The Logic Café
|Connective Name||Resulting Sentence Type||Component Names||Typical English Versions||English Statement||Symbolization in SL|
|&||Ampersand||Conjunction||Conjuncts||"and", "both ... and ... "||Agnes and Bob will attend law school.||A&B|
|>||Horseshoe||Conditional||Antecedent, Consequent||"if ... then ... "||If Agnes attends, then Bob will.||A>B|
|~||Tilde||Negation||Negate||"it's not the case that", "not"||Agnes will not attend law school.||~A|
|v||Wedge||Disjunction||Disjuncts||"or", "either... or... "||Either Agnes or Bob will attend law school.||AvB|
|=||Triple Bar||Biconditional||Bicomponets1||"if and only if", "just in case"||Agnes will attend law school just in case Bob will.||A=B|
Our most important task is to say just what objects count as sentences of our language SL. We do this by showing how sentences are constructed.
We will call the simplest sentences of our language the atomic sentences of SL. These building blocks include 'A','B','C','D'..., 'Z' (except we won't use the 'V' to avoid confusion with the wedge 'v'). Also, you may add a numeral subscript to any of the upper case letters: so 'A1' or 'L16' also count as atomic sentences of SL. That way, you'll never run out of atomic sentences.
The atomic sentences, as their name implies, are used as the basic building blocks for SL sentences. You use them to construct longer sentences, molecular sentences, like the following four sentences:
(Notice that we do not yet drop outside parentheses.) The rough idea behind the definition of an SL sentence can be put as follows:
One can construct a molecular sentence of SL either by taking any one sentence which is already constructed and adding a tilde on its left or by taking any pair of already constructed sentences, writing a binary connective between them, and surrounding the result with parentheses. (Atomic sentences count as "already constructed".)
Now, let's put the definition of a sentence of SL a bit more precisely. First, the lexicon of SL is just the collection of basic symbols we use to construct sentences. The lexicon includes (a) the binary connectives '&','v','>', and '=', (b) the unary connective '~', (c) parentheses '(' and ')' for punctuation, and finally (d) the collection of all atomic sentences, the set of all uppercase Roman letters except 'V' (possibly subscripted).
We may now more precisely define just what counts as a sentence of SL. We give what is sometimes called an "inductive" or "recursive" definition. But all this means is that we define what counts as a sentence by showing how to construct one from basic parts.
Here's the definition. First,
i) All atomic sentences count as sentences of SL.
Second we say how we can build more complex sentences from any sentences which have already been built.
ii) If P is any sentence of
SL, then so is ~P.
When you construct a sentence by these rules, you may apply both of these clauses again and again. For example, think about following these rules to construct the sentence, (*):
To construct this sentence following these rules, you would first apply the first clause i), noting that 'A' and 'B' are sentences of SL because they are atomic sentences. Then, think of setting P='A' and Q='B' so that clause iii) says that '(A&B)' is a sentence of SL. Also, 'C' is an atomic sentence, so is a sentence of SL by i). Thus by ii) '~C' is a sentence of SL. Finally, we can put the two sentences just highlighted together by yet another application of iii) to form '((A&B)>~C)' as desired.
Further Syntactical Concepts
The sentence (*) is an "if...then..." sentence, a conditional. In other words, (*) means that
if A and B are both true, then C is not.
Or to put the point in English, these sentences could express something like
If Agnes and Bob will both attend law school, then Carola will not.
Because this expresses a conditional, we will say that (*)'s main connective is the horseshoe. It's made up of antecedent '(A&B)' and consequent '~C' which we will call the two "immediate components". Its most basic sentential components we'll call its "atomic components" which in (*)'s case are 'A', 'B', and 'C'.
Now we need to carefully define these terms.
The sentential components of a sentence of SL are all components used in the building process in order to construct that sentence.
The atomic components of a sentence of SL are all atomic sentences used to construct that sentence.
The main connective of a sentence of SL is the last occurrence of a connective used to construct it.
The immediate component or components of a sentence of SL is (are) the sentential component(s) used in the final stage of its construction.
For example, to build (*) one first builds '(A&B)' and '~C', then finally puts these together with the horseshoe. Thus our definitions lead to the conclusion that (*)'s main connective is indeed the horseshoe and its antecedent and consequent are the immediate components.
Relaxing the Definition of a Sentence
To save just a little time and to make our SL constructions easier to read we allow two alterations to the above definition:
Metalanguage: English used to describe SL
Several final notes need to be made about our use of English to talk or write about the language SL. When we do this, we use English as a "metalanguage": a language to refer to another language. In order to talk or write about SL and its expressions, we need names for those expressions: the standard convention used throughout the Logic Cafe is to put an expression in quotes to name that expression. Thus we mention an expression rather than use it. We may also mention sentences when we put them on display inside (say) a blue box. Finally, we can talk about expressions of SL in general by using variables. We have already seen the variables 'P', 'Q', 'R' which are stand-ins for arbitrary SL sentences. Such variables are often called metavariables.
