ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Symbolic Logic Cheat Sheet - Premises, Conclusions, and Validity in Propositional Logic

Updated on August 28, 2012

Before I Begin

I wrote this while taking a Logic class to help with tests and homework. If any of this is inaccurate or not helpful please let me know and I will change it immediately. This is a very brief note sheet but I hope it is helpful to anyone else studying logic.

Symbols and What They Mean

& →Conjunction read as: and eg. (P & Q) both have to be true to be true overall

v Disjunction read as: or eg. (P v Q) one has to be true to be true overall

Conditional read as: if x ⊃ then y eg. (P⊃Q) Q has to be true or both P and Q have to be false to be true overall. If P is true and Q is false then it is false overall.


(x) Universal Quantifier read as: For every x…

(∃x) Existential Quantifier read as: There exists some (or at least one) x that…

Premises

UI (Universal Instantiation): You may instantiate a universally quantified premise with any individual constant. This rule is wide open. For universallyquantified premises only à (x).

If Premise is (x)(Sx ⊃ Px) → (Sa ⊃Pa) UI

EI (Existential Instantiation): You may instantiate an existentially quantified premise with an individual constant if, but only if, that individual constant does not appear anywhere else in the instantiation except in a premise that was derived by UI or in a conclusion that was derived by EG. For existentially quantified premises only → (∃x).

If Premise is (∃x)(Sx & Px) → (Sa & Pa) EI

However if some premise has REIT and contains constants you can’t repeat them in an EI no matter where they come up in the argument such as this:

Sa → Sa REIT

(∃x) (Sx & Px) → Sb & Pb EI (Have to put b or something else because a is used in REIT)

————————————————————————————————

∴(∃x) Px → Pb EG (because it is existentially quantified (∃x))

Conclusions

UG (Universal Generalization): You may instantiate a universally quantified conclusion with an individual constant if, but only if, that individual constant does not appear anywhere else in the instantiation except where it was derived by UI. This is for universally quantified conclusions only → (x).

Example:

(x) (Sx⊃Px) → Sa ⊃ Pa UI

(x) (Px⊃Qx) → Pa ⊃ Qa UI

——————————————————————————————————

(x) (Sx⊃Qx) → Sa ⊃ Qa UG

EG (Existential Generalization): You may instantiate an existentially quantified conclusion with any individual constant. This is for existentially quantified conclusions only → (∃x).

Example:

Sa & Pa → Sa & Pa ReIt

Pa → Pa ReIt

——————————————————————————————————

(∃x)(Px &Sx) → Pa & Sa EG

REIT: You may reiterate any premise or conclusion with an individual constant. Only label REIT if the constant is already used in the original argument. See above.

Validity

Validity in PL(PL Valid): An argument that is true on all relevant lines in a truth table. (Relevant lines are where premises are true. If the conclusion is that is good. If the conclusion is false that means the argument is invalid.) To set up a truth table set up a chart for each variable (P,Q,R,S…) Doubling the amount of T’s and F’s for each row..eg

First Column → First half Ts, Second half Fs.

Second Colum First quarter Ts, Second quarter Fs Third quarter Ts Fourth quarter Fs

Third Column First eighth Ts, Second eighth Fs, Third eighth Ts, Fourth eighth Fs, and so on

Fourth Column First 16th Ts, Second 16th Fs, Third 16th Ts, Fourth 16th Fs, and so on.

So if there is one variable (P) then there will be one column containing P that is T and F. If there are two variables, there will be two columns, one column will contain P and one will contain Q and there will be 4 spots for Ts and Fs. And so on.


Validity in RQL: An argument is valid in RQL if and only if it has a proper instantiation whose form is valid. It is easy to show validity, however it is more difficult to show that it is not valid in PL. To test for this use this procedure.

Step 1: Reiterate all sentences that have individual constants (but no quantifiers)

(Apply REIT to premises and conclusions that get it.)

Step 2: Instantiate each existential premise and universal conclusion. In each case, use a constant that does not yet appear in the instantiation you are constructing.

(Apply EI and UG to premises(EI) and conclusions(UG) that get it. If any constants are already in here from REIT use a different constant.)

Step 3: Instantiate each universal premise and existential conclusion with each constant occurring in the instantiation after step 1 and step 2. (If no constant occurs, use a new constant. If two constants occur, construct one proper instantiation for each constant.)

(Apply UI and EG to premises(UI) and conclusions(EG) that get it.)

If this procedure yields a proper instantiation whose form is valid in propositional logic, then the original argument is valid in RQL. If not then this argument is invalid in RQL.

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://corp.maven.io/privacy-policy

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)
ClickscoThis is a data management platform studying reader behavior (Privacy Policy)