LESSON #12: LOGICAL & TAUTOLOGICAL CONSEQUENCE; TAUTOLOGICAL CONSEQUENCE IN FITCH; LOGICAL AND TAUTOLOGICAL EQUIVALENCEPOWERPOINT SLIDE #1Last time we discussed the relations between logical possibility, logical necessity, and tautologies and saw that these stand in subset relations (most importantly for today’s purposes, tautologies are a subset of logical necessities or logical truths (i.e., sentences that are true in all worlds), where tautologies are specifically true in all worlds strictly because of the way the Boolean connectives structure the complex sentence.POWERPOINT SLIDE #2Another important and related concept is that of logical equivalence – when two sentences have the same truth values in every possible circumstance or world, which is the same as to say they have the same truth conditions. (Note that the book points out that logical equivalence between two sentences can also be viewed as a two-way logical consequence relationship: each sentence is a logical consequence of the other, so that whenever the one is true, the other must be true, and vice versa.)But just as we saw that truth tables cannot spot every logical truth/necessity (because truth tables are ‘blind’to the meaning of the atomic sentences represented by the capital letters), so truth tables cannot spot all logical equivalences between sentences.POWERPOINT SLIDE #3It may be easiest to first consider which logical equivalences a truth table can spot: logical equivalences that are due solely to the way the connectives structure the sentences. For example, consider the truth table on p. 106 comparing the two sentences of the first DeMorgan law. [show how such a comparison truth table is set up, with a separate column for each sentence, and all their collective atomic sentences listed once each in the reference columns.] POWERPOINT SLIDE #4When two sentences are logically equivalent in this way such that their main connectives show the exactsame pattern of truth values in all rows, we call them tautologically equivalent. Note that THIS DOES NOT MEAN THE TWO SENTENCES ARE TAUTOLOGIES!!!! You can see this in the truth table in the previous slide, where both sentences are False in the first row (so neither is a tautology). We call them tautologically equivalent NOT because they’re tautologies but because their equivalence derives solely from the meanings of their truth-functional connectives—and this is the same thing that (in other cases) makes a logical truth (more specifically) a tautology.POWERPOINT SLIDE #5As mentioned earlier, truth tables cannot detect every form of logical equivalence. Specifically, truth tables cannot detect logical equivalence that depends on the meanings of the predicates involved. For example consider the following truth table …POWERPOINT SLIDE #6This table compares the truth values of two sentences that are logically equivalent but that have different values in rows 2 and 3 under their main connectives. This is because the truth table cannot detect the fact that those two rows represent logically impossible worlds …POWERPOINT SLIDE #7We see from this that tautological equivalence is a strict form of logical equivalence; thus, all tautologically equivalent sentences are logically equivalent, but not vice versa. (Recall how tautologies were a proper subset of logical necessities in slide 1.)POWERPOINT SLIDE #8Finally, we return to the matter of logical consequence, and we find that just as necessity relations have a superset-subset relation (i.e., tautologies are a proper subset of logical necessities), and just as equivalence relations stand in superset-subset relation (i.e., tautological equivalence is a proper subset of logical equivalence), so also consequence relations stand in a superset-subset relation: tautological consequence is a proper subset of the larger category of logical consequence. POWERPOINT SLIDE #9We discussed logical consequence the first week of class: One claim (B) is a logical consequence of another (A) if the first claim (B) must be true whenever the other (A) is true.So, if in some world A can be true but B false, this demonstrates that B is not a logical consequence of A. (Recall how in earlier lessons we demonstrated the invalidity of an argument in basically the same way; that’s because the conclusions of invalid arguments are not logical consequences of the premises.)POWERPOINT SLIDE #10One special sort of logical consequence: Tautological consequence – a sentence Q is a tautological consequence of another sentence P if—in a joint truth table for P and Q—in every row where the main connective for P is True, the main connective for Q is also True (but not necessarily vice versa). That is, as shown by the truth table (which is only sensitive to the meanings of the truth-functional connectives), Q is true in every world where P is true, though P is not necessarily true in every world where Q is true.POWERPOINT SLIDE #11The book gives the following example, showing that A ᴠ B is a tautological consequence of A ᴧ B, but not vice versa:That is, in every row where A ᴧ B is true (which is only one row, the first one), A ᴠ B is also true. So, that means that A ᴠ B is a tautological consequence of A ᴧ B. But the reverse is not true—because in the 2nd and 3rd rows of truth-values, A ᴠ B is true but A ᴧ B is not. So, A ᴧ B is not a tautological consequence of A ᴠ B.POWERPOINT SLIDE #12Keep in mind that not every logical consequence of a sentence is a tautological consequence of it. Logical consequence is the bigger notion, just like logical necessity was a superset of tautological necessity. The key is what the consequence derives from: If only from the meanings of the atomic sentences or predicates, then there is only logical consequence between the two sentences, not tautological consequence.POWERPOINT SLIDE #13The second sentence here is a logical consequence of the first, but not a tautological consequence, because we cannot see the logical consequence anymore once we blind ourselves to the meanings of the atomic sentences.a = b ᴧ b = c P ᴧ Qa = c (can see consequence) R (cannot see consequence)POWERPOINT SLIDE #14You can also check whether a sentence is a tautological consequence of a set of other sentences, which comes in handy for checking whether the conclusion of an argument is a tautological consequence of theargument’s set of premises. Work up a truth table for all of the
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