Philosophical logic

Despite its name, philosophical logic is neither a kind of logic nor simply to be identified with the philosophy of logic(s)—the latter being the philosophical examination of systems of logic and their applications. Though the subject of philosophical logic is hard to define precisely, it may loosely be described as the philosophical elucidation of those notions that are indispensable for the proper characterization of rational thought and its contents—notions like those of reference, predication, truth, negation, necessity, definition, and entailment. These and related notions are needed in order to give adequate accounts of the structure of thoughts—particularly as expressed in language—and of the relationships in which thoughts stand both to one another and to objects and states of affairs in the world. But it must be emphasized that philosophical logic is not concerned with thought inasmuch as the latter is a psychological process, but only in so far as thoughts have contents which are assessable as true or false. To conflate these concerns is to fall into the error of psychologism , much decried by Frege .

No single way of dividing up the subject-matter of philosophical logic would be agreed upon by all of its practitioners, but one convenient division would be this: theories of reference, theories of truth, the analysis of complex propositions, theories of modality (that is, of necessity, possibility, and related notions), and theories of argument or rational inference. These topics inevitably overlap, but it is roughly true to say that later topics in the list presuppose earlier ones to a greater degree than earlier ones presuppose later ones. The order of topics in the list reflects a general progression from the study of parts of propositions , through the study of whole and compound propositions, to the study of relations between propositions. (Here we use the term ‘proposition’ to denote a thought content assessable as true or false—something expressible by a complete sentence.)

Theories of reference are concerned with the relationships between subpropositional or subsentential parts of thought or speech and their extra-mental or extra-linguistic objects—for instance, with the relationship between names and things named, and with the relationship between predicates and the items to which they apply. According to some theories, a name refers to a particular thing by virtue of its being associated with some description which applies uniquely to that thing. Other theories hold that the link between name and thing named is causal in nature. (Theories of either sort are intimately bound up with questions concerning identity and individuation .) As for predicates—where a predicate may be thought of as what remains when one or more names are deleted from a sentence—these are variously held to carry reference to universals, concepts, or classes . Thus the predicate ‘ … is red’, formed by deleting the name from a sentence like ‘Mars is red’, is held by some philosophical logicians to stand for the property of redness, by others to express our concept of redness, and by yet others to denote the class of red things. Monolithic theories of reference are unpromising, however. Even if some names refer by way of description, other names and namelike parts of speech—such as demonstratives and personal pronouns—plausibly do not. And even if some predicates stand for universals, others—such as negative and disjunctive predicates—can scarcely be held to do so.

Truth and falsehood—if indeed they are properties at all—are properties of whole sentences or propositions, rather than of their subsentential or subpropositional components. Theories of truth are many and various, ranging from the robust and intuitively appealing correspondence theory —which holds that the truth of a sentence or proposition consists in its correspondence to extra-linguistic or extra-mental fact —to the redundancy theory at the other extreme, according to which all talk of truth and falsehood is, at least in principle, eliminable without loss of expressive power. These two theories are examples, respectively, of substantive and deflationary accounts of truth , other substantive theories being the coherence theory , the pragmatic theory , and the semantic theory , while other deflationary theories include the prosentential theory and the performative theory (which sees the truth-predicate ‘ … is true’ as a device for the expression of agreement between speakers). As with the theory of reference, a monolithic approach to truth, despite its attractive simplicity, may not be capable of doing justice to all applications of the notion. Thus the correspondence theory, though plausible as regards a posteriori or empirical truths, is apparently not equipped to deal with a priori or analytic truths , since there is no very obvious ‘fact’ to which a truth like ‘Everything is either red or not red’ can be seen to ‘correspond’. Again, the performative theory, while attractive as an account of the use of a sentence like ‘That's true!’ uttered in response to another's assertion, has trouble in accounting for the use of the truth-predicate in the antecedent of a conditional, where no assertion is made or implied.

Whichever theory or theories of truth a philosophical logician favours, he or she will need at some stage to address questions concerning the value of truth—for instance, why should we aim at truth rather than falsehood?—and the paradoxes to which the notion of truth can give rise (such as the paradox of the liar). In the course of those inquiries, fundamental principles thought to govern the notion of truth will inevitably come under scrutiny—such as the principle of bivalence (the principle that every assertoric sentence is either true or false). A rejection of that principle in some area of discourse is widely supposed to signify an anti-realist conception of its subject-matter.

Propositions and sentences can be either simple or complex (atomic or compound). A simple sentence typically concatenates a single name with unitary predicate, as, for example, in ‘Mars is red’. (Relational sentences involve more names, as in ‘Mars is smaller than Venus’, but a sentence like this is still regarded as simple.) One way in which complex sentences can be formed is by modifying or connecting simple ones; for instance, by negating ‘Mars is red’ to form the negation ‘Mars is not red’, or by conjoining it with ‘Venus is white’ to form ‘Mars is red and Venus is white’. Sentential operators and connectives , like ‘not’, ‘and’, ‘or’, and ‘if’, are extensively studied by philosophical logicians. In many cases, these operators and connectives can plausibly be held to be truth-functional —meaning that the truth-value of complex sentences formed with their aid is determined entirely by the truth-values of the component sentences involved (as, for example, ‘Mars is not red’ is true just in case ‘Mars is red’ is not true). But in other cases—and notably with the conditional connective ‘if’—a claim of truth-functionality is less compelling. The analysis of conditional sentences has accordingly become a major topic in philosophical logic, with some theorists seeing them as involving modal notions while others favour probabilistic analyses.

