Consider two kinds of norms for playing chess. On the one hand, there are prescriptions for playing chess well, for carrying out the Queens Indian defense, or for playing an endgame with two rocks. These are norms that are applicable only in specific situations, or only given certain goals or interests: they are “hypothetical” in the sense of Kant’s “hypothetical imperatives”. On the other hand, there are rules of the game. These norms apply to chess playing as such, because they are constitutive of chess playing. […] Unless these norms are binding on one’s moves, one is not playing chess, but some other game. They are “categorical” norms for chess. We can make the same distinction among norms for thought (which, following Kant, I will take as a blanket term for all concept use, including judging, inferring, supposing, asserting, and even perceiving insofar as it involves the application of concepts). On the one hand, there will be norms for certain kinds of concept use (asserting, laying down legal decisions), or for the use of certain kinds of concepts (physical concepts, chemical concepts, moral concepts). These will be “hypothetical” norms for thought, norms with limited or conditional application. On the other hand, there may be norms for concept use as such: norms that are constitutive of concept use. If there are any such norms, they will be “categorical” norms for thought, and they will be universally and unconditionally applicable. MacFarlane, What does it mean to say that Logic is Formal?, p.52

The laws of logic […] are defined [by Kant] as the “necessary laws of understanding and reason in general, or what is one and the same, of the mere form of thought as such”. By “necessary forms of understanding”, Kant means “... those [laws] without which no use of the understanding would be possible at all”, that is, the norms constitutive of thought. Similarly, in the first Critique he says that general logic “... contains the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding”. In my terminology, this is a demarcation of logic by its 1-formality. MacFarlane, What does it mean to say that Logic is Formal?, p.53

It is important to be clear about the sense in which logical laws are, on this tradition, normative for thought as such. The point is not that nothing can count as thought unless it conforms to the laws of logic. That would make logical error impossible, and it would make nonsense of Kant’s claim that logic is a normative discipline, since the way we ought to think would turn out to be the only way we could think. When Kant says that we cannot think except “according to” the laws of logic, he means that our thought must be responsible to the laws of logic for its assessment. […] What makes it [a thought] a thought is not that it conforms to the laws of logic but that the laws of logic are normative to it. To say that the laws of logic are norms for thought as such, then, is not to say that it is impossible to think illogically, but only that it is impossible to think illogically and be thinking correctly. MacFarlane, What does it mean to say that Logic is Formal?, p.54

[…] it does follow from logic’s being normative for thought as such that it is necessary in a strong sense. We can think about possible worlds or situations in which the physical laws of the actual world do not hold: “where animals speak and stars stand still, where men are turned to stone and trees turn into men …” (Frege). Such thought violates certain contingent norms for thought - norms for correct thought about animals, stars, forces, and so on - and thus does not count as correct thought about these. But considered merely as thought […] it does not violate the norms for thought as such. What makes it possible for us to think correctly about such counterfactual worlds, then, is that we can prescind from some of the norms of thought - by acknowledging our thought as counterfactual, as concerning mere possibility - while continuing to acknowledge others. In the limiting case of logical possibility, we prescind from all contingent forms of thought, acknowledging only the norms of thought as such. But if we try to prescind from these norms, too - say, by thinking about a possible world in which contradictions are true - then no norms remain to which our concept use can be held responsible. In this case it is no longer recognizable as concept use at all, since concept use is essentially evaluable as correct or incorrect. We can correctly think about what the world would be like if the laws of physics were different, but not about what it would be if the laws of logic were different. This is the sense in which the norms for thought as such are necessary: it is impossible to think at all, even counterfactually, without being constrained by them. MacFarlane, What does it mean to say that Logic is Formal?, p.55

In the limiting case of logical possibility, we prescind from all contingent forms of thought, acknowledging only the norms of thought as such. MacFarlane, What does it mean to ay that Logic is Formal?, p.55

We can correctly think about what the world would be like if the laws of physics were different, but not about what it would be if the laws of logic were different. This is the sense in which the norms for thought as such are necessary: it is impossible to think at all, even counterfactually, without being constrained by them. MacFarlane, What does it mean to ay that Logic is Formal?, p.55

