PL 120 Symbolic Logic I
Hints about putting words into symbols
We often speak as if the matter of putting words into symbols is quite like translating from one natural language to another. It is not. Natural languages have degrees of flexibility and ambiguity that would be disastrous in a formal language. When translating from one natural language into another, it is important to try to retain the ambiguities and subtleties of the original. When putting sentences from a natural language into a formal language, we need to remove ambiguities. Logic provides us with many conventions for removing ambiguities, but some of those conventions clearly do not reflect the way we actually use certain expressions in everyday speech. Accordingly, it is better to speak of symbolizing and expression than of translating it into symbols.
- Statement Letters. Remember, we use capital letters as statement letters to represent simple statements. Usually the statement letter we choose is the first letter of a significant word in the simple statement (e.g., we might use C to stand for "The cat is on the mat."), but the statement letter stands for a statement, not a word. One word may occur in several different statements, and it is crucial to distinguish which statement letters represent which statements. SO, we should NOT use C to represent "The cat is grey" after we have already used it to represent "The cat is on the mat." We need a unique statement letter for each simple statement.
- Negations. Negation is a logical operation, and any statement in which a negation occurs is a compound statement, not a simple statement. Accordingly, it is wrong to symbolize "The cat is not on the mat" as M. Rather, it should be symbolized as ~C. Whenever one of the English expressions indicating negation occurs, your symbolization should have a tilde to represent it (or an equivalent expression).
- Exclusive disjunction. Remember that "▼" in our notation expresses inclusive disjunction: "p ▼ q" means that either p is true or q is true or both. The exclusive disjunction of p and q asserts that either p is true or q is true but not both. The natural, but long-winded, way to express exclusive disjunction, then, is "(p ▼ q) · ~(p · q)".
Notice, however, that exclusive disjunction is really saying that p and q have different truth-values; if one of them is true, then the other isn't, and vice versa. The way to say they have different truth values is to deny their equivalence: "~(p ↔ q)".
For example, when a menu says "cream or sugar", it uses an inclusive "or", because you may take one, the other, or both. But when it says "coffee or tea", it uses an exclusive "or", because you are not invited to take both.
- Conjunction. We express conjunction with many words other than "and", including "but," "moreover," "however," "although", and "even though". In English these expressions sharply contrast the two conjuncts, saying in effect "if you believe the first conjunct, then you will be surprised by the second." But they still assert conjunction. The contrast between the conjuncts is not logically relevant; validity never turns on it.
- Sometimes "and" does not join whole propositions into a compound proposition. Sometimes it simply joins nouns: "Bert and Ernie are brothers" (or are they just room-mates?). This cannot be paraphrased, "Bert is a brother and Ernie is a brother," for that does not assert that they are brothers to each other.
- Sometimes "and" joins adjectives: "The leech was long and wet and slimy." This, however, can be paraphrased, "The leech was long, and the leech was wet, and the leech was slimy."
- Unless. "Unless" is used ambiguously in ordinary English, and many people get into trouble when they try to use "unless" in a technical sense (Click here to see Congressmen trying to use "unless"). Sometimes "unless" is used to express inclusive disjunction, and sometimes exclusive disjunction. For example, "I'll go to the party unless I get another offer" means that I'll go if nothing else comes along. In many contexts it also means that I might go anyway; the second offer might be worse. So I'll go or I'll get another offer or both (inclusive disjunction). Consider by contrast, "The Spurs will win the series unless Shaq recovers". In many contexts this means that if I learn Shaq has recovered, then I'll change my mind and bet against the Spurs. So either the Spurs win the series or Shaq recovers (exclusive disjunction). The convention that I suggest for dealing with "unless" is simply to treat it as indicating disjunction. Symbolize "unless" as "or". Yes, this violates our ordinary usage, but this is one of those cases where preserving ambiguity is not a goal of logic.
