Our system uses 11 different equivalency rules. These are: We have introduced 10 new rules, the equivalence rules. They are summarized on a separate page as a summary of the equivalence rules. In addition to the inference rules, you now have all the rules you need to provide proof. The exercises listed below will give you a lot of practice in applying the rules. In the next section, we present two strategies that make evidence a little easier. Syntactically, (1) and (2) can be derived from each other via the rules of contraposition and double negation. Semantically (1) and (2) are true in exactly the same models (interpretations, evaluations); namely, those in which Lisa in Denmark is fake or Lisa in Europe is true. After eliminating the double negation from the sequence of line 1, we applied the contraposition to line 6.
Next, we used modus ponens to derive ~A. Question: Can you find a derivative of ~A from the same premises that does not use the counterposition? We could have used modus tollens instead. Our rule base becomes more powerful as we add new rules, and you`ll find that there are often many ways to achieve the same result. Also note that we applied the contraposition to the entire wff on line 1. Here is a quick derivation where we only apply the rule to part of the line: Here are some additional rules. After listing them, let`s look at a derivation that incorporates them all. We have already used the word “distribution” in logic twice: once we have opposed collective preaching to distributive preaching to understand the error called composition. We all know that “America is a rich country” does not mean that “all Americans are rich.” From a true preaching of property, being rich to the whole, which consists of all Americans, it does not follow that one can “distribute” this property to all the individuals who are its parts. The other time we encountered the cast was in determining the validity of categorical syllogisms: we saw two rules related to the idea that any member of the class named by a term is referenced. To test the logical equivalence of 2 statements, create a truth table that contains all the variables to evaluate, and then verify that the resulting logical values of the 2 statements are equivalent.
Each of our equivalence rules can be verified as legitimate with a truth table. Consider the equivalence rule known as the DeMorgan transform: again, equivalence rules express only a portion of equivalences between wffs. As long as it is usually listed, you can use it in a proof. There are, of course, many equivalencies that are not set out in our equivalence rules, and you cannot use such equivalences in your evidence. Why not? Such “proof” would not be PL proof because it uses rules that are not part of PL`s theory of proof. Your “proof” might represent proof in another formal system, but it is not proof in PL. To take advantage of our equivalence rules, we need to map the formula we want to change on one side of the equivalence rule, and then use that mapping to create a surrogate instance on the other side of the rule. A final and very important observation is that these rules apply everywhere. Wherever you want, wherever you see the corresponding main operator of the rule – whether it is the main operator of the line in the proof or in a subordinate clause inside the line. For example, see if you can tell where DM should be applied to this, and what the result will be: Can we construct a proof of P from ~~P? If you look at the inference rules, you won`t find an inference rule that allows you to draw this conclusion.
It is clear that the rules of inference must be completed. We call the supplementary rules “equivalency rules” and there is an important reason not to group them with the inference rules. The equivalence rules we list below are logically equivalent pairs of wff. They work like this: wherever you find one of the pairs, you can deduce the wff, which replaces the second member of the pair with the first. It is very important that you do not confuse this formula as an expression of the so-called “distributive property” in mathematics with the distribution of negative signs (for negative numbers). These are not sums or figures, but meanings. You can use a truth chart to make sure that “Neither. Again” does not mean “neither this nor that”. You can also convince yourself of this by imagining a situation in which “neither… ni” is really important to you! You will not be fooled by someone who misinterprets it. For example, let`s say you`re planning a party and someone asks if you`re going to invite Scott and Rhonda, “Are you crazy? I`m not going to invite either of them! (meaning “neither”). Now, if your friend asks, “Then which one aren`t you going to invite?” (because he thinks “ni” – ~ (S v R) – means (~S v ~R), you will answer something like “Do you have a brain?” Only in a classroom, you wouldn`t know that “neither nor” is not the same as “neither one nor the other.” Like all forms of statement for which equivalence rules exist, each of them can be replaced by the other.
As for how you see them on the page, you could say they “work” both ways, left-to-right or right-to-left. CONTRA is an equivalence between a condition and the condition we receive by changing and denying the anteriority and consequence. Here`s how it can work in a derivative. Note that we also use DN in this derivative. In logic and mathematics, the statements p {displaystyle p} and q {displaystyle q} are logically equivalent if they have the same logical value in each model. [1] The logical equivalence of p {displaystyle p} and q {displaystyle q} is sometimes expressed as p ≡ q {displaystyle pequiv q} , p :: q {displaystyle p::q} , E p q {displaystyle {textsf {E}}pq} , or p ⟺ q {displaystyle piff q}, depending on the notation used. However, these symbols are also used for material equivalence, so the correct interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are inextricably linked. Logical equivalence is different from material equivalence. The formulas p {displaystyle p} and q {displaystyle q} are logically equivalent if and only if the statement of their material equivalence ( p ⟺ q {displaystyle piff q} ) is a tautology.
[2] Here we have a disjunction whose left disjunction is a condition. We are free to apply CONTRA only to the conditional, as long as we reproduce the rest of the line as it was before the transformation. This is an important “freedom”. Again, you can apply equivalence rules to parts of wffs as long as you reproduce the entire wff, the equivalent part of which is a part, on a new line. The material equivalence of p {displaystyle p} and q {displaystyle q} (often written p q {displaystyle pleftrightarrow q} ) is itself another declaration in the same object language as p {displaystyle p ↔ } and q {displaystyle q}. This statement expresses the idea “p {displaystyle p} if and only if q {displaystyle q} `”. In particular, the logical value of p q {displaystyle p ↔ leftrightarrow q} can change from one pattern to another. In logic, there are many common logical equivalences, often listed as laws or properties. The following tables illustrate some of them. The following table can be used to reduce compound statements to simpler shapes.
For the statement variables p, q, and r, a tautology t, and a contradiction c, the following logical rules apply: Let`s see how this works by listing some of the equivalence rules, starting with the double negation rule: There are four other equivalence rules. We list them below and comment on them briefly. The following truth table indicates that equivalence is legitimate. Now, a derivation that illustrates the use of these rules: Proving with all these rules takes a lot of practice. I will provide a range of exercises from easy to more difficult, and I expect you to work on them for at least an hour a day. You can write to each other or to me with questions, you can come and visit me with questions. Try them, and if you have a problem, move on or ask for help. Don`t waste unproductive time, that`s the most important thing. Learn all the rules of inference.