PHIL 109 · Introduction to Formal Reasoning and Decision Making

2 Translations

2.1 From English to PL

Recall that we constructed the artificial language of PL in order to give a toy model of validity for arguments. In order to do that, we need to learn to translate between English and PL.

The following table gives you a preliminary translation guide for our five connectives of PL:

In order to translate a sentence from English to PL we need to first identify whether it is an atomic or a molecular sentence. If it is an atomic sentence, then our job is easy: we just pick an atomic sentence letter to stand for it. If it is a molecular sentence, then our job is more complex: first we need to identify which atomic sentences the molecular sentence is made up of (and assign them corresponding sentence letters), next we have to identify which connectives the sentence uses and which sub-sentences those connectives connect, and finally we connect the sub-sentences of PL to each other with the corresponding connectives.

Let us start with the easy case:

Leona is a cat.

That sentence is atomic: it doesn’t have any other sentences as parts. Therefore, our translation will be pretty easy:

Dictionary:

$C$ = Leona is a cat.

But consider now this rather more complicated sentence:

Leona is not a cat or she is a Maine Coon if and only if Lucas likes to play with her if she is purring.

The first thing we need to do is to identify the atomic sentences that make up that molecular sentence. To do that it will actually help to start with the second step and identify the connectives we see. In this case, we have “not”, “or”, “if and only if” and “if”. Notice that Leona is not a cat is not an atomic sentence, but a molecular one. Negation is the only unary connective, which forms a molecular sentence out of a single sentence. We have therefore identified our first atomic sentence, which is the same as before, “Leona is a cat”. We also have “She is a Maine Coon”, “Lucas likes to play with her” and “She is purring”. Those sentences have pronouns (“she”, “her”), and in isolation it will not be clear who those pronouns are referring to. Therefore, when building our dictionary we should replace those pronouns with the name of the person or animal they are referring to (in this case, “Leona”). So now we have our dictionary:

Dictionary:

$C$ = Leona is a cat.
$M$ = Leona is a Maine Coon.
$L$ = Lucas likes to play with Leona.
$P$ = Leona is purring.

Next we need to figure out how those atomic sentences are connected to each other via our connectives. Clearly, the negation is affecting $C$, so that we should somewhere have $\neg C$ as the translation of “Leona is not a cat”. Next, we need to figure out which one is the main connective of the whole sentence. It cannot be the negation, because that has scope over just one atomic sentence, not the whole molecular one. The other three connectives all have some claim to being the main one. For instance, we could perhaps take the disjunction to be the main connective of the sentence. But if whoever wrote (or uttered) this sentence wanted disjunction to be the main connective, he could have done a much better job of conveying his intentions by writing instead:

Either Leona is not a cat, or else she is a Maine Coon if and only if Lucas likes to play with her if she is purring.

Given the sentence as we have it, the best interpretation is that disjunction is not the main connective. We are left with two options: the biconditional or the conditional. The conditional here is indicated by an “if” unaccompanied by a “then”. How are we to interpret it? As we will see soon, this happens often (“If it’s raining, the picnic is cancelled”). In general, what follows the “if” is the antecedent of the conditional (unless we have an “only if”, in which case it is the consequent). So $P$ will be the antecedent of the conditional. What is its consequent? If the conditional were the main connective, the consequent would have to be “Leona is not a cat or she is a Maine Coon if and only if Lucas likes to play with her”. That would be a very strange thing to want to say. Let’s try the biconditional as the main connective. The right-hand side of this biconditional is “Lucas likes to play with [Leona] if she is purring”, which we will translate as $P \rightarrow L$. And what would be its left-hand side? “Leona is not a cat or she is a Maine Coon”, of course. Putting it all together, we have:

$(\neg C \lor M) \leftrightarrow (P \rightarrow L)$

This exercise has brought to the fore a very important lesson: natural language sentences are often ambiguous. We want to interpret the sentence so as to reflect the intentions of the speaker or writer. Thankfully, most of the time the ambiguities can be resolved by paying attention to the context in which the sentence is used. Sometimes, however, there will be unresolvable ambiguities. In that case, either the user of the sentence simply did a bad job, or (more often) we are just missing important context that helps with disambiguation.

2.2 Translation notes for connectives

We have above a table with basic information about translating from English to PL. However, natural languages all have different ways of transmitting information, and this applies to our five connectives. Let’s look at some of those.

2.2.1 Negation

As we saw before, negation in English can be expressed by a simple “not”, as in “Leona is not a cat”. Remember that contractions, such as “didn’t” and “can’t”, also contain a “not”. There are also more bombastic expressions of negations, such as “it’s not the case that”, or “it is false that”.

