Monday 21 October 2013

Formal Logic Isn't a General Science of Validity

'Tell me how you are searching, and I will tell you what you are searching for.' - Wittgenstein.


A curious feature of the philosophical landscape is that the most fundamental and influential discussions of several key issues in the philosophy of formal logic occur, in large part, in introductory logic textbooks. There's something Wild-West-ish and exciting about this, even if it stems partly from regrettable neglect of these issues. While proficient philosophically-trained logic teachers all impart a certain core of technical understanding, they differ greatly on what they tell their students formal logic is all about.

Here I want to give an argument for a thesis in the philosophy of formal logic:

(NGV) Formal logic is not aptly regarded as having, among its aims, that of giving a general account of validity.

This argument, if any good, should be of considerable interest, since the view which (NGV) denies – call it (GV) – is widely held, and plays a large role in our collective understanding of the nature of formal logic. I will use as a foil N.J.J. Smith's 2012 textbook Logic: The Laws of Truth, which endorses (GV).

My argument in compressed form is as follows: there are glaring lacunae in formal logic construed as a general account of validity; if that were one of its aims, it would be blatantly neglecting certain basic phenomena which it is meant to account for. But no one with any close relationship to formal logic cares, and we cannot put this down to laziness or stupidity. Therefore, it is inappropriate to construe formal logic as having a general account of validity among its aims. 

What I Mean by 'Formal Logic' and 'Validity' 

Before explaining the argument further with an example, a brief word about what I mean by 'formal logic' and 'validity'. 'Formal logic' means the formal, mathematized discipline of logic – syntax, formal semantics, proof-theory. Observations about the validity of natural language arguments, even if validity can be aptly conceived as turning on the form of propositions, are not themselves bits of formal logic, on this usage. By 'validity' I mean the thing which gets explicated sometimes as 'necessary truth preservation in virtue of form', or 'necessary truth-preservation in virtue of the meanings of logical (or subject-matter neutral) vocabulary', but also in other ways. A key point is that the valid inferences are a subset of the deductive (in a common sense of 'deductive'). We may deduce 'There is a vehicle' from 'There is a car', but this is not a valid inference in the relevant sense, since it turns on 'vehicle' and 'car'. 'There is a car, therefore there is a car or a vehicle', on the other hand, is valid. We say that the conclusion of a valid argument is a logical consequence of the premises. (Often what I am calling 'validity' here gets called 'logical validity', or even 'narrowly logical validity'. This may not be a bad practise, but here I follow the logicians who use 'validity' in this already-narrow way.)

There are different ways of spelling out what validity consists in. 'Valid' may also be a vague or indeterminate predicate (if, for example, the form/content distinction or the distinction between logical and non-logical vocabulary is vague or indeterminate), but that doesn't matter.

This should suffice to identify the relevant basic intuitive idea of validity, an idea which plays a major role in thinking about logic. It is not our purpose here to investigate the notion of validity, or the nature of validity, any further. We just needed to get it clear enough for the purpose of giving the argument for (NGV).


Here is a clear expression of a conception involving (GV) - the view that formal logic is aptly regarded as having, among its aims, that of giving a general account of validity - in The Laws of Truth (pp.20-21, Chapter 1: Propositions and Arguments):

When it comes to validity, then, we now have two goals on the table. One is to find a precise analysis of validity. (Thus far we have given only a rough, guiding idea of what validity is: NTP [necessary truth-preservation - TH] guaranteed by form. As we noted, this does not amount to a precise analysis.) The other is to find a method of assessing arguments for validity that is both
1. foolproof: it can be followed in a straightforward, routine way, without recourse to intuition or imagination—and it always gives the right answer;
2. general: it can be applied to any argument.
Note that there will be an intimate connection between the role of form in the definition of validity (an argument is valid if it is NTP by virtue of its form) and the goal of finding a method of assessing arguments for validity that can be applied to any argument, no matter what its subject matter. It is the fact that validity can be assessed on the basis of form, in abstraction from the specific content of the propositions involved in an argument (i.e., the specific claims made about the world—what ways, exactly, the propositions that make up the argument are representing the world to be), that will bring this goal within reach.

