The most obvious answer to the question [of what logic is about] is that logic studies relations of implication and compatibility between claims. […] To be sure, logic does concern itself with relations of implication and compatibility between claims. The problem is that it is not the only science that does so. A logician can tell you that (1) Julie is a girl who loves every boy. implies (2) Every boy is loved by some girl. But to learn whether (3) This substance turns litmus paper red. implies (4) This substance is an acid, one has to ask a chemist, not a logician. MacFarlane, What does it mean to say that Logic is Formal?, p.1-2
It might be thought that logical implication can be characterized as implication that is knowledgeable a priori, necessary, and/or analytic (Alternatively, we might say that logic is the study of implication in the strict sense, and that implication in the strict sense must be knowable a priori, necessary and analytic.). All three criteria seem to weed out the inference from (3) to (4). We cannot know a priori that (3) implies (4): we must do experiments in order to see that there is a connection between turning litmus red and acidity. Nor is the inference from (3) to (4) necessary, in any deeper sense than “demanded by physical law”: if the laws of nature were different, acids might not turn litmus red. Finally, our understanding of the meanings of “acid”, “litmus paper” and “red” does not give us a sufficient basis for accepting that (3) implies (4). But there are three problems with characterizing logic as the study of a priori, necessary, and/or analytic implication. First, this approach presupposes that we have a reasonably clear understanding of a priori knowledge, (broadly logical) necessity, and/or analyticity notions many philosophers regard as dubious or at least unclear. Perhaps the very idea that there is a non-arbitrary dividing line between logic and other disciplines stands or falls with these notions. But before drawing that conclusion, it seems worth exploring other alternatives. Second, such demarcations threaten to make the scope of logic much broader than it has traditionally been taken to be. For example, the inference from (6) The ball is red. to (7) The ball is colored. is surely analytic, a priori and necessary, if anything is; but it is usually not held to be logically valid. Again, if the natural numbers exist necessarily, then (8) There are infinitely many objects. is a necessary truth; but it is often held that logic alone cannot justify any existence claims. I will not multiply examples. This objection is not merely an appeal to tradition. There are important philosophical positions - notably Kantianism - that cannot even be articulated unless there is a conceptual space between the logical and the necessary and a priori. Third, while it may be true that logic is analytic, knowable a priori, and necessary, it would be nice if we could explain why logic has these features. By building on the notions of analyticity, a prioricity, or necessity into the very concept of logic, we preclude meaningful discussion of whether and why logic is analytic, knowable a priori, or necessary. We also preclude using the notion of logic in an account of analyticity or necessity, on pain of circularity. Yet historically it has usually been the notion of logic that elucidates analyticity and necessity, not vice versa. For example, Kant appeals to logic to clarify the difference between analytic and synthetic judgements, Frege defines analytic truths as those whose justification requires only logic and definitions, and Leibniz invokes the notion of logical compatibility in explaining the notion of a possible world. [An exception to this usual order of explanation is Carnap, who uses the notion of analyticity to explicate logicality and necessity.] For all of these reasons, we ought to seek a demarcation of logic that does not appeal to the notions of analyticity, a prioricity, or necessity. MacFarlane, What does it mean to say that Logic is Formal?, p.4-6
Logicism
[…] mathematical concepts are definable in terms of logical concepts, mathematical truths are reducible to logical truths, and mathematical knowledge is really logical knowledge. MacFarlane, What does it mean to say that logic is formal?, p.7
A popular way to attack logicism is to argue that the “logic” to which mathematics has been reduced is really just more mathematics. Poincare charges that Russel’s “logical” principles are really intuitive, synthetic judgements in disguise: “We regard them as intuitive when we meet them more or less explicitly enunciated in mathematical treatises; have they changed character because the meaning of the word logic has been enlarged and now we find them in a book entitled ‘Treatise on Logic’”? Quine, a more sympathetic critic, draws a similar conclusion: “Frege, Whitehead and Russell made a point of reducing mathematics to logic; Frege claimed in 1884 to have proved in this way, contrary to Kant, that the truths of arithmetic are analytic. But the logic capable of encompassing this reduction was logic inclusive of set theory.” Applied to the full-fledged logicism that identifies numbers with classes, Quine’s criticism is just - and nearly universally accepted. [Note] Frege’s class existence axiom, a plausible candidate for a logical law, was shown to be inconsistent, and all known consistent theories of classes require existence assumptions that seem to lack the “special character” of logic. For Frege, this spelled the death of logicism. Although Russell persisted in maintaining the logicist theses, he acknowledged that his axioms of infinity and reducibility are not logical and must be taken as hypotheses. MacFarlane, What does it mean to say that logic is formal?, p.8
