Logical foundations: is there one true logic?
Position: Logical monism
Why are we now moving to logic before anything else? The reason will become clearer as we unpack the four theses that make up our position:
Ontological theses (about logic’s existence):
Logical objectivity: there are objective logical facts and structures.
Logical realism: these logical facts and structures exist.
Metaphysical theses (about logic’s place in the world’s structure):
Logical primitivism: logic is more fundamental than general ontology and metaphysics.
Logical monism: there is one logic above them all.
Historically, logic has been seen as the study of valid inference (for instance in natural-language arguments), or else of the laws of thought. It is tempting to conclude that logical facts are then dependent on our language or our minds, and indeed various forms of psychologism or constructivism have long been in vogue. Frege detected the mark of that mindset in Kant’s views, who saw logical facts as purely formal and an example of synthetic a priori knowledge, only attainable via some form of intuition inherent to minds. Frege saw such appeals to intuition as intolerable, the cause for shaky foundations of logic. He preferred to sharply distinguish between logic’s prescriptive role (outlining normative laws of thought) and its descriptive role as regards what he calls the laws of truth. Logical facts, logical constants (like truth-values, connectives or quantifiers) and the concept of logical consequence are then all objective.
Strictly speaking, this objectivity doesn’t entail realism: the separate thesis that logical objects exist in an ontological sense, in the same way that you or I exist. Yet I will endorse logical realism. The objective nature of logic means that it is not grounded in features of the physical or the mental realms. Yet, logical facts make their presence felt in those realms, for instance if one accepts that there are laws of physics (I do), or that the laws of thought stem from the "laws of truth". This suggests that the fundamental structure of the world is at least partially grounded in logical laws, which woud impose that all relevant constants and structures be existent. There are ways around that conclusion, but in a fashion that should be familiar by now, I will forego ontological parsimony in favor of explanatory power, and accept that logical facts, constants and structures exist. The combination of objectivity with this realist thesis thus yields a form of logical Platonism (i.e. we are starting to populate an abstract domain of existence that is separate from the physical and mental spheres).
Although it wasn’t always couched in these openly Platonist terms, logical realism has been prevalent since the turn of the twentieth century amongst representatives of such different traditions as Frege, Husserl, or Heidegger. One significant difference emerged, however, about the primacy of logic over a more general ontology. Heidegger is as stridently critical of psychologism as Frege, and as insistent that logic should be part of "first philosophy". But to him, logical laws must be grounded into a "metaphysics of truth": the connections and relations it sets out to manipulate cannot be understood without a science of "being qua being" (being in itself). Frege takes the opposite stance: metaphysics, like natural science, shall be consistent with logical principles, and requires thinking (in an abstract sense); therefore it necessarily presupposes logic. More prosaically we can see that difference pervade through the style of each man: Heidegger explains metaphysics through his famously dense prose, whilst Frege aims to formalize as much as he can. The position that logic grounds ontology, and then that it is legitimate for the present paragraphs to precede any discussion of general ontology, I call logical primitivism.
Lastly, we must consider whether we can talk about "logic" or only "logics". Classical logic had been dominant from Aristotle’s time until Frege, at which point predicate logic took over as the prevalent force. Many others have been devised since: second-order, intuitionist, many-valued, paraconsistent, etc. The point of logical monism is not to question their consistency or their usefulness: these are perfectly serviceable formal systems. Under the prescriptive mission of the discipline, we can recognize that some of these non-standard logics are better representations of natural language inference, and thus provide better norms for arguments. Under its descriptive role, some concrete systems might be more adequately explained in terms of a tailor-made logic than of classical logic— see the use of quantum logic in quantum theory. Mathematically, the difference between alternative logics can be reduced to a different definition of logical consequence, or a different set of logical constants; although some may be more general than others, there is no case to be made for one true mathematical logic. Ontologically speaking, we must paint a different picture. Using, in a Fregean fashion, logic as the underpinning of ontology tells us that logic ought to carve reality at the joints: it represents the most fundamental structure of the world. Having distinct logics cohabiting in a pluralist model directly entails that contradictions and vicious regresses must exist. It is possible to accomodate those in a dialetheic way: that is, to re-phrase the contradictions via a higher-level meta-language, in such a way that the principle of explosion (that anything follows from a contradiction) is avoided. But this meta-language must itself be grounded in a logical system. That is quite a common move: for instance possible-worlds metaphysics allows to reframe modal logic in the language of standard predicate logic. Isn’t it the case then, via a recursive principle, that there must be an ultimate meta-language that we can acknowledge as our one true logic? That is what I contend, and why I am a logical monist.