One of the philosophically most important sections of the book is 14.4, 'Expressive Power', where four kinds of propositions are discussed which may seem to create difficulties for first-order logic with identity (FOL=) construed as a general account of validity (what follows is summary and paraphrase, not quotation):

(1) 2 + 2 = 4. (Translatable into FOL=, but the result is considerably more complex, and this may make us hope for something more elegant.)

(2) She will see the doctor but she hasn't yet. (Quantification over times is required to translate this adequately into FOL=, and, like with (1), we might hope for something else. And there is something else: tense logic.)

(3) Propositions involving vague predicates. (FOL= requires predicates to have definite extensions. But there is fuzzy set theory and other formal tools which could deal with these.)

(4) 'There are finitely many Fs'. (Can't be expressed in FOL=, but there are extensions in which it can be.)


Note carefully that Smith is not claiming that classical first-order logic is enough for a general account of validity. He admits, for example, that it cannot adequately represent propositions of type (4). But none of this casts any real doubt on (GV).

I want to bring a different kind of example into the mix. It will perhaps seem a bit boring or basic, but that's actually what makes it philosophically important.

Example: From 'Only' to 'All' 

Consider this argument:

Only horses gallop, therefore all gallopers are horses.

This is at least as non-trivial as 'A and B, therefore A', which formal logic deigns to capture.


I think this is, in a way, a more philosophically instructive sort of "problem for classical logic", in that it doesn't seem like much of a problem, and this gives us a way of seeing that (NGV). The argument isn't "fancy", it doesn't seem to make us want new systems, and the problem-propositions involved are translatable in a loose sense with no worries at all. But there is no way of capturing the inference in first-order logic. You either wind up with the premise the same as the conclusion, or a premise which differs from the conclusion (e.g. by using existential quantification), but no more similar to the natural language premise than the formal conclusion is, and so not capturing the original inference at all. 

This translatability in a loose sense is important. Since we can quite easily see that only Xs Y if and only if all Y'ers are X's, we just translate 'only' propositions using ordinary first-order quantifiers. But the logical insight we needed for that was at least as real and non-trivial as that which we need to see that 'A and B' implies 'A'.

We must ask the logician: if you're bothering to codify things like 'A and B, therefore A', and you're quite generally interested in the validity of arguments in natural and other languages, why are you so content to leave 'Only horses gallop, therefore all gallopers are horses' unaccounted for by your science?

The fact that we are so content shows (GV) to be misleading, indeed false. In doing formal logic, we are actually doing something quite different from: trying to construct a general account of validity.

It seems like what we are really doing is something more like: developing artificial languages fulfilling certain desiderata, arguments in which can be easily inspected for validity. (Quine, regimentation.) Or perhaps, getting a sense of what validity turns on, without trying to give a general account. (Realistically, it's probably both of these plus many others, right down to aesthetic or even social motivations.) At least, if we were doing either of those things, our formal neglect of the Only-to-All argument would be quite intelligible.

Again: if we're so keen on a general accoubt of validity, how can we be so ready to not care about having no formal account of 'Only Xs Y, therefore all Y'ers are Xs'? Because it is so trivial? That can't be right, since things we do try to capture are no less trivial. No, there is no way out - the fact that we don't care shows that we really aren't so keen on a general formal theory of validity. That idea doesn't capture the real life of formal logic.

We might tell ourselves we want a general account of validity, but then when we're underway, we think 'Why bother formalizing "only", when we've already formalized "all" and can use that?' - the fact that we take that attitude gives the lie to the idea that we're really after a general account of validity.

We sense that it would be fairly uninstructive to formalize 'only' once we've got 'all' etc. - we should let that help guide our thinking about what we're really doing when we do formal logic (hence the Wittgenstein quote at the beginning).

