ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Help on Paraphrasing Sentences and Arguments into Sentenial Logic

Updated on January 10, 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 explore it.


Sentential logic (SL) uses sentences as the building blocks for its work in proving logical arguments, which can have far-reaching implications for truth. Each of the sentences in SL can only be true of false, but not both at the same time and not nothing either (28). Throughout SL, we want to link sentences together to create arguments which can be used to find new conclusions about given premises. To make such links, we use sentential connectives such as “and,” “or,” “although,” “unless,” “before,” and “if and only if” (Bergmann 29).

When we link two or more simple, or singular, sentences with a sentential connective, we have created a compound sentence (29). Such compounds are powerful because they rely on the simple sentences within them to give them their total truth value that can exceed the parts that make it. When a sentential connective is performing in such a way that the compound’s truth value is based off the truth values of the simple sentences that make up that compound, we call it truth functional (29).


In SL, we use capital letters to abbreviate sentences in an effort to look at the compound structure and derive from it some conclusions which we can ultilize. The letters allow us to present that information in an easy-to-manage manner. Rewriting the same sentence over and over is tiresome, so just abbreviate it with a letter that makes sense. When we have a sentence that is abbreviated, we refer to the letter that abbreviates it as an atomic sentence. Any combination of letters is referred to as a molecular sentence (30).

So how do we go about converting sentences so we can work with them in SL? Generally, we need to paraphrase the sentence into its appropriate form. If it has a sentential connective, we need to split it up into simple sentences, and then use an appropriate letter relating to the information in the sentence to make it atomic. Take note that we cannot always convey all the information we desire in the paraphrase. That is okay, just make the closest approximation you can (32-3). For example, “All my dogs are brown” could be simplified with a D for dogs or a B for brown. Usually the other sentences we are paraphrasing will push us to choose certain letters over others. If the other sentence had been, “None of my dogs are puppies,” then we would definitely use B for the first sentence and P for the second, the P representing puppies.



It is nice to know what a paraphrase will look like once in molecular form. Anything that can be replaced with, “Neither A nor B,” which is the same as, “Both it is not the case that A and it is not the case that B,” will take on the form ~A ^ ~B in SL, with the tilde meaning a negation of the sentence and the carat meaning, “And.” It is interesting to note that “neither A nor B” can also be paraphrased as “it is not the case that either A or B,” which in SL is ~ (A v B), with the wedge meaning, “Or.” So, ~A ^ ~B is truth equivalent to ~ (A v B) (39). This is one of De Morgan’s Laws. We will frequently encounter more equivalent statements in SL.

Another example of what a paraphrase will look like in SL is “not both A and B.” This could be paraphrased as “it is not the case that both A and B,” which in SL is ~ (A ^ B). “Not both A and B” could also be paraphrased as “either it is not the case that A or it is not the case that B,” which in SL is ~A v ~B. Hence, ~ (A ^ B) is equivalent to ~A v ~B (39-40). This is the other De Morgan’s Law.


In general, when you paraphrase: (51)

  1. Anything that you find to be already a simple sentence can be put directly into SL
  2. Use those connectives when necessary to join atomic sentences
  3. Ensure that clarity is preserved! Parenthesis can be our friend
  4. If dealing with an argument, be sure to have premises separated from conclusion
  5. If a sentence repeats a main idea, do not include it twice

When we work with an argument, we have multiple sentences that may have different levels of reliability. If sentence A guarantees the truth of sentence B but B does not ensure A, then we say that B is the weaker sentence. If instead B had led to truth of A but A did not guarantee truth of B then B would be the stronger sentence. What we aim for are equivalent sentences, where both lead to the truth of the other (62).

Finally, it is important to make the distinction between an object language and a meta-language. The object language is what we are talking about, but the meta-language is used to do the actual talking. This distinction is necessary because we will be aiming to make generalizations of sentences and need to have a broader plane to work with. Those broader statements in the meta-language are often bolded as a meta-variable, while sentences in the object language are put in quotations. (67-9).

Works Cited

Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book. New York: McGraw-Hill Higher Education, 2003. Print. 28-30, 32, 33, 39, 40, 51, 62, 67-9.

© 2013 Leonard Kelley


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, 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:

    Show Details
    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 or domains, for performance and efficiency reasons. (Privacy Policy)
    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)
    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.
    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)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)