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January 21, 2010

4:07 AM

So-called material implication cannot adequately do the job of ‘de re’ conditionals

Sixth post in the ongoing series on important innovations in logic theory to be found in my works. The present post is intended as a continuation of the preceding one regarding ‘de re’ conditioning. Although it presents no major innovation, it is needed here to put some novel order in the concept of conditioning, and thus to highlight certain failures of understanding displayed by many logicians since the late 19th Century.

Of course, implication in various guises was used in human discourse long before it was discussed by philosophers. But so far as we know, according to logic historians, the notion of implication was first elucidated by Diodorus (Cronus of Megara, d. circa 307 BCE), who defined it as a sequence of events (or concepts or propositions) such that the first (the antecedent) is always followed by the second (the consequent). Soon after, his disciple Philo (known as the Megarian, though his origin is not sure) advocated a simpler definition of implication, which eschewed the specification ‘always’. For the former, implication (the hypothetical form ‘If P, then Q’) was a modal relation, meaning that the conjunction ‘P and not-Q’ never occurs; whereas for the latter, it sufficed to simply deny that conjunction (effectively, at a given point in time). As closer scrutiny makes clear, Philo was a bad student who should have listened more carefully to his teacher, Diodorus.

Nevertheless, modern logicians (at least those mathematically inclined) considered implication as referring to Philo’s form of implication (later called material implication). This was the situation at least until Clarence I. Lewis revived Diodorus’ form of implication (now called strict implication), in 1918 and more forcefully in 1932. Even after this, many logicians have continued to formulate logic theories or teachings with reference to implication as mere negation of conjunction instead of as impossibility of conjunction (of P and not-Q, given ‘If P, then Q’). They do not yet realize that without appeal to strict implication, we would be hard put to express the difference between disproof (‘proving the contradictory’) and non-sequitur (‘showing the conclusion does not follow from the premises’). Such refinement of discourse is impossible using material implication.

If we compare the ‘truth-tables’ for strict and material implication, they would seem superficially the same as regards the positive aspect ‘if P, then Q’. The truth of P implies that of Q and the falsehood of Q implies that of P. But when we look at the negative aspect, their difference becomes glaring. In strict implication, the negation of ‘if P, then Q’ is ‘if P, not-then Q’, which leaves the respective truths and falsehoods of P and Q open and does not allow us to infer from P the truth or falsehood of Q or from Q the truth or falsehood of P. Whereas in material implication, the negation of ‘if P, then Q’ is simply ‘P and nonQ’, i.e. it tells us categorically that P is true and Q is false!

However, this is only half the story. Already in antiquity to some extent (if only implicitly), and especially since the Scholastics (who seem to have coined the terms), a distinction was made between ‘de dicta’ (or de dicto) and ‘de re’ modalities. The former related to the states of our knowledge (including speculations and hypotheses) about things, whereas the latter concerned the things themselves. That is, more specifically in the present context, the former concerned the logical mode of modality (epistemology), while the latter was about natural, temporal, extensional or similar ontological modes (notably that related to volition, the personal mode).

Many modern logicians after Lewis, whether out of ignorance or naivety, or a stubborn desire to simplify complex issues, have tended to conflate de dicta conditioning with strict implication and de re conditioning with material implication. That is evident in the terminology used – strict means formal, in accordance with logical discourse (i.e. de dicta), and material means contentual, relating to things (i.e. de re); and in the kind of examples they give to illustrate material implication (e.g. ‘if it rains, the match will be called off’). But of course this is quite wrong. Both strict and material implication are (stronger and weaker) forms of logical conditioning. Material implication cannot adequately do the job of ‘de re’ forms of conditioning. This is made evident in Future Logic, part IV, where the latter are examined in detail.

The poverty of modern assumptions in this respect is evident as soon as we try using them to interpret or explain commonly used expressions like ‘when and if’, for example. If ‘when’ and ‘if’ both belong to the logical mode of modality, their conjunction is incomprehensible. Whereas, when we understand the difference between modes of conditioning and between connection and basis of conditioning, such conjunction becomes clear. The ‘when’ tells us that a natural or temporal modality of connection between the theses applies, while the ‘if’ tells us that the base of the antecedent is uncertain, i.e. is merely a logical possibility (and not a de re one). The formal possibilities of material implication cannot be compared to the precision and complexity of discourse made possible by ‘de re’ conditioning. That our practice is more in accord with the latter is easily demonstrated by examples.

Much of this confusion has remained hidden from public view, and to the logicians themselves, due to excessive reliance on symbolic logic.

For more details on this topic, see FUTURE LOGIC, PART III (CHAPTERS 20-32), on logical conditioning. Note in particular chapter 24.3, regarding the strict-material implication distinction.


See also: RUMINATIONS, chapter 4.8, on the strict-material implication distinction; and VOLITION AND ALLIED CAUSAL CONCEPTS, chapter 6.5, regarding ‘personal conditioning’.

2 Comment(s).

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July 6, 2010 @ 2:05 AM

Posted by Jessica:

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February 6, 2010 @ 10:25 AM