That is my argument for (NGV) and against (GV).

Other Kinds of Examples
The above example may be resisted on the grounds that 'only' has been studied from within Montague semantics, and perhaps from other perspectives. My reply to that, in the first instance, is that these contributions were not generally thought of as part of formal logic, nor have they come to be incorporated into it. I think that's the right reply, but it isn't entirely satisfying - it might seem overly legalistic and trusting in the status quo. Fortunately, other examples are to hand.

What other kinds of examples are there, besides the Only-to-All argument, which show the same thing? A couple more I have thought of are:

- 'All men are mortal, therefore everything is such that its being a man materially implies its being mortal', or 'There are men, therefore there is something which is a man'. That is, arguments which take us from basic natural quantifier-constructions to something of the quantifier-variable form. These arguments embody logical insights due to the inventors of quantification theory (such as Frege and Peirce), and yet it seems we don't care about giving formal accounts of the arguments themselves.

- Arguments involving truth-functional compounds where the premise is written in some non-standard notation (such as: Venn diagrams, Gardner-Landini shuttle diagrams, Wittgensteinian ab-notation, truth-tables construed as propositional signs), and the conclusion is written with regular connectives (or vice versa).

These two kinds of examples differ from the Only-to-All argument in involving technically-formed propositions, whereas 'All gallopers are horses' and 'Only horses gallop' are forms that come naturally to us.


  1. It seems that all your examples are of the form "X, therefore Y," where X and Y are sentences with the same logical form. So they are really of the form "A, therefore A," which is already covered by first-order logic.

    "Only horses gallop, therefore all gallopers are horses": "Only horses gallop" translates to "all gallopers are horses."

    "There are men, therefore there is something which is a man": "There are men" translates to "there is something which is a man." Alternatively, both translate to "some things are men" or "some thing is a man."

    Likewise for the "all men are mortal" example. So it seems to me that formal logic does include every distinct valid form of argument, but that a general account of validity would include formal logic plus an account of how to translate logically equivalent sentences into one another. It's understandable that the latter tends to ignored--translation is usually trivial. If it were to be studied, though, it would properly be a part of informal logic, since it depends on the peculiarities of natural languages.

    1. What are the criteria for an adequate translation, though? I realize that these cases are usually treated in the way you indicate, but why are those acceptable translations?

      It cannot be enough for a translation to be acceptable, from the point of view of assessing formal logic from the point of view of which valid argument forms it captures, that it yields something logically equivalent to the object of translation. For if that were true, all deductive argumentation could be captured with the form 'A and B, therefore B'.

      For example, 'All men are mortal and Socrates is a man, therefore. Socrates is a mortal'. But why can't we translate the premise, 'All men are mortal and Socrates is a man' as 'All men are mortal and Socrates is a man, and Socrates is mortal', which is after all logically equivalent to the premise? That would make the original argument an instance of the 'A and B, therefore A' form. But intuitively it isn't one.

      If being logically equivalent were enough for being a good translation, then all valid arguments could be captured with conjunction elimination alone (just translate the premise in such a way that the conclusion is one of its conjuncts). But surely it shouldn't be that easy for logic to "include every distinct valid form of argument": something has gone wrong here.

      The more logical wisdom you allow or presuppose for the task of translation, the less formal logic has left to do - and where do we draw the line?

      When you say 'it seems to me that formal logic does include every distinct valid form of argument, but that a general account of validity would include formal logic plus an account of how to translate logically equivalent sentences into one another. It's understandable that the latter tends to ignored--translation is usually trivial', I think you are overlooking this tension between being more liberal with 'acceptable translation' and trivializing logic, and being stricter with translation and making logic harder. Furthermore, I think that on most natural, non-ad-hoc standards of translation, formal logic, if it isn't trivialized, will be seen to have lacunae of the sort I've tried to point to.

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