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 symbolsLettersFive sentential connectivesOpened and closed brackets

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

Five sentential connectives :

~ (tilde, or the negation sign) & (ampersand, or the conjunction sign) ∨ (the wedge, or the disjunction sign) → (the arrow) ↔ (the double-arrow)

Opened and closed brackets: ( )

Often used as separation of statements

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”

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.

Well formulated formulas

Nothing more than a grammatical expressionA WFF is like a sentence which is why these connectives are called “sentential connectives”.

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.

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.

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

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

Inconsistency A WFF that is false under all assignments

of truth-values to its sentences letters

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).

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.

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.

Sentential connectiveTo put two things together

that make sense, you are creating a sentence using truth tables and WFFs.

Other uses for negationNegation 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.

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.

Disjunction

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

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.

References Picture from http://www.csus.edu/ Activity and Information obtained

from http://philosophy.hku.hk/think/sl/