ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Material Conditionals and Bi-Conditionals

Updated on October 19, 2016
1701TheOriginal profile image

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly improve it.

In symbolic logic, we have special types of sentences that have truth value patterns that can be extended to many other sentences and allow us to draw new conclusions. Some examples include conjunctions and disjunctions, which contain “and” and “or.” These link two sentences together and making “both” necessary to be true (“and”) or just needing one or both of them to be true (“or”). We can also negate a sentence by taking its opposite truth value as the overall value we are seeking. But what about statements that require a condition to be true for a conclusion to be true also? That is what do we do with “if A, then B?” What do we call this?

Material Conditionals

This is known as a material conditional (Bergmann 41). We can think of “if A then B” as being paraphrased as “either it is not the case that B or it is the case that A." This is true because since A implies B we know that if B does not happen then neither does A but if A happens then B follows. We can symbolize this as ~ B v A, with the "~" a negation and “v” meaning “or.” We can also create a new symbol, an arrow, to signify an “if … then…” statement represented above. This looks like A -> B, though some textbooks use a horseshoe “⊃” instead of an arrow. A, the “if” sentence, is known as an antecedent while B, the “then” sentence, is known as the consequent. The truth table for conditionals is easy to remember: the conditional is false only if true premises lead to a false conclusion (41-2).

Now, a fair warning about paraphrasing sentences into conditionals. If you see “A only if B,” do not think that B is the antecedent just because it is after the if. In these types of paraphrases, B is the consequent and A is the antecedent, or "if A, then B." Also, when we see “A unless B,” we can paraphrase that as either a disjunction ("A or B") or as a conditional ("if A, then B"), depending on the context of the statement. Logic keeps us on our toes, as you can see, so stay alert and be mindful about what you are doing (44).

Material Bi-Conditionals

Now, if I have the conditional “if A, then B,” what happens when I switch the order of the precedent and the antecedent? Would the statement “if B, then A” be true because “if A, then B” was true? Not necessarily, for if A was false and B was true, then “if A, then B” would be true but “if B, then A” would be false. By switching the order, we got to be careful because the truth of the conditional changes. But what if both “if A and B” and “if B, then A” were true at once?

That would be known as a material bi-conditional, and can be symbolized as the conjunction of two conditionals, or (A -> B) ^ (B ->A). To paraphrase into a bi-conditional, we could go with “either both it is A and B or both is not B and that it is not A.” This is because of what we demonstrated above in the conditional section. If they both have different truth values, then one of the conditionals would be false, so both the antecedent and the consequent must have the same truth value. We can symbolize the paraphrase as (A ^ B) v (~A ^ ~B). Typically, however, we symbolize a bi-conditional as a two-headed arrow, or A <-> B, though some books may use a triple bar instead (45-6). You can also paraphrase “A if and only if B” as a bi-conditional statement.

Works Cited

Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book. New York: McGraw-Hill Higher Education, 2003. Print. 41, 42, 44-6.

© 2013 Leonard Kelley

Comments

    0 of 8192 characters used
    Post Comment

    No comments yet.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: "https://hubpages.com/privacy-policy#gdpr"

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)