There are other ways of forming complex sentences than by connecting simpler ones, the most important being through the use of quantifiers—expressions like ‘something’, ‘nobody’, ‘every planet’, and ‘most dogs’. The analysis and interpretation of such expressions forms another major area of philosophical logic. An example of an important issue which arises under this heading is the question how existential propositions should be understood—propositions like ‘Mars exists’ or ‘Planets exist’. According to one approach, the latter may be analysed as meaning ‘Something is a planet’ and the former as ‘Something is identical with Mars’ (both of which involve a quantifier), but this is not universally accepted as correct. Another issue connected with the role of quantifiers is the question how definite descriptions—expressions of the form ‘the so-and-so’—should be interpreted, whether as referential (or namelike) or alternatively as implicitly quantificational in force, as Russell held.

The fourth topic in our list is theories of modality , that is, accounts of such notions as necessity , possibility, and contingency, along with associated concepts such as that of analyticity. One broad distinction that is commonly drawn is that between de re and de dicto necessity and possibility, the former concerning objects and their properties and the latter concerning propositions or sentences. Thus, a supposedly analytic truth such as ‘All bachelors are unmarried’ is widely regarded as constituting a de dicto necessity, in that, given its meaning, what it says could not be false. But notice that this does not imply that any man who happens to be a bachelor is incapable of being married—though should he become so, it will, of course, no longer be correct to describe him as a ‘bachelor’. Thus there is no de re necessity for any man to be unmarried, even if he should happen to be a bachelor. By contrast, there arguably is a de re necessity for any man to have a body consisting of flesh and bones, since the property of having such a body is apparently essential to being human.

As for the question how, if at all, we can analyse modal propositions, opinions vary between those who regard modal notions as fundamental and irreducible and those who regard them as being explicable in other terms—for instance, in terms of possible worlds , conceived as ‘ways the world might have been’. (Although this appears circular, in that ‘possible’ and ‘might’ are themselves modal expressions, with care the appearance is arguably removable.) For instance, the claim that every man necessarily has a body made of flesh and bones might be construed as equivalent to saying, of each man, that he has a body made of flesh and bones in every possible world in which he exists. However, we should always be on guard against ambiguity when talking of necessity, because it comes in many different varieties—logical necessity, metaphysical necessity, epistemic necessity, and nomic necessity being just four.

Modal expressions give rise to special problems in so far as they often appear to create contexts which are non-extensional or ‘opaque’ ( extensionality)—such a context being one in which one term cannot always be substituted for another having the same reference without affecting the truth-value of the modal sentence as a whole in which the term appears. For example, substituting ‘the number of the planets’ for ‘nine’ in the sentence ‘Necessarily, nine is greater than seven’, appears to change its truth-value from truth to falsehood, even though those terms have the same reference. (No such change occurs if the modal expression ‘necessarily’ is dropped from the sentence.) How to handle such phenomena—which also arise in connection with the so-called propositional attitudes , such as belief—is another widely studied area of philosophical logic.

Finally, we come to questions concerning relations between propositions or sentences—relations such as those of entailment , presupposition, and confirmation (or probabilistic support). Such relations are the subject-matter of the general theory of rational argument or inference , whether deductive or inductive . Some theorists regard entailment as analysable in terms of the modal notion of logical necessity—holding that a proposition p entails a proposition q just in case the conjunction of p and the negation of q is logically impossible. This view, however, has the queer consequence that a contradiction entails any proposition whatever, whence it is rejected by philosophers who insist that there must be a ‘relevant connection’ between a proposition and any proposition which it can be said to entail. ( Relevance logic .) The notion of presupposition, though widely appealed to by philosophers, is difficult to distinguish precisely from that of entailment, but according to one line of thought a statement S presupposes a statement T just in case S fails to be either true or false unless T is true. For instance, the statement that the present King of France is bald might be said to presuppose, in this sense, that France currently has a male monarch. (Such an approach obviously requires some restriction to be placed on the principle of bivalence.) As for the notion of confirmation, understood as a relation between propositions licensing some form of non-demonstrative inference (such as an inference to the truth of an empirical generalization from the truth of observation statements in agreement with it), this is widely supposed to be explicable in terms of the theory of probability—though precisely how the notion of probability should itself be interpreted is still a matter of widespread controversy.

No general theory of argument or inference would be complete without an account of the various fallacies and paradoxes which beset our attempts to reason from premiss to conclusion. A ‘good’ argument should at least be truth-preserving, that is, should not carry us from true premisses to a false conclusion. A fallacy is an argument, or form of argument, which is capable of failing in this respect, such as the argument from ‘If Jones is poor, he is honest’ and ‘Jones is honest’ to ‘Jones is poor’ (the fallacy of affirming the consequent), since these premisses could be true and yet the conclusion false. (Strictly, this only serves to characterize a fallacy of deductive reasoning.) A paradox arises when apparently true premisses appear to lead, by what seems to be a good argument, to a conclusion which is manifestly false—a situation which requires us either to reject some of the premisses or to find fault with the method of inference employed. An example would be the paradox of the heap (the Sorites paradox): one stone does not make a heap, nor does adding one stone to a number of stones which do not make a heap turn them into a heap—from which it appears to follow that no number of stones, however large, can make a heap. This paradox is typical of many which are connected with the vagueness of many of our concepts and expressions, a topic which has received much attention from philosophical logicians in recent years. This is again an area in which the principle of bivalence has come under some pressure.

Although philosophical logic should not be confused with the philosophy of logic(s), the latter must ultimately be responsive to considerations addressed by the former. In assessing the adequacy and applicability of any system of formal logic, one must ask whether the axioms or rules it employs can, when suitably interpreted, properly serve to articulate the structure of rational thought concerning some chosen domain—and this implies that what constitutes ‘ rationality ’ cannot be laid down by logicians, but is rather something which the formulators of logical systems must endeavour to reflect in the principles of inference which they enunciate.