The Definition of Logical Validity

What does it mean for a conclusion to follow logically from other sentences? Logical consequence is at the basis of any rational reasoning. Textbooks generally give a precise and formal account of logical consequence, but rarely provide a justification for this account. In this monograph the foundations of logical consequence are examined.

According to the semantic definition of logical consequence or validity, an argument is logically valid if, and only if, the conclusion is true under all interpretations under which also all premisses are true. The semantic definition is only a general pattern, and Volker Halbach presents a specific way of spelling out this definition. In contrast to the predominant approaches, truth is taken to be a primitive notion, which is not reduced away by a mathematical definition. An interpretation of a sentence is obtained by replacing non-logical terms uniformly with arbitrary other terms of the same grammatical kind. This conception of interpretations is in line with naïve and straightforward understandings of interpretations that hark back at least to the middle ages. The resulting definition of logical validity combines two advantages: first, it is universal in the sense that it applies not only to a restricted language, but to the entire in which the definition is stated. This is in contrast to definitions of logical validity in higher-order languages. Secondly, it admits the intended interpretation, that is, the interpretation of sentences at their face value without any re-interpretation. Thus, logical consequence is trivial preserving truth. The usual model-theoretic definition lacks this property, although it has always taken to be fundamental to logical consequence and warrants its usefulness in philosophy.