To specify the semantics for SL, we simply give truth conditions: we say under what conditions a sentence is true and under what conditions it is false. In SL we are only interested in connectives used truth functionally:
A connective is used truth functionally to form a sentence from components if and only if that sentence's truth value depends only on the truth value of the components. Otherwise, it is used non-truth functionally.
If a connective (like those of SL) are always used truth functionally, we say the connective itself is truth functional.
Now, suppose we have a sentence P of SL with a (truth functional) main connective. Then our definition just above says that giving truth conditions for P is only a matter of saying how its truth value depends on the truth values of its immediate components.
For example, take P = 'A&B'; this is true just in case both of P's immediate components ('A' and 'B') are true. The best way to describe the truth function for each connective is in terms of the...
Truth Table Definitions for the Connectives
Truth conditions are usually given by way of tables. The following table summarizes the five separate tables given in tutorial 2. Like the tables we saw in the tutorial, read these from left to right along any one row. For example, in row one we are thinking about the possibility in which both P and Q are true: in that case only ~P is false.
Notice that the column under ~P looks a little different from what we say in tutorial 2. What you saw there was:
You should take a moment to see that there is no difference. Both say that '~P' is false exactly when 'P' is true.
You will need to memorize the table for all five connectives; you should do so immediately! To help, keep the following in mind. Conjunctions are true in only one row. Disjunctions and conditionals are false in only one row. The biconditional works a lot like the equal sign it resembles! Finally, the negation simply "reverses" truth values.
In the tutorials, you will find many examples of "symbolization", i.e., of translating a compound English sentence into a molecular sentence of SL with roughly the same meaning.
(**) For example, anything in English of the form "P only if Q" may be symbolized as 'P>Q'. Or equivalently, it may be symbolized as '~Q>~P'.
What does this mean exactly? First, the hybrid form, "P only if Q", is shorthand for an English sentence "___ only if ___" where the blanks can be filled in by sentences symbolized by some SL sentences P and Q respectively. For example,
(***) There is fire only if oxygen is present
or "F only if O". This is logically equivalent to either "If there is fire then oxygen is present" or "If there is no oxygen, then there is no fire". So, you may symbolize (***) as either
This is all (***) comes to. But make sure you understand it before looking at the following review of the many examples discussed in the tutorial.
|Equivalent English Forms (Each table element -- i.e., box -- below gives English forms instances of each of which can be symbolized by a sentence of any SL form on its right. Please note that there are many more English forms than can be covered below.)||Equivalent SL Forms (Each table element below gives SL sentence-forms to guide in translating English sentences of forms found on the left. Please note that this is an incomplete list of possible symbolizations.)||Example Applications (Each of the table elements below shows a way to apply the table elements on their left.)|
If P, then Q.
|If there is fire, then there is Oxygen" or "There is fire only if there is oxygen" may both be symbolized as 'F>O' or equivalently as '~O>~F'.|
|P if Q.
P provided Q.
P is a necessary condition for Q.
|"Water is a necessary condition for life" or "there's water if there's life" may both be symbolized as 'L>W' or equivalently as ~W>~L'.|
|P if and only if Q.
P just in case Q.
P is necessary and sufficient for Q.
|"An argument is sound if and only if it is both valid and has true premises" may be symbolized as either 'S=(V&T)' or '[S>(V&T)] &[(V&T)>S]|
|"Sandra is both brave and careful", "Sandra is brave, moreover she is careful' or "Sandra is brave but careful" can all be symbolized as 'B&C' or 'C&B'.|
Either Q or P.
P or Q.
Q or P.
At least one of P, Q.
|"Either the other team will score and tie up the game, or we win!" can be symbolized as '(S&T)vW'.|
|P unless Q.
Q unless P.
Unless P, Q.
Unless Q, P.
|"We win unless the other team scores" can be symbolized as 'WvS'.|
|Neither P nor Q.
Not-P and not-Q.
|"They neither scored not tied the game" may be symbolized as either '~(SvT)' or '~S&~T'.|
|It's not the case that both P
Not both P and Q.
Either not-P or not-Q.
|"Sandra is not both brave and careful" may be symbolized as either '~(B&C)' or '~Bv~C'.|
There are lots of ways to indicate negation. Words like "not", "it's not the case that" are obvious examples. The prefixes "un" or "not" are sometimes indicative of negation. See the tutorial for more... .
We use parentheses in SL to group sentential components and so show the sentence's structure. English has many means to do this. Consider the following sentence of English:
The Russian president will be reelected if and only if either the opposition bows out or the gods intervene.
Is this symbolized as the following?
Yes, this is correct. But how do we know the parentheses group 'BvI'? Why not
If the second translation were intended, then the English sentence in question would have begun with an "Either". This would have indicated that the main connective was the "or". Instead, the word "either" comes just before "the opposition bows out" and groups that with "the gods intervene" to form the disjunction. (Thus the word "either" works much like a left hand parenthesis.)
Other ways to indicate a main connective at the beginning of a sentence are to use "Both", "Either", or "If". Commas are also frequently used to set off two immediate components of the main connective. Examples? How about: "If one of the British and French win, then both the Americans and the Russians loose." This would be
All of these basic rules must be used together when you get to more complicated sentences. But practice first on the simpler cases.