What is (not) Logic, and what is it (not) Good for?
To the memory of Hans Wussing (1927-2011),
historian and historiographer of mathematics
Building on a previous article on assertion and negating a proposition, an attempt is made to characterise logical knowledge from other kinds, especially mathematics. The tradition of stressing forms of proposition is revived, showing that logic is dependent on the context to which it is applied. Attention is paid to several theories that overlap with both logic and foundational branches of mathematics; they include set theory, model theory, axiomatisation and metamathematics.
Logic is concerned with the real world just as truly as zoology, though with its
more abstract and general features.
Bertrand Russell [1919, 169]
In a previous article [Grattan-Guinness 2011d], named ‘GGan’ here, two features of the assertion of a proposition (that is, the assignment to it of a truth-value) in classical two-valued logic, and various modes of negating it, were examined in some detail and applied to various contexts. Here several features of that study are extended in an attempt to distinguish logic from other kinds of knowledge, especially mathematics.
The uses made in GGan of several technical terms in and around logic are maintained here. So also is the preference to take the proposition rather than the term as the primary notion of logical knowledge. Otherwise the position put forward here is intended to be neutral about various philosophical positions and issues. Of especial pertinence is the question of whether logical knowledge is primarily concerned with (in)valid deduction and truth transmission or with the processing of information [Sagüillo 2009].
To avoid excessive length, the discussion is usually limited to propositions; the consequences for sentences as propositions in a language and to statements as utterances of sentences are not normally explored. The intentions of the utterer include persuading others to share his beliefs and knowledge of what he knows to be true or untrue, improving the cogency of an argument by converting it to a line of reasoning that is already well known, detecting errors in the logic of an argument, using words such as ‘true’ and ‘untrue’ metaphorically, exploiting equivocations, ambiguities and jokes, and even resorting to deliberate lying. There are important issues here, called ‘argumentations’, well captured in [Corcoran 1989] and [Walton 1989, 1996]; they complement the discussion proffered here.
Given any proposition R, the asserted proposition ‘It is true that R’ (symbolised ‘+R’) is the ‘affirmation’ of R, while ‘It is untrue that R’ (‘–R’) is its ‘denial’. R is a proposition about some states of affairs in a context. This can be anything: plasma physics, or structures in piano sonatas, or making wine at home, or the publication of [Russell 1919], or …. . An important special case of self-reference occurs when the context is logic itself. The key question is: what role does logic play in these assertions?
Logical knowledge is notoriously elusive, difficult to detach from its applications. This characterisation starts by proposing four main departments of logic (proposition, propositional function, deduction, assertion), all construed in a way that distinguishes them from other kinds of knowledge, especially “neighbouring” theories such as collections (including set theory), metamathematics, model theory and some “close-by” branches of mathematics such as arithmetic and abstract algebras. The focus remains upon two-valued logic (called ‘bivalent’ and named ‘L2’), comprising both propositional and functional calculi; but some version of our considerations obtains in any other logic, at least ones with explicit organisation and prominent symbolism.
 A much more elaborate philosophical and historical version of this article and its predecessor is in preparation as [Grattan-Guinness 2011c].
 Other issues include intension versus extension, Platonism versus empiricism versus nominalism versus a priorism versus psychologism versus formalism [Weir 2010], analysis versus synthesis [Otte and Panza 1997], and the status of universals. Some of these issues may well bear less upon logic than upon its applications.