meaningful.

Both attitudes, especially when taken stridently or dogmatically, can seem stupid. The denier of meaning can seem like a shirker. Or like someone who doesn't want to play a game and justifies it by acting superior.  The denier can seem crude - like they have this stupid hammer that they wield instead of taking the trouble to understand something (or at least admitting that they can't or won't). 

But the affirmer of meaning can also seem stupid and crude. As though they are blind, perhaps even wilfully blind, to subtle problems and difficulties. In the context of doing philosophy, this may seem like especially bad form, in that such problems and difficulties are just the thing that a philosopher should presumably try to be alive to.

It is quite clear that both tendencies have their pitfalls, and that both can be justified in particular cases. But the question of which attitude is right overall, or in a particular area, or in a particular vexed case even, seems peculiarly intractable, in something like the way moral and aesthetic issues can be intractable.  

I have become interested in thinking about this from the point of view of the consequences of the different attitudes, focusing on output rather than input. That is, rather than inquiring into when it is right to deem something meaningful, let us try to think about what might ensue given an attitude to the meaningfulness question. And that may then shed light back on the input question. 

A liberal attitude to the meaningfulness question can be refreshing, in that ideas and views get engaged with and clarified, in a way that they don't seem to if you're being skeptical about meaning. For example, someone who is very suspicious about the meaningfulness of the questions and views in what is nowadays called temporal ontology, e.g. whether the past and the things in it exist, is not, at least while being suspicious, likely to note certain (apparent) logical relationships between these views and others. For instance, the relationship to the question of whether true statements (perhaps of a certain kind) are all made true by some existing thing. Noticing these things seems to require that we, at least for a time, don't stand back and worry about meaningfulness, but come forward and handle these ideas - i.e. that we treat them as meaningful, and do the things that we do when we try to find out whether things are true or false.

But a warier attitude to meaningfulness can also be refreshing, leading us to take (in Wittgenstein's phrase) a wider look around. Fresh considerations may arise in this way, where from inside the meaning-granting perspective they would not. For example, someone suspicious about the way questions in temporal ontology are debated in philosophy seminars and publications may find themselves wondering what is really going on here, what is really animating all this discussion. They may think suspicious thoughts about the emotional associations of some of these ideas playing more of a role than the broadly scientific, "theory choice" mode of discussion of analytic metaphysics would suggest. And such thoughts may be fruitful. 

Understanding this better may help us avoid stupidities on both sides, making us more nimble and sophisticated. We don't have to pick a posture and stay in it forever. Even if a question or a view is ultimately dubious as regards meaningfulness - or needs to be transformed based on a better understanding of what is really at stake - taking the liberal attitude and working with the question or view for a stretch, i.e. treating it as meaningful enough, may yield insights which survive the transformation, or can be transferred to other places. And we may be more prepared to let this go on if we realise that it doesn't preclude us from taking the other viewpoint, i.e. getting suspicious and sniffing around more widely. And the same going the other way. One thing is that, even if the question is ultimately just really hard and not at all meaningless or in need of transformation, we may not know that, and so encouraging some suspicious minds to have a sniff around may be good. Another thing is that it may lead to a more subtle kind of improvement, where we may not radically reformulate our questions and views, but fold in new criteria and considerations that come from a wider look around.