LESSON 10: LOGICALTRUTH & TAUTOLOGIESAssigned reading pp. 93-103POWERPOINT SLIDE #1LOGICAL POSSIBILITY:A sentence-claim is logically possible if there is some (i.e., at least one) logically conceivable circumstance (situation or world) in which the claim is true. For example, “Jimmy Fallon is the president of the United States” Though it’s not true in our world, the ‘real’ world, we can at least imagine a ‘possible’ world in which it might be true; that is, there is nothing logically contradictory about the notion of Jimmy Fallon being president, say, in some parallel universe more or less like ours except with Fallon in the Oval Office.POWERPOINT SLIDE #2Proving logical possibility is simple: If you can conceive of a world (situation or state of affairs) in which the sentence is true, then you’ve demonstrated that it is logically possible. (Of course, the world has to abide by the laws of logic for this to count. So, for example, imagining a world in which multiple names of a single object are distributed onto other objects in the world would violate the indiscernibility of identicals and not be a ‘logical’ world for our purposes.)POWERPOINT SLIDE #3Logical possibility can also be interpreted more narrowly as being specific to a particular world or set of worlds, such as the worlds that can be built with the Tarski’s World program (i.e., each world a particularblock-inhabited checkered board). For example: Cube(a) ᴧ Larger(a,b)Tarski’s World has it rules and limits, so we won’t find many imaginable situations realized in Tarski’s World (e.g., you’ll never see a Tarski’s World world with tetrahedrons riding dancing ponies). Sentences that can be made true by at least one conceivable orientation of blocks in Tarski’s World are called in the book TW-possible. POWERPOINT SLIDE #4To prove that a sentence is TW-possible, all you have to do is construct a world in Tarski’s World that makes that sentence true.For example, to show that Cube(a) ᴧ Larger(a,b) is TW-possible, you can simply construct aworld with a cube in it labeled a that is bigger than some other object labeled b in the same world).POWERPOINT SLIDE #5LOGICAL NECESSITY: In the most general sense, a sentence-claim is logically necessary if it is true in every logically possible circumstance (in every logically conceivable world, at least in worlds that can still be considered ‘logical’).Such a sentence is called a logical truth. For example, “Jimmy Fallon is Jimmy Fallon” is a logical truth, as long as we assume that the name ‘Jimmy Fallon’ in each case refers to the same individual (and we can assume this since equivocation of terms isn’t allowed in FOL: that is, each individual constant can refer only and always to one individual, so as to rule out any ambiguity).POWERPOINT SLIDE #6Like logical possibility, logical necessity can be interpreted more narrowly as being necessity relative to a particular world or set of worlds, like those constructible in Tarski’s World. Sentences that are true in every conceivable orientation of blocks in Tarski’s World are called TW-necessary. This may include some sentences that are necessary in the world merely because of the particular features or limits of that world.For example, the sentence Tet(a) ᴠ Cube(a) ᴠ Dodec(a) is necessarily true in any Tarski’s World world simply because Tetrahedron, Cube, and Dodecahedron are the only three shapes of objects available in the program.Obviously, in most non-TW worlds (like our ‘real’ world), many other shapes of objects are possible, so a parallel sentence like “Object a is a tetrahedron or a cube or a dodecahedron” would not be logically necessary in those non-TW worlds (e.g., object ‘a’ could refer to my pet goldfish, which isn’t a tet or cube or dodec—but my pet goldfish can’t exist within any of the Tarski’s World worlds).There are two other kinds of necessarily-true sentences in Tarski’s World that do not depend for theirnecessity on the unique properties of the TW program. Instead, these sentences are necessary for strictly logical reasons that hold both inside and outside of Tarski’s World. POWERPOINT SLIDE #7At this point before going further we should look at Figure 4.1 on p. 102 of your textbook.**** SHOW FIGURE 4.1 FROM TEXTBOOK ON THE SCREENFigure 4.1 diagrams the relationship between these different kinds of necessity (and possibility). One important thing to remember about Figure 4.1 (so as not to get confused) is that it describes only the sentences of Tarski’s World. (as work through the diagram, refer to the sample sentences of FOL at each concentric level)…(TW-) LOGICAL POSSIBILITIESSo, working from the outside in, outside the broadest circle in Figure 4.1 there is an unlabeled area that includes every sentence that can possibly be truly stated of one or more ‘worlds’ using the Tarski’s World program. This is the area we have already talked about as being what is TW-possible.TW NECESSITIESJust inside this circle is a subset circle that contains all the sentences stateable with Tarski’s World that are necessarily true and labeled “Tarski’s World Necessities” (= TW-necessary, as we’ve already discussed). These sentences will always be true in TW no matter how you move the blocks around, resize them, or reshape them. Within the outermost part of this circle are sentences like Tet(a) ᴠ Cube(a) ᴠ Dodec(a) that we discussed a moment ago—sentences that are necessarily true in TW because of the peculiarities of the structure of TW (i.e., in this case because there are only three shapes allowed in TW).LOGICAL NECESSITIES (IN TW)Within this circle is a smaller subset of sentences that Figure 4.1 simply calls “Logical Necessities”, meaning that the sentences within this circle are logically necessary or necessarily true for logical reasons that hold not just in Tarski’s World but in all other logical worlds as well (including the ‘real’ world and any other logically conceivable possible world). Keep in mind that even though logical necessity is a concept that holds outside of Tarski’s World,Figure 4.1 is nonetheless only describing the sentences of Tarski’s World, so the diagram can represent logically necessary sentences as being a subset of (i.e., a smaller group within) TW-necessary sentences. To avoid confusion, the circle labeled “Logical Necessities” in Figure 4.1 might be better
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