- And/or. Logicians have little patience with this barbarous locution. People who say "and/or", seem to mean inclusive disjunction. "Bring a scarf and/or a hat" means "bring a scarf or bring a hat or both", and so we symbolize it simply as, "S ▼ H". Of course, if this is what such people mean to say, then it is what they should say.
- Neither, nor. "Neither p nor q" means that both p and q are false. Therefore translate it "~p · ~q" or "~(p ▼ q)". These two formulas are equivalent by DeMorgan's Theorem.
- Not both / both not. Many students confuse "not both" with "both not". If p and q are "not both" true, then we are denying their conjunction: "~(p · q)". One of them may be true, just not both. So this is equivalent to "~p ▼ ~q". On the other hand, if p and q are "both not" true, then we are denying each of them; they are both false: "~p · ~q". Neither of them may be true; so this is equivalent to "~(p ▼ q)". The best cure for confusing "not both" with "both not" is familiarity with DeMorgan's Theorems.
- Conditionals. "p → q" symbolizes a wide variety of English expressions, for example, "if p, then q", "if p, q", "p therefore q", "p hence q", "q if p", "q provided p", "q follows from p", "p is the sufficient condition of q", and "q is the necessary condition of p". The least intuitive is "p only if q". See the next two tips.
- But be careful. Some nearly synonymous expressions, like "q because p", "q since p", "because p, q", "since p, q", and even some instances of "p therefore q", are not genuine cases, or at least not merely cases, of material implication. They may seem so because they make p into a condition or reason for q. But in "if p, then q" we are non-committal about the truth of p, whereas most speakers who assert "q because p" and its variants are asserting the truth of p. To capture this aspect of the proposition's meaning, use conjunction, "q · p". To capture the implication claim as well, use both conjunction and material implication, "p · (p → q)".
- Implication and Entailment. There is a strong tendency to symbolize "p implies q" or "p entails q" as "p → q". We should resist this tendency. Implication and entailment are logical relations that exist between statements or sets of statements. The conditional is a kind of statement. It is fully possible that a conditional is true even though no one would claim that the antecedent implies or entails the consequent. Consider the conditional "If it is raining, then I will take a hat". While that conditional may be true, the argument "It is raining" therefore "I will take a hat" is not valid (test it with a truth table). Treating 'implies' as indicating a conditional generates paradoxes of material implication.
- Necessary and sufficient conditions. We say that p is a sufficient condition of q when p's truth guarantees q's truth. By contrast, q is a necessary condition of p when q's falsehood guarantees p's falsehood. In the ordinary material implication, "p → q", the antecedent p is a sufficient condition of the consequent q, and the consequent q is a necessary condition of the antecedent p.
- Satisfy yourself of this by reflecting on modus ponens and modus tollens. Given "p → q", modus ponens tells us that the truth or presence of p suffices to give us q. Hence the antecedent is the sufficient condition of the consequent. Similarly, modus tollens tells us that the falsehood of q (the truth of ~q) guarantees us the falsehood of p (the truth of ~p). Hence the consequent is the necessary condition of antecedent.
- Or, satisfy yourself of this by reflecting on an example: "If Socks is a cat, then Socks is a mammal." Being a cat is a sufficient condition of being a mammal. Being a mammal is a necessary condition of being a cat.
- The fact that material implication expresses sufficient and necessary conditions in this way can be a great help in translation. Ask yourself about a difficult English sentence: what is being asserted to be the sufficient condition of what here? What is being asserted to be the necessary condition of what here? When you find the sufficient condition, make it the antecedent. When you find the necessary condition, make it the consequent.
- If p is both necessary and sufficient for q, then we must say "p ↔ q" (material equivalence).
- Only if. We translate "p only if q" as "p → q". This is surprising to many people because "if" usually cues the antecedent. Rather than say that "if" sometimes cues the consequent, it is better to say instead that "only if" differs from "if", and "only if" cues the consequent.
If you understand necessary and sufficient conditions, this translation should make more sense: "p only if q" clearly asserts that q is a necessary condition of p. The necessary condition of something is the consequent of that something in a material conditional.
Modus tollens assures us that p → q asserts that p is true only if q is true, or that q is the necessary condition of p. For under modus tollens, from p → q and ~q we can validly infer ~p.
- If and only if. Tip 9 showed why "p only if q" is translated "p → q". It should already be clear why "p if q" should be translated "q → p" ("if" cues the antecedent). So if we say "p if and only if q" we are asserting both "p → q" and "q → p", which amounts to "p ↔ q".
- Remember that "p ↔ q" means that p and q have the same truth-value, not necessarily the same meaning. So it may be correct to translate an English sentence into "p ↔ q" even if its components differ in meaning.
- Many logicians, mathematicians, and philosophers abbreviate "if and only if" as "iff". Hence "p iff q" should be translated as "p ↔ q".
- Just when. Sometimes in English we say that p is true "just when" q is true. (Or perhaps this locution is only common among logicians and mathematicians.) This means that p is true when and only when q is true, or that p if and only if q, and should be translated "p ↔ q".
- Even if. "P even if q" means "p whether or not q" or "p regardless of q". Therefore one perfectly acceptable translation of it is simply "p". If you want to spell out the claim of "regardlessness", then you could write "p · (q ▼ ~q)". The two translations are equivalent. (A proposition conjoined to any tautology has the same truth-value as the original proposition.)
- Truth-functionality. All our operators are truth-functional. So if an operator in English is not truth-functional, don't translate it with one of our operator symbols. If the English operator has multiple meanings, one truth-functional and others not, then only translate it with one of our operator symbols if you want the truth-functional core of meaning and are willing to discard the rest. Two examples:
- "And" in English sometimes expresses temporal succession, not just conjunction. "She cursed like a sailor and hung up." Here the "and" should be translated as " · " because it does express conjunction; but our translation will no longer make it clear which act was performed first. One function of the "and" in English is to do so, but that function is not truth-functional and cannot be captured by our operators. "And" sometimes also functions as a slovenly substitute for the infinitive, as in "Try and make me." Here the meaning is even further from truth-functional conjunction.
- Conditional statements in English (if..., then...) often express causation or definition rather than implication. For example, we use "if..., then..." to express a definition when we say, "if it's ice, then it's frozen", and to express causation when we say, "if we put ice in boiling water, then it will melt". But the horseshoe operator expresses neither definition nor causation, only implication. Moreover, not all senses of logical implication in English are truth-functional. In particular, implications in which the antecedent and consequent must be "relevant" to one another are not truth-functional. The horseshoe only expresses truth-functional implication.
- Punctuation. Parentheses and brackets are the punctuation marks of our logical language. Unlike algebra, logic has no fixed order of operations. Whenever you have two or more dyadic operators in a single compound expression, you will need parentheses in order to prevent ambiguity. For example, "p → q ▼ r" is ambiguous. It could mean "(p → q) ▼ r", or "p → (q ▼ r)", which are not equivalent.
- Negation is the only one of our operators with a pre-defined scope: namely, the first meaningful proposition (simple or compound) to its right. Hence "~p → q" is not ambiguous; it means "(~p) → q", not "~(p → q)".
- Disjunction is both commutative and associative. So "(p ▼ q) ▼ r" is equivalent to "(q ▼ p) ▼ r" and "p ▼ (q ▼ r)". As a result, omitting parentheses from disjunctive strings does not create ambiguity. One may write "p ▼ q ▼ r". The same is true of conjunctive strings. (However, ANY string that mixes disjunctions and conjunctions requires parentheses to prevent ambiguity.)
- Omitting necessary parentheses is a common mistake in translation. Make one last check before you think your translation is finished.
- Unnecessary parentheses sometimes violate the basic rules of WFF formation. When you mean "p", writing "(p)" or "((p))" violates the rules of WFF formation. The same is true with "p → q", "(p → q)", "((p) → (q))", and so on. According to our rules, parentheses are used ONLY when we have a dyadic connecive.