Principled vs Pragmatic Demarcation of Logic
First, pragmatic approaches are guided by what Warmbrod calls a “requirement of minimalism”: … logical theory should be as simple, as modest in its assumptions, and as flexible as possible given the goal of providing a conceptual apparatus adequate for the project of systematization. In practice, the minimalist constraint dictates that the set of terms recognized as logical constants should be as small as possible (Warmbrod 1999:521). While pragmatic demarcations seek to minimize what counts as logic, principled demarcations are inclusive. They count as logical any notion (rule, system, etc.) that has the favored property. It is simply irrelevant whether a notion (rule, system) is required for a particular purpose: its logicality rests on features that it has independently of any use to which we might put it. MacFarlane, What does it mean to say that Logic is Formal?, p.16-17
The literature on demarcating logic contains examples of both general approaches (Principled: Kneale, Tarski, Dummett, Peacocke, Hacking, McCarthy, Sher, Dosen; Pragmatic: Harman, Tharp, Quine, Wagner, Shapiro, Warmbrod) and it is important to recognize three differences between them. MacFarlane, What does it mean to say that Logic is Formal?, p.16
First, it is useful to distinguish between two general strategies for demarcating logic. The first is to identify some favored property (formality, topic-neutrality, conservativeness of introduction and elimination rules, permutation invariance, etc.), or perhaps some combination of properties, as a necessary and sufficient condition for logicality. Call approaches like this principled demarcations. The second strategy is to start with a particular job - for example, to serve as a “framework for the deductive systematization of scientific theories”, or a framework for the characterization of structures - then find something that is capable of doing that job, and identify it as logic. Call approaches like this pragmatic demarcations. MacFarlane, What does it mean to say that Logic is Formal?, p.16
A second difference is that pragmatic approaches tend to be holistic, in a way that principled approaches usually are not. Since it is the whole logical system that does the job, systemic properties - decidability, completeness, compactness, the Lowenheim-Skolem property - play a greater role in pragmatic demarcations than local features of particular notions or inference rules (see Quine 1986 ch.5, Tharp 1975, Wagner 1987). Although nothing about the idea of a principled demarcation excludes appeal to systematic properties, principled demarcations tend to appeal to local properties of particular notions or inference rules (e.g. topic-neutrality, necessity, introducibility through conservative inferential definitions), rather than systemic properties. MacFarlane, What does it mean to say that Logic is Formal?, p.17
The third (and most important) difference between pragmatic and principled demarcations has to do with the consequences one can draw for their verdicts about logicality. A demarcation tells us that this implication is a logical consequence, that one non-logical: but what follows from this classification? Not much, if the demarcation is pragmatic; potentially quite a bit, if the demarcation is principled. To see why, consider the significance of a demarcation of logic for the evaluation of logicism. What would be the upshot of showing that some significant part of mathematics is reducible to logic, where logic is given a pragmatic demarcation? We would have shown that the minimal conceptual apparatus that is significant for some purpose (say, deductive systematization of theories) already allows us to do a considerable amount of mathematics. Such a result would not be uninteresting, but it would not shed much light on traditional questions in the philosophy of mathematics (what is the source of objectivity of mathematics? does mathematics have special objects? how do we come by our mathematical knowledge?). If, on the other hand, we had a principled demarcation of logic, then a logical thesis would have clear philosophical significance: whatever privilege feature we use to pick out logic could be immediately transferred to the reduced body of mathematics. For example, if logical implication were demarcated in part by its necessity (in some sense), then the reduced body of mathematics would have been shown to be necessary (in the same sense). […] MacFarlane, What does it mean to say that Logic is Formal?, p.17-18