2.2.2 Conjunction and its English cousins

The most common English expression translated into PL as “$\land$” is, of course, “and”. But there are important differences between “and” and “$\land$” that we should be aware of. To begin with, “and” is often used to express temporal order in addition to conjunction. For instance, when we say “John walked in and took a seat” what we mean can be more explicitly expressed by saying “First John walked in, and then he took a seat.” The “$\land$” of PL cannot express this temporal order, for an interesting reason. The reason is that temporal order is not truth-functional. Suppose that it is true both that John came in and that he took a seat. It may in that case be true that he came in first and took a seat afterwards, but it may also be false. Even setting aside the ambiguity of “taking a seat”, which can mean picking up a chair in addition to sitting on it, it may be that John first sat and then came in (hopping while seated in his chair, perhaps). So, the truth values of “John came in” and “John took a seat” do not settle the truth value of “First John came in and afterwards he took a seat” or of “John came in and took a seat” in its obvious interpretation. That means that “and” in English sometimes has a non-truth-functional interpretation, and we know that PL only deals in truth-functional connectives.

Besides “and”, other English constructions can be translated into PL by “$\land$”. One of them is “but”. But “but” carries with it some additional information that is not part of its truth conditions, and that therefore is not captured by “$\land$”. Consider, for instance, the sentence “She loves him, but she won’t marry him.” The “but” is not just conjoining “She loves him” with “She won’t marry him”, but rather also introducing a clashing contrast between them (suggesting that there is something surprising about the fact that she won’t marry him given that she loves him). Nevertheless, “She loves him, but she won’t marry him” is true just in case she loves him and she won’t marry him, so as far as truth conditions go “but” behaves just like “and”. The contrast is an additional element that the word conveys by convention (philosophers call this a conventional implicature), not a contribution to the sentence’s truth conditions. Another English conjunction that works in a similar way is “yet” (“he is thin yet strong”). Some textbooks will tell you that the following should also be translated by “$\land$”: “moreover”, “however”, and “nonetheless”. However, these English expressions often function as discourse connectives rather than sentence connectives. That is to say, they are often used to introduce a new sentence, one that contrasts in some way with the previous one, rather than to add a conjunct. This nevertheless highlights an interesting fact: there is little difference between separately asserting two sentences and asserting their conjunction (at least as far as truth conditions go).

2.2.3 Disjunction—inclusive and exclusive

The “$\lor$” of PL has a clear English companion in “or” (sometimes accompanied by “either”). Recall, however, that we said that “$\lor$” is an inclusive disjunction. What that means is that a disjunction is true whenever at least one of its disjuncts is true, including when both are. With the English “or” this is sometimes not the case. Suppose that the waiter tells you: “You can have soup or salad with your entree”. Usually, that means that you cannot have both (for the same price). That use of the English “or” is an exclusive one, meaning that the sentence is false when both of its disjuncts are true. Thus, it would be wrong to translate that sentence as “$P \lor Q$”. However, the exclusive “or” of English can be translated into PL—just not with a simple “$\lor$”.¹

2.2.4 Conditional

The conditional connective of PL, “$\rightarrow$”, has given rise to notorious philosophical disputes that center on whether it is a faithful rendition of the English indicative conditional, often expressed with an “if $\ldots$ then $\ldots$” construction. Recall from our truth table for the conditional that it is false exactly in one case: when its antecedent (the formula to the left of the arrow) is true and its consequent (the formula to the right of the arrow) false. This means, in particular, that a conditional of PL is automatically true whenever its antecedent is false or its consequent true. Does that happen also with the English conditional? For instance, in a context where it is assumed that it is not raining, is it true that if it is raining then Mbappé is a better soccer player than Messi? It may seem that the answer here is a clear “No”, regardless of what you think of the truth-value of the consequent. But things are complicated.

One clear reason why no one would utter the sentence “If it is raining then Mbappé is a better soccer player than Messi” is that English conditional constructions usually express some kind of connection between antecedent and consequent, and there is no discernible connection in this case. Notice that if this is the correct explanation for why we find that sentence odd, it has little to do with whether the antecedent is true or false. What does rain have to do with the comparison between Mbappé and Messi?² Notice that conditionals where a connection is plausible do not sound bad. Suppose we have just woken up, and we do not know whether it is raining or not. I say “If it is raining, then the streets are wet”. We then look outside the window and notice that it is not raining. My previous assertion seems still to be perfectly correct. Still, this implied connection between antecedent and consequent is not something that can be captured in PL, because it is another example of an aspect of the meaning of the English conditional that goes beyond the truth conditions captured by the truth-table of $\rightarrow$ (on many views, it is not part of the truth conditions at all).

Besides the “If $\ldots$ then $\ldots$” construction, conditionals can be expressed in other ways in English. For instance, I can say “The streets are wet, if it is raining”. This expresses the same proposition as “If it is raining, then the streets are wet”, just with the antecedent coming after the consequent. However, if I say “The streets are wet only if it is raining” I express a different proposition, one where the antecedent and consequent have changed places. “If it is raining, then the streets are wet” is false exactly when it is raining and the streets are not wet, whereas “The streets are wet only if it is raining” is false exactly when the streets are wet and it is not raining. That second sentence, for instance, is false when it is not raining but the streets are wet because they are being cleaned, for example, whereas the first sentence would not be false in that case.

English contains conditionals other than the indicative one. Consider, for instance, the following one: “If Argentina had won the World Cup in 2026, Juan would have been unhappy”. This is a subjunctive conditional, and clearly does not behave like the indicative conditional—nor like the conditional of PL. It is generally accepted that the subjunctive conditionals like that one are not truth-functional connectives, and so cannot be translated into PL.

2.2.5 Biconditional

The biconditional of PL is simply a conditional going in both directions. If you look at its truth-table, what a biconditional says is that both of its sides share a truth-value: either they are both true or both false. In other words, a biconditional is false exactly when one of its sides is true and the other false.

As far as its expression in English goes, “$\ldots$ if and only if $\ldots$” is the canonical way of stating a biconditional. Other expressions are “$\ldots$ just in case $\ldots$” and “$\ldots$ exactly when $\ldots$”—although it is fair to say that one rarely encounters biconditionals in natural languages outside of mathematical or logical contexts.

2.2.6 Some other expressions

A few other English expressions come up often enough that they deserve a mention. In each case, the translation below is the standard one, but as always you should check that it fits the context in which the sentence is used.

Unless. “$\varphi$ unless $\psi$” says that if $\psi$ does not happen, then $\varphi$ does. So it is translated as $\neg\psi \rightarrow \varphi$, which has the same truth-table as $\varphi \lor \psi$. For instance, “Bob will not like the strawberries unless they are ripe” can be translated as $\neg R \rightarrow \neg L$ or, equivalently, as $\neg L \lor R$. Notice that this treats “unless” as an inclusive disjunction: nothing is said about what happens when $\psi$ is true.³

Neither … nor. “Neither $\varphi$ nor $\psi$” says that $\varphi$ is false and $\psi$ is false, so it is translated as $\neg\varphi \land \neg\psi$, or equivalently as $\neg(\varphi \lor \psi)$. Do not confuse it with “not both $\varphi$ and $\psi$”, which is translated as $\neg(\varphi \land \psi)$ and is compatible with one of them being true.

Provided that, as long as, on the condition that. These all introduce the antecedent of a conditional, just like “if”: “$\varphi$ provided that $\psi$” is translated as $\psi \rightarrow \varphi$.

The only way. “The only way for $\varphi$ to happen is for $\psi$ to happen” says that $\varphi$ cannot happen without $\psi$, so it is translated as $\varphi \rightarrow \psi$, exactly like “$\varphi$ only if $\psi$”.

Try it in the Logic Lab Translate sentences step by step; the lab picks new ones each time.

Exercises

1. How would you define exclusive disjunction?

   Show solution

   Exclusive or: $(\varphi\lor\psi)\land\lnot(\varphi\land\psi)$, equivalently $\lnot(\varphi\leftrightarrow\psi)$.

Translate the following English sentences into PL. Define any atomic sentences that you are going to use. Practice translating in the Logic Lab

2. Alyssa will play basketball if the weather is nice and she doesn’t have work.

   Show solution

   $(N\land\lnot W)\to B$.  ($N$: the weather is nice; $W$: Alyssa has work; $B$: Alyssa plays basketball.)

3. Charly graduates if and only if she’s satisfied her curriculum requirements and she has enough credits.

   Show solution

   $G\leftrightarrow(S\land E)$.  ($G$: Charly graduates; $S$: she has satisfied her curriculum requirements; $E$: she has enough credits.)

4. Deion will fall if the stairs are slippery or wet.

   Show solution

   $(S\lor W)\to F$.  ($S$: the stairs are slippery; $W$: the stairs are wet; $F$: Deion falls.)

5. Erin likes to watch TV shows and movies but does not like to play basketball.

   Show solution

   $(T\land M)\land\lnot B$.  ($T$: Erin likes to watch TV shows; $M$: she likes to watch movies; $B$: she likes to play basketball.)

6. Francis won’t go to the party just in case neither his mother nor his father gives their permission.

   Show solution

   $\lnot P\leftrightarrow(\lnot M\land\lnot D)$.  ($P$: Francis goes to the party; $M$: his mother gives permission; $D$: his father gives permission.)

7. Gianna goes hiking if the weather is good, but, if the weather isn’t good, she plays video games.

   Show solution

   $(G\to H)\land(\lnot G\to V)$.  ($G$: the weather is good; $H$: Gianna goes hiking; $V$: she plays video games.)

8. The only way Hadrian will get a trophy is if either his soccer team or his tennis team wins its championship.

   Show solution

   $T\to(S\lor N)$.  ($T$: Hadrian gets a trophy; $S$: his soccer team wins its championship; $N$: his tennis team wins its championship.)

9. Ida will be successful if she works hard or if she is lucky.

   Show solution

   $(W\lor L)\to S$.  ($W$: Ida works hard; $L$: she is lucky; $S$: she is successful.)

10. Joel gets sick if he eats ice cream or tomatoes without taking his medication.

    Show solution

    $((I\lor T)\land\lnot M)\to K$.  ($I$: Joel eats ice cream; $T$: he eats tomatoes; $M$: he takes his medication; $K$: he gets sick.) The sentence is ambiguous; on the other reading it is $(I\lor(T\land\lnot M))\to K$.