Relating Logic is a logic of relating connectives – just as Modal Logic is a logic of modal operators. The basic idea behind a relating connectives is that the logical value of a given complex proposition is the result of two things: (i) the logical values of the main components of this complex proposition; supplemented with (ii) a valuation of the relation between these components. The latter element is a formal representation of an intensional relation that emerges from the connection of several simpler propositions into one more complex.

Although the simplest model for a relating logic is a pair consisting of a valuation function and a relation, the situation may get more complicated. We can use multi-relating models to represent more types of relations between sentences. In addition, the valuation of relationships between sentences may not be binary but may be many-valued or more subtly graded. Furthermore, we can mix relating semantics with possible world semantics, equipping all worlds with additional valuations of complex sentences.

The solution that relating logics offers seems to be quite natural, since when two (or more) propositions in natural language are connected by a connective, some sort of emergence occurs. In fact, the key feature of intensionality is that adding a new connective results in the emergence of a new quality, which itself does not belong to the components of a given complex proposition built by means of the same connective. An additional valuation function determines precisely this quality. When examining reasoning in logic, we usually consider affirming a logical relationship between the premises and the conclusion so that any situation which assigns a meaning of true to the premises, must assign a meaning of true to the conclusion. However, in many cases, there are non-logical relationships that can greatly contribute to the recognition of reasoning. Consider the following standard example:

 The inference (♢) can be seen as an instance of the transitivity of classical material implication if we read “if. . . then. . .” as the material implication. Thus, in a classical setting, in which the truth values of the implications are determined only by the truth values of the subformulas, the inference (♢) is correct. But clearly, the conclusion of (♢) is bizarre as there is no direct logical connection between the thief trying to rob your house and the thief’s running away! Our challenge is to accommodate extra, non-logical, relationships such as “causation”, to block (♢). This is exactly what relating semantics provides. But “causation” is just one example of a non-logical relationship. In many inferences, similar relationships of a non-logical nature also appear. These include not only causal relationship but also temporal, analytical, content-based, preferential, structural relationships etc. These are intensional relationships, because they are irreducible to the properties of their elements.

The aim of the tutorial is to outline relating semantics in three parts: (a) the history of relating semantics and the motivation behind it, (b) the application of relating semantics to define systems of philosophical logic, and (c) the foundations of proof theory for relating logic (axiomatization, tableaux, natural deduction).

Some selected references to the topic are: related to history and motivations: [20], [28], [6], [7], [28], [30], [38], [39], [15], [36], [8], [2], [24]; related to applications to philosophical logic: [11], [1, 25], [14], [21], [34], [22], [32], [33], [40], [37], [22], [23], [16], [18], [9], [13], [12], [29], [10]; related to proof theory: [6], [7], [27], [3], [19], [4], [5], [35], [26], [31], [17]

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Accessed: 2022-12-10.