[…] in characterizing logic as general, Kant clearly intends to be distinguishing it from mathematics. In the Jasche Logic, for instance, Kant defines general logic as the science of the necessary rules of understanding, “without which no use of the understanding would be possible at all”, contrasting these with various contingent rules for understanding, “without which a certain determinate use of the understanding would not occur”. One of his examples of such a “particular, determinate use of the understanding” is the “use of the understanding in mathematics”. Arithmetic and algebra, then, are not “general” in the sense that logic is. Though they may not have their own special objects, and though their concepts may not distinguish between the particular identities of objects, their laws are applicable only to a “particular, determinate use of the understanding”, not to all uses of the understanding. Because the content of the concept of magnitude depends on sensibility, while thought is intelligible apart from sensibility, the norms governing this concept (i.e., the laws of arithmetic and algebra) cannot be norms of thought as such. […] It appears, then, that when Kant speaks of the generality of logic, he means its 1-formality (constitutive normativity for thought as such), not its 2-formality. MacFarlane, What does it mean to ay that Logic is Formal?, p.86-87

This approach [2-formality] starts with the idea that what makes content specific is its concern with particular individuals. It is clear that the concept ‘horse’, the relation ‘is taller than’, and the quantifier ‘every animal’ have specific content, because they all distinguish between Lucky Feet, on the one hand, and the Statue of Liberty, on the other. - “Lucky Feet is a horse” is true; “The Statue of Liberty is a horse” is false. - “The Statue of Liberty is taller than Lucky Feet” is true; “Lucky Feet is taller than the Statue of Liberty” is false. - The truth that “Every animal is healthy” depends on whether Lucky Feet is healthy, but not on whether the Statue of Liberty is healthy. On the other hand, the concept ‘is a thing’, the relation ‘is identical with’, and the quantifier ‘everything’ do not distinguish between Lucky Feet and the Statue of Liberty. In fact, they do not distinguish between any particular objects. As far as they are concerned, one object is as good as another and might just as well be switched with it. MacFarlane, What does it mean to say that Logic is Formal?, p.56-57

Notions with […] indifference to the particular identities of objects might reasonably be said to abstract from specific content - to be “formal”. Tarski seems to have something like this in mind when he says: “since we are concerned here with the concept of the logical, i.e., formal, consequence, and thus with a relation which is to be determined by the form of the sentences between which it holds, this relation cannot be influenced in any way by empirical knowledge, and in particular by knowledge of the objects to which the sentence X or the sentences of the class K refer. The consequence relation cannot be affected by replacing the designations of the objects referred to in these sentences by the designations of any other objects” (Tarski 1936: 414-5) Call this notion of formality “2-formality”. MacFarlane, What does it mean to say that Logic is Formal?, p.57

One can capture the indifference of logical notions to the particular identities of objects by demanding that they be invariant under the group of all transformations of the objects in a domain. […] For present purposes, this approach can best be illustrated through examples. Unary first-order quantifiers can be modeled semantically as functions from sets (predicate extensions) to truth values. Now consider the quantifier “all chickens”, which takes every set that contains all the chickens in the domain to True and every other set to False. Suppose the domain of objects includes two chickens and two cows. Let K be a set containing two chickens, so that “all chickens” takes the value True on K. If we permute the objects in the domain by switching the chickens for the cows, so that K is now a set containing two cows, then “all chickens” takes the value False on K. Since the truth of sentences containing “all chickens” is sensitive to permutations of the domain, “all chickens” is not permutation-invariant. On the other hand, the numerical quantifier “at least three things”, which takes every set with three or more members to True and every other to False, is permutation-invariant. Permutation invariance can be regarded as a precise technical gloss on the idea of indifference to the particular identities of objects. Whereas “all chickens” is sensitive to the difference between chickens and cows, permutation-invariant notions - like quantifiers and identity - are insensitive to particular identities of the objects to which they apply. […] MacFarlane, What does it mean to say that Logic is Formal?, p.58-59

We motivated 2-formality as a way of thinking about abstraction from specific content. Thus the claim that logic is 2-formal leaves open the possibility that logic is concerned with general content and very general facts about the world. 3-formality […] precludes any concern with content. […] a logic that abstracts from the contents of some concepts (“specific” ones, like “horse” or “red”), but not from the contents of others (“general” or logical ones, like existence, identity, and conjunction), does not count as 3-formal. MacFarlane, What does it mean to say that Logic is Formal?, p.60

We motivated 2-formality as a way of thinking about abstraction from specific content. Thus the claim that logic is 2-formal leaves open the possibility that logic is concerned with general content and very general facts about the world. 3-formality […] precludes any concern with content. […] a logic that abstracts from the contents of some concepts (“specific” ones, like “horse” or “red”), but not from the contents of others (“general” or logical ones, like existence, identity, and conjunction), does not count as 3-formal. MacFarlane, What does it mean to say that Logic is Formal?, p.60

What is left to consider when we abstract entirely from semantic content of a thought? Some philosophers would say “nothing”. But some have held that there is more to thought than the concepts it employs: there is also the way they are put together. On Kant’s view, for example, “... all judgements are functions of unity among our representations”, and the various modes of unity determine the possible forms of judgements. “The matter of the judgement consists in the given representations that are combined in the unity of consciousness in the judgement, the form in the determination of the way that the various representations belong, as such, to one consciousness”. General logic, in Kant’s view, abstracts from the matter and considers only the form of judgements: it considers only the way in which concepts are united in judgements. “General logic … abstracts from all content of knowledge, that is, from all relation of knowledge to the object, and considers only the logical form in the relation of any knowledge to any other knowledge; that is, it treats of the form of thought in general.” For example, general logic treats “all horses are mammals” simply as the unification of two concepts in a universal, affirmative, categorical, and assertoric judgement. It abstracts entirely from the content of concepts. The way in which the concepts are united in thought is not, for Kant, a further constituent of the thought (a “binding” concept), but a feature of the thought’s form. MacFarlane, What does it mean to say that Logic is Formal?, p.60-61

To claim that logic is 3-formal presupposes that we can distinguish between the constituents and the form of thought, in such a way that the latter can be understood apart from the former. […] Many twentieth century philosophers accept the distinction, demarcate logic by its concern with the form of thought or conceptual inquiry, and draw the same consequences as Kant: 1. that logic alone tells us nothing about the world; 2. that the world does not constrain logic, so that logical knowledge and knowledge about the world are fundamentally different; and 3. that “logical truths”, if there are such things, do not state facts. MacFarlane, What does it mean to say that Logic is Formal?, p.62

“In this distinction we [The Vienna Circle] had seen the way out of the difficulty which had prevented the older empiricism from giving a satisfactory account of the nature of knowledge in logic and mathematics. ... Our solution, based on Wittgenstein’s conception, consisted in asserting the thesis of empiricism only for factual truth. By contrast, the truths in logic and mathematics are not in need of confirmation by observations, because they do not state anything about the world of facts, they hold for any possible combination of facts” (Schilpp, 1963, The Philosophy of Rudolf Carnap). Carnap acknowledges the influence of Wittgenstein’s Tractatus here, and Wittgenstein does seem to be advocating for a version of the view that logic is 3-formal: “The propositions of logic are tautologies... The propositions of logic therefore say nothing… Theories which make a proposition of logic appear substantial are always false” (TLP, §6.1, 6.11, 6.111) MacFarlane, What does it mean to say that Logic is Formal?, p.63-64

I am not persuaded that 3-formality does not imply 2-formality. It seems to me a stronger version of it as it abstracts entirely from semantic content. (See MacFarlane, What does it mean to say that Logic is Formal?, p.67-68 for a relevant argument).

Although Kant thinks that logic can be formal, he does not take formality to be part of the essence of logic. That a general logic must also be formal is a substantive claim that requires justification, not a matter of semantics. I call this claim Kant’s Thesis: General (i.e., 1-formal) logic must also be formal (i.e. 3-formal). That is, “the universal, and necessary rules of thought in general can concern merely its form and not in any way its matter” (Jasche Logic, p.12). MacFarlane, What does it mean to say that Logic is Formal?, p.91-92