It seems to me that the philosophical projects that give point and purpose to demarcating logic in the first place require a principled rather than a pragmatic demarcation. The philosophical interest of projects like logicism depends on logicality having some philosophically interesting essence hence on logic being picked out by a distinguishing property or properties, not by its suitability for some task. […] Hence the focus of this investigation is on principled demarcations. MacFarlane, What does it mean to say that Logic is Formal?, p.19
Aris Goudouras
I don’t really understand the difference between systemic and local properties
Historical Importance of Formality in Logic
[…] the universal and necessary rules of thought in general can concern merely its form and not in any way its matter. Accordingly, the science that contains these universal and necessary rules is merely a science of the form of our cognition through the understanding, or of thought. Kant, Jasche Logic (Logic: A Manual for Lectures), p.12
Logic inquires into the form of thought, as separable from and independent of the matter of thought of. De Morgan, On the Syllogism (1858)
What is of concern to logic is not the special content of any particular relation, but only the logical form. Frege
Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, “purely formal”. Russell, Introduction to Mathematical Philosophy, p.198
The progress achieved by axiomatics consists in its having neatly separated the logical-formal form from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject-matter of mathematics, which is not concerned with the intuitive or other content associated with the logical-formal. Einstein, Geometrie and Erfahrung, p.28
[…] since we are concerned with the concept of the logical, i.e., formal, consequence, and thus with a relation which is to be uniquely 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, On the Concept of Logical Consequence, p.414-415
These passages exemplify a tradition, going back at least to Kant and encompassing philosophers and logicians with wildly different philosophical views, of demarcating logic by its formality. I call this tradition logical hylomorphism. Logical hylomorphism is a tradition or current, not a thesis, because its advocates understand the formality of logic in very different ways. MacFarlane, What does it mean to say that Logic is Formal?, p.21

MacFarlane lists various representative passages which he considers represent a current in favor of formality in the history of philosophy.

This is not to say that we should ignore all our intuitions about logicality. But before invoking these intuitions in justifying or criticizing a proposed demarcation of logic, we ought to seek their sources in the philosophical tradition. By studying the history of conceptions of logicality we can see why philosophers have the intuitions they do. Knowing this, we can proceed to ask which intuitions we still have reasons to have, which go together, and which come from incompatible traditions. Historical reflection is a way to make our “brute intuitions” less “brute”. MacFarlane, What does it mean to say that Logic is Formal?, p.25-26
[…] Our intuitions about logicality are not a kind of perception of an external reality: they are historical artifacts, a product of our logical and philosophical educations. To the extent that there is intersubjective agreement about them, it should be attributed to a shared tradition, not access to a tradition-independent reality. As Stewart Shapiro and others have pointed out, the very idea of “pretheoretical logical intuitions” is dubious. […] All our intuitions about logicality bear the stamp of theory. If we have the intuition that logic must be “formal”, this is not because of some extrasensory perception of the essence of logic, but because we have encountered the idea so often in the course of our philosophical training. MacFarlane, What does it mean to say that Logic is Formal?, p.26
Logic, it is often said, is distinctively formal: it concerns itself with those relations of implication and consistency that turn only on “forms” or “formal features” of thoughts or statements, abstracting from the “matter” or “content”. MacFarlane, What does it mean to say that Logic is Formal?, p.50
3 Notions of Logical Formality according to MacFarlane
1-formality (Normativity for thought as such)
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
Aris Goudouras
Is this idea supported by the Tractatus too?
2-formality (Indifference to particular identity)
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
3-formality (Abstraction from semantic content)
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
Aris Goudouras
It seems to me that, indeed, if we are not allowed to think of even these very general concepts like identity and conjunction, there is indeed nothing left, so 3-formality sounds too strong to achieve.
Aris Goudouras
General logic seems to be synonymous with Structure for Kant. But how can one describe differences in structure/form if we are not even allowed to have a vocabulary for it (e.g. how do we understand a judgement as assertoric)? It seems to me that it’s just not possible to abstract away from semantics; we need them to at least describe logical forms / structure.
Kant holds logic to be formal in all three senses. MacFarlane, What does it mean to say that Logic is Formal?, p.66