Thursday, October 19, 2006

How Not to Think About Formal Logic

I teach a course on Critical Thinking each semester and we're now getting into some formal logic - just scratching the surface, really. But each time I notice that some students catch on very quickly and just accept the principles as I lay them out; others catch on quickly but immediately want to challenge the principles, especially when it comes to translating ordinary English into standard logic form; still others seem to never catch on, in some cases because of a mental block toward anything that seems even quasi-mathematical. This has gotten me thinking about different attitudes toward formal logic, of which it seems to me that there is one healthy attitude and at least three different sorts of unhealthy attitude.

The first unhealthy attitude is indifference. Formal logic, who cares? If I can't use it to bake a loaf a bread, what good is it? Fair questions, but easily answered by the fact that setting our reasonings out in explicit, formal terms is often indispensable for making them clear enough for our reasoning to reach any significant degree of rigor. Just imagine where disciplines like mathematics and physics would be today if no one had ever bothered to formulate mathematical axioms and physical laws in explicit, formal terms. Certainly those fields would be nowhere nearly as advanced, and we would be without many of the technological developments (e.g., computers) made possible by those advances.

The second unhealthy attitude that of fear and loathing. For some students applying any kind of formal regimen to their thinking is about as fun as pulling teeth. The formal rules seem so abstract, so alien, and if they can't immediately bring things back into concrete and familiar terms, they feel lost and adrift in a unending sea of incomprehensible symbols. In response to this, I would simply point out that the basic rules of formal logic are not as alien as the formal mode of representation might make them seem. Formal logic is really just applied set theory. And that, in turn, is really just a matter of comparing and contrasting groups of things, something we do all the time. "These two groups are really identical (e.g., triangles and 3-sides polygons)." "This group is a subgroup of that one (e.g., squares and rectangles)." "That group only partially overlaps with this one (e.g., heroes and celebrities)." "These two groups are wholly distinct (humans and fish)." Etc. Anyone who can classify ordinary objects in commonsense ways already has all of the conceptual tools they need to do basic formal logic.

The third unhealthy attitude is an uncritical zeal. When some people first encounter formal logic, they are so impressed by the power it gives us to make nice, tight formal proofs, that they learn the rules and immediately apply them with abandon, often in a rote "plug and chug" way. Take an ordinary English statement, formalize it in terms of the logical resources that have been learned, construct a formal proof, and then translate the conclusion back into ordinary English. The main problem with this is that the translation from ordinary English to logical formalization is sometimes problematic. The rigor and power of formal logic come from the fact that it tightly regiments and limits the expressive power of ordinary language. There's a tradeoff here. By sacrificing expressive power we gain in rigor, but we also court the danger that imposition of standard logical forms on ordinary language cuts out something really important and thus distorts the meaning of what we're trying to express. In short, we must bear in mind that formal logic has intrinsic limits. We can develop more and more complex formal systems to account for additional nuances of meaning, but there is no way to capture the full expressive power of ordinary language in formal terms. Syntax cannot replace semantics.

The healthy attitude toward formal logic is, therefore, one of critical respect - respect for the power of formal logic as a tool that can help us make our ideas clear and construct rigorous proofs and disproofs, but a respect that is critical because it bears in mind the limitations of formal logic and is cautious in application.


At 10/22/2006 1:11 PM, Blogger Macuquinas d' Oro said...

Hi Alan,

Nice post and completely in agreement with my classroom experience.

One thing I found useful, particularly in studying conditional inferences with my students, was to show them at the same time how to model such inferences in richer formal systems that use non-truth functional connectives. Recognizing that such systems are available options give them a perspective on both the limitations and advantages of a much simpler TF logic. That way we also get away from the fiction of pretending that p>q or pvq are translations of natural language sentences and their connectives.

At 10/22/2006 4:59 PM, Blogger Alan Rhoda said...

Thanks for the comment, Mac. I think you're suggestion is a good one.

At 10/22/2006 5:00 PM, Blogger Alan Rhoda said...

Oops, that should have been "your", not "you're".


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