DAY 30: FIRST-ORDER VALIDITY AND CONSEQUENCE (10.2)Assigned reading: sec. 10.2POWERPOINT SLIDE #1We learned last lesson that even with quantifiers now installed in our First-Order Logic, we can still identify tautological relations by using the method of converting a sentence into Truth-Functional Form (TFF). If we can see that a sentence must always be true even when it’s put into TFF, then we know that it’s a tautology. So, for example,Sentence’s original form Truth-Functional Form (TFF)1. (Small(b) ᴧ ∀x Cube(x)) Cube(b) (A ᴧ B) C *logical truthNote that the above sentence is clearly a logical truth because we can see from its original form that it must always be true (i.e., it says that “If b is small and everything is a cube, then b is a cube”). But notice that when we convert it to TFF (by replacing the atomic sentences in it and the quantified phrase with capital letters), we can no longer ‘see’ that the sentence must always be true. This proves that the sentence is not a tautology.Now contrast the above sentence with the sentence below:2. (Small(b) ᴧ ∀x Cube(x)) Small(b) (A ᴧ B) A *tautologyAs with first sentence, we can see that this second sentence is a logical truth because we see that it must be true in ever world, but this time when we convert the sentence into TFF, we can still see from the TFF that it must always be true (note that there are two instances of ‘A’ in the TFF of the second sentence, because the atomic sentence ‘Small(b)’ shows up twice in the original form so must be replaced by the same letter each time in TFF). The fact that we can still see the logical necessity of this sentence even when it is converted to TFF proves that this second sentence is not only a logical truth, it is more specifically a tautology.POWERPOINT SLIDE #2We can use Truth-Functional Form (TFF) in precisely the same way to distinguish mere logical consquence from tautological consequence. In the case of the first argument below, we can see that the conclusion of the argument is a consequence of its premises (i.e., that if the premises are true, then the conclusion must also be true) only when the argument is in its original form. As soon as we convert it to TFF, we can no longer see the consequence. This proves that the argument involves logical consequence only (not tautological consequence).∀x (¬Tet(x) Small(x)) A ¬Tet(a) B *logical consequence onlySmall(a) CContrast the above case with a second argument. Here, we can see that the conclusion of the argument is a consequence of its premises in both the argument’s original form and also when the argument is converted into TFF. Therefore, we know that the conclusion of the argument is not merely a logical consequence of its premises, it is more specifically a tautological consequence of the premises. Note that in this case we can say that the argument is not just logically valid, it is also tautologically valid.∀x ¬Tet(x) Small(a)) A B∀x ¬Tet(x A *tautological consequenceSmall(a) BPOWERPOINT SLIDE #3Finally, TFF can be used in the same way to distinguish between cases of logical equivalence and cases of tautological equivalence. The method and logic is exactly the same as in the previous two powerpoint slides …¬Ǝx (Small(x) ᴠ Large(x)) ó ¬Ǝx Small(x) ᴧ ¬Ǝx Large(x) ¬A ó ¬B ᴧ ¬CWe lose our ability to see the equivalence relation in the first case (above), proving that the equivalence seen in the original pair of senences is only logical equivalence. In constrast, with the pair of sentences below we continue to see the that the pair of sentences is equivalent even when the sentences are converted to TFF, proving that the sentences are not only logically equivalent, they are tautologically equivalent as well:¬ (Ǝx Small(x) ᴠ Ǝx Large(x)) ó ¬Ǝx Small(x) ᴧ ¬Ǝx Large(x) ¬(A ᴠ B) ó ¬A ᴧ ¬BPOWERPOINT SLIDE #4Remember that tautological relations (i.e., tautologies, tautological consequence, and tautological equivalence) are a subset of those relations that are merely logical or logically necessary (i.e., logical truths, logical consequence, and logical equivalence). In order to identify merely logical relations, you must be able to see all aspects of sentence meaning and structure. That is to say, you must not ‘blind’ yourself to anything in the sentence.In contrast, to identify tautological relations, all you need to be able to see is how the Boolean connectives structure the sentence(s). That is, you can see tautological relations even when you’ve blinded yourself to the meanings of atomic sentences and the content of quantified phrases.POWERPOINT SLIDE #5In other words, the necessary nature of logical relationships (i.e., why they express relations that are true of sentences in all worlds) potentially depends on any or all aspects of sentence meaning and structure. The necessity seen in tautological relationships, on the other hand, depends only on how the Boolean connectives structure the sentence(s).POWERPOINT SLIDE #6But 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 . . .It turns out that there is a middle option of this sort . . . POWERPOINT SLIDE #7First-Order (FO) relations are those that are based 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 (except for the identity predicate). POWERPOINT SLIDE #8Examples: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)The above two sentences are FO logical truths, or what are commonly (but confusingly) called FO validities – meaning they are necessarily true in all worlds because of the way that the truth-functional connectives, the quantifiers, and the identity predicate operate (but NOT because of the meanings of the other predicates besides identity). We know they are FO validities because when we replace the non-identity predicates with meaningless capital letters (making ourselves “blind” to the meaning
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