Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Review: logical & tautological relations(Small(b) ᴧ ∀x Cube(x)) Cube(b) convert to TFF(A ᴧ B) C *logical truth(Small(b) ᴧ ∀x Cube(x)) Small(b) convert to TFF(A ᴧ B) A *tautologyLogical & tautological consequence: ∀x (¬Tet(x) Small(x)) A ¬Tet(a) BSmall(a) C ∀x ¬Tet(x) Small(a) A B ∀x ¬Tet(x) ASmall(a) BWith both arguments in their original forms we can see the conclusion is a logical consequence of the premises. However, when we convert the arguments to TFF, we can still see the consequence relation only with the second argument, so only the second argument involves tautological consequence.Logical & tautological equivalence:¬Ǝx (Small(x) ᴠ Large(x)) ¬Ǝx Small(x) ᴧ ¬Ǝx Large(x) ¬A ¬B ᴧ ¬C¬ (Ǝx Small(x) ᴠ Ǝx Large(x)) ¬Ǝx Small(x) ᴧ ¬Ǝx Large(x) ¬(A ᴠ B) ¬A ᴧ ¬BSimilarly, we can see from their original forms that each of these pairs of sentences is a logical equivalence . However, once we convert each pair of sentences into TFF, we can still see the equivalence relation only with the second pair of sentences, proving that only this second pair exhibits tautological equivalence.Can see the equivalence in both originalsCANNOT see the equivalence in TFFCAN see the equivalence in TFFKeep in mind that tautological relations are a subset of logical (or logically necessary) relations. TFF is simply a method of blinding ourselves to certain aspects of sentence meaning and structure.logicalCan see (merely) logical relations only without any blinders on.So, to identify a (merely) logical relation you need to see everything:the meanings of all the atomic sentences, all the quantified phrases,and all the Boolean connectives. tautologicalCan see tautological relations even when blind to the atomicsentences and quantified phrases. So, you can identify tautologicalrelations by just seeing how the connectives structurethe sentence(s).In other words . . .logicalThe necessity of logical relations potentially dependson any or all aspects of sentence meaning and structure. tautologicalThe necessity of tautological relations dependsonly on the way the Boolean connectivesstructure the sentence(s).But what if there were a middle option as well? That is, a type of relation whose necessity depended not only on how the connectives structure the sentences but on other aspects of sentence meaning/structure as well … but not on all aspects of meaning/structure . . .Logical ??? tautologicalA less strict form of necessity relation … First Order (FO) necessityFO relations are those in which the logical necessity involved depends only on the operations of the truth-functional connectives, the quantifiers, and the identity predicate. Put differently, FO relations are those we can still identify when we make ourselves “blind” to all the predicates of FOL other than the identity predicate.Demonstrating an FO validity by converting sentences into what we’ll call ‘FO Checking Form’:FO Checking Form:∀x Cube(x) Cube(b) ∀x R(x) R(b)(Cube(b) ᴧ b=c) Cube(c) (R(b) ᴧ b=c) R(c)We know that each of the above two sentences is an FO validity because we can still see that the sentences are logical truths (i.e., true in every world) even when we’ve blinded ourselves to the meanings of all predicates except the identity predicate.Note that only the predicate and not its argument(s) is replaced in FO Checking Form.Don’t replace the identity predicate!Examples of logical truths that are not FO validities:∀x SameSize(x,x) ∀x R(x,x)(Small(b) ᴧ SameSize(b,c)) Small(c) (R(b) ᴧ S(b,c)) R(c) We can no longer see that these sentences are logical truths when we convert them to FO Checking Form; that is, when we blind ourselves to the meanings of every predicate except for identity.Another way to demonstrate (sometimes even more clearly) that a sentence is not a FO validity is to create a counterexample interpretation by substituting new, random predicates in place of the original ones (we’ll call this variation ‘FO Checking Form—Random Predicate Method’)∀x y (SameSize(x,y) ↔ SameSize(y,x))∀x y (Punch(x,y) ↔ Punch(y,x))Notice that with the new predicate ‘punch’, we can no longer see that the sentence is necessarily true (i.e., just because x punches y doesn’t necessarily mean that y will punch back). This shows that the meaning of the original predicate SameSize contributes to the logical necessity of these sentence, so the sentence is a logical truth but not a FO validity.With true FO validities, you’ll fail to be able to create a counterexample (i.e., the necessity will still show even with the random predicate inserted):∀x Cube(x) Cube(b)∀x Smart(x) Smart(b)∀x Mortal(x) Mortal(b)∀x Purple(x) Purple(b)∀x Smurf(x) Smurf(b)∀x Philosopher(x) Philosopher(b)etc. …Notice that we still see these are necessarily true. If you have a true FO validity (as here), then it will be impossible to create a counterexample by switching out the predicates (i.e., impossible to ‘lose’ the necessity this way). logical relationsCan see these are always true only without any blinders on. first-order relationsCan see these are always true even when blind to all predicates (other than the identity predicate). tautological relationsCan see these are always true even whenblind to everything except connectives. logical relationsThe necessity of logical relations potentially dependson any and all aspects of sentence meaning and structure. first-order relationsThe necessity of FO relations potentially depends only on(1) how the Boolean connectives structure the sentence,(2) the quantifiers, and (3) the identity predicate. tautological relationsThe necessity of these depends only on howthe connectives structure the sentence. Logical truthsCan see these are true in every world only without any blinders on. First-order validitiesCan see these are true in every world even when blind to all predicates (other than the identity predicate). tautologiesCan see these are true in every world evenwhen blind to everything except connectives.Demonstrating FO consequence:∀x
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