1 / 23

Sentential Logic

Sentential Logic. Brooks DiBenedetto and Marie Deynes. Sentential logic- simple system of logic. Better yet, a set of rules that tell us how to make use of special symbols to construct sentences and do proofs. Three main symbols. Letters Five sentential connectives

anne
Download Presentation

Sentential Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Sentential Logic Brooks DiBenedetto and Marie Deynes

  2. Sentential logic- simple system of logic. Better yet, a set of rules that tell us how to make use of special symbols to construct sentences and do proofs.

  3. Three main symbols • Letters • Five sentential connectives • Opened and closed brackets

  4. Letters: A, B, C, etc. These capital letters are used to translate sentences. If we run out of sentence letters we can always add subscripts to make new ones using extensions, like A1

  5. Five sentential connectives : ~ (tilde, or the negation sign) & (ampersand, or the conjunction sign) ∨ (the wedge, or the disjunction sign) → (the arrow) ↔ (the double-arrow)

  6. Opened and closed brackets: ( ) Often used as separation of statements

  7. Truth Tables • Provide one systematic method for determining the validity of arguments in sentential logic. It helps show the truth values and how the complex ones depend on something that is much more simpler. • There are only two possible answers in truth tables, “true or false”

  8. Truth table activity 1.All squares are red if and only if all squares are green. 2.If there is no red square, then there is a triangle. 3.Either there is a green circle, or there are no yellow squares.

  9. Well formulated formulas Nothing more than a grammatical expression A WFF is like a sentence which is why these connectives are called “sentential connectives”.

  10. Relationship of negation sign and WFFs • Negation signs always connect to one single WFF to make a longer WFF, and is called a one-place connective. • You can connect well formulated formulas to each other without a negation sign but then its called binary or 2-place connectives.

  11. Rules of formation : • All sentence letters are WFFs. • 2.If P is a WFF, then ~P is a WFF. • 3.If P and A are WFFs, then (P&A), (PvA), (P→A), (P↔A) are also WFFs. • 4.Nothing else is a WFF.

  12. Logical Properties and relations Once people are experts at interpreting truth tables, they are used to put well formulated formulas according to their logical status: • Tautology • Inconsistency • Contingency

  13. Tautology A WFF that is true under all assignments of truth-values to its sentence letters

  14. Inconsistency • A WFF that is false under all assignments of truth-values to its sentences letters

  15. Contingency • A WFF that is not inconsistent and not a tautology. In other words, there is at least one assignment under which it is true, and at least another assignment under which it is false (P&Q), (P∨Q), ~(P→~Q).

  16. Formalization [Premise 1] The pollution index is high. [Premise 2] If the pollution index is high, we should stay indoors. • [Conclusion] We should stay indoors. • This argument is of course valid, as it is an instance of modus ponens. To use the methods of SL to show that it is indeed valid, we need to translate it from English into the language of SL. This process of translation is called formalization.

  17. Translation Scheme • A translation scheme in SL is simply a pairing of sentence letters of SL with statements in natural language. In carrying out formalization you should always write down the translation scheme first. Translation scheme : P : The pollution index is high. Q : We should stay indoors.

  18. Sentential connective • To put two things together that make sense, you are creating a sentence using truth tables and WFFs.

  19. Other uses for negation Negation Suppose “P” translates the sentence “God exists”. Then “~P” can be used to translate these sentences : • God does not exist. • It is not the case that God exists. • It is false that God exists.

  20. Conjunction “(P&Q)” can be used to translate the following : • P and Q. • P but Q. • Although P, Q. • P, also Q. • P as well as Q.

  21. Disjunction “(PQ)” can be used to translate the following: • P or Q. • Either P or Q. • P unless Q. • Unless Q, P.

  22. Conditional “(P→Q)” can be used to translate the following: • If P then Q. • P only if Q. • Q if P. • Whenever P, Q. • Q provided that P. • P is sufficient for Q. • Q is necessary for P.

  23. References • Picture from http://www.csus.edu/ • Activity and Information obtained from http://philosophy.hku.hk/think